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Graph Modification Algorithms

Neeldhara Misra, profile picture
Neeldhara Misra

The document explores graph modification problems, focusing on the goal of transforming an input graph to belong to a specified class or property with minimal changes. It discusses various modification operations, such as vertex deletions and edge additions, and their implications on different graph classes and properties. Additionally, the document touches on algorithmic complexity and problem-solving approaches related to graph classification and modification.

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Graph Modification Problems A Modern Perspective
Graph Modification Algorithms
Setting the stage: Definitions and Preliminary Observations
Setting the stage: Definitions and Preliminary Observations Brief excursions into specific examples
Setting the stage: Definitions and Preliminary Observations Brief excursions into specific examples Current Trends & Future Directions
Ingredients of a typical Graph Modification Problem
Graph Modification Algorithms
Input Graph, G
Input Graph, G Graph Class or “property”, ᴨ
Goal: Transform G so that it belongs to ᴨ with “minimum damages”. Input Graph, G Graph Class or “property”, ᴨ
Goal: Transform G so that it belongs to ᴨ with “minimum damages”. Input Graph, G Graph Class or “property”, ᴨ
Goal: Transform G so that it belongs to ᴨ with “minimum damages”. Input Graph, G Graph Class or “property”, ᴨ
Goal: Transform G so that it belongs to ᴨ with “minimum damages”. vertex deletions Input Graph, G Graph Class or “property”, ᴨ
Goal: Transform G so that it belongs to ᴨ with “minimum damages”. vertex deletions edge deletions Input Graph, G Graph Class or “property”, ᴨ
Goal: Transform G so that it belongs to ᴨ with “minimum damages”. vertex deletions edge deletions edge additions Input Graph, G Graph Class or “property”, ᴨ
Goal: Transform G so that it belongs to ᴨ with “minimum damages”. vertex deletions edge deletions edge additions edge contractions Input Graph, G Graph Class or “property”, ᴨ
Goal: Transform G so that it belongs to ᴨ with “minimum damages”. vertex deletions edge deletions edge additions edge contractions edge editing Input Graph, G Graph Class or “property”, ᴨ
Goal: Transform G so that it belongs to ᴨ with “minimum damages”. vertex deletions edge deletions edge additions edge contractions edge editing Input Graph, G Vertex Cover Edgeless Graphs
Goal: Transform G so that it belongs to ᴨ with “minimum damages”. vertex deletions edge deletions edge additions edge contractions edge editing Input Graph, G Feedback Vertex Set Acyclic Graphs
Goal: Transform G so that it belongs to ᴨ with “minimum damages”. vertex deletions edge deletions edge additions edge contractions edge editing Input Graph, G Minimum Fill-In Chordal Graphs
Goal: Transform G so that it belongs to ᴨ with “minimum damages”. vertex deletions edge deletions edge additions edge contractions edge editing Input Graph, G Cluster Editing Cluster Graphs
Goal: Transform G so that it belongs to ᴨ with “minimum damages”.
Goal: Transform G so that it belongs to ᴨ with “minimum damages”. Minimum ᴨ-Completion
Goal: Transform G so that it belongs to ᴨ with “minimum damages”. Minimum ᴨ-Completion Minimum ᴨ-Supergraph
Goal: Transform G so that it belongs to ᴨ with “minimum damages”. Minimum ᴨ-Completion Minimum ᴨ-Deletion Minimum ᴨ-Supergraph
Goal: Transform G so that it belongs to ᴨ with “minimum damages”. Minimum ᴨ-Completion Minimum ᴨ-Deletion Minimum ᴨ-Supergraph Maximum ᴨ-Spanning Subgraph
Goal: Transform G so that it belongs to ᴨ with “minimum damages”. Minimum ᴨ-Completion Minimum ᴨ-Deletion Minimum ᴨ-Editing Minimum ᴨ-Supergraph Maximum ᴨ-Spanning Subgraph
Goal: Transform G so that it belongs to ᴨ with “minimum damages”. Minimum ᴨ-Completion Minimum ᴨ-Deletion Minimum ᴨ-Editing Minimum ᴨ-Supergraph Maximum ᴨ-Spanning Subgraph Closest ᴨ-Graph
Goal: Transform G so that it belongs to ᴨ with “minimum damages”. Minimum ᴨ-Completion Minimum ᴨ-Deletion Minimum ᴨ-Editing Minimum ᴨ-Supergraph Maximum ᴨ-Spanning Subgraph Closest ᴨ-Graph Minimum ᴨ-Vertex Deletion
Goal: Transform G so that it belongs to ᴨ with “minimum damages”. Minimum ᴨ-Completion Minimum ᴨ-Deletion Minimum ᴨ-Editing Minimum ᴨ-Supergraph Maximum ᴨ-Spanning Subgraph Closest ᴨ-Graph Minimum ᴨ-Vertex Deletion Maximum ᴨ-Induced Subgraph
Goal: Transform G so that it belongs to ᴨ with “minimum damages”. Minimum ᴨ-Completion Minimum ᴨ-Deletion Minimum ᴨ-Editing Minimum ᴨ-Supergraph Maximum ᴨ-Spanning Subgraph Closest ᴨ-Graph Minimum ᴨ-Vertex Deletion Maximum ᴨ-Induced Subgraph Completion to Minimum Max-Clique
Goal: Transform G so that it belongs to ᴨ with “minimum damages”. Minimum ᴨ-Completion Minimum ᴨ-Deletion Minimum ᴨ-Editing Minimum ᴨ-Supergraph Maximum ᴨ-Spanning Subgraph Closest ᴨ-Graph Minimum ᴨ-Vertex Deletion Maximum ᴨ-Induced Subgraph Completion to Minimum Max-Clique Modification with Restrictions, eg, the Sandwich problem
Goal: Transform G so that it belongs to ᴨ with “minimum damages”. Minimum ᴨ-Completion Minimum ᴨ-Deletion Minimum ᴨ-Editing Minimum ᴨ-Supergraph Maximum ᴨ-Spanning Subgraph Closest ᴨ-Graph Minimum ᴨ-Vertex Deletion Maximum ᴨ-Induced Subgraph Completion to Minimum Max-Clique Modification with Restrictions, eg, the Sandwich problem Restricted Classes of Input
Graph Class or “property”, ᴨ
Non-Trivial ᴨ admits infinitely many graphs, and also excludes infinitely many graphs. Graph Class or “property”, ᴨ
Non-Trivial ᴨ admits infinitely many graphs, and also excludes infinitely many graphs. Monotone If G belongs to ᴨ, then all subgraphs of G also belong to ᴨ. Graph Class or “property”, ᴨ
Non-Trivial ᴨ admits infinitely many graphs, and also excludes infinitely many graphs. Graph Class or “property”, ᴨ Monotone Hereditary If G belongs to ᴨ, then all subgraphs of G also belong to ᴨ. If G belongs to ᴨ, then all induced subgraphs of G also belong to ᴨ.
Non-Trivial ᴨ admits infinitely many graphs, and also excludes infinitely many graphs. (Bipartite, Planar) Graph Class or “property”, ᴨ Monotone Hereditary If G belongs to ᴨ, then all subgraphs of G also belong to ᴨ. If G belongs to ᴨ, then all induced subgraphs of G also belong to ᴨ.
Non-Trivial ᴨ admits infinitely many graphs, and also excludes infinitely many graphs. (Bipartite, Planar) Graph Class or “property”, ᴨ Monotone Hereditary If G belongs to ᴨ, then all subgraphs of G also belong to ᴨ. If G belongs to ᴨ, then all induced subgraphs of G also belong to ᴨ. (Complete graphs, Forests)
Non-Trivial ᴨ admits infinitely many graphs, and also excludes infinitely many graphs. (Bipartite, Planar) Graph Class or “property”, ᴨ Monotone Hereditary If G belongs to ᴨ, then all subgraphs of G also belong to ᴨ. If G belongs to ᴨ, then all induced subgraphs of G also belong to ᴨ. (Complete graphs, Forests) Connectivity, Biconnectivity, Trees, Stars, Eulerian, etc.
Graph Modification Algorithms
Lewis & Yannakakis [1980] The vertex-deletion problem is NP-complete for non-trivial, hereditary properties.
Lewis & Yannakakis [1980] The vertex-deletion problem is NP-complete for non-trivial, hereditary properties. Yannakakis [1979] The connected vertex-deletion problem is NP-complete for non-trivial properties that hold on connected induced subgraphs.
Lewis & Yannakakis [1980] The vertex-deletion problem is NP-complete for non-trivial, hereditary properties. Yannakakis [1979] (Edgeless, Acyclic, etc.) The connected vertex-deletion problem is NP-complete for non-trivial properties that hold on connected induced subgraphs.
Lewis & Yannakakis [1980] The vertex-deletion problem is NP-complete for non-trivial, hereditary properties. Yannakakis [1979] (Edgeless, Acyclic, etc.) The connected vertex-deletion problem is NP-complete for non-trivial properties that hold on connected induced subgraphs.( Trees, Stars, etc.)
Graph Modification Algorithms
Edge-Deletion or Completion problems are sometimes easier than their vertex-deletion counterparts.
Edge-Deletion or Completion problems are sometimes easier than their vertex-deletion counterparts. Add edges to make the input graph a cluster graph.
Edge-Deletion or Completion problems are sometimes easier than their vertex-deletion counterparts. Add edges to make the input graph a cluster graph. Remove edges to make the input graph edgeless.
Edge-Deletion or Completion problems are sometimes easier than their vertex-deletion counterparts. Add edges to make the input graph a cluster graph. Remove edges to make the input graph edgeless. Remove edges to make the input graph acyclic.
Edge-Deletion or Completion problems are sometimes easier than their vertex-deletion counterparts. Add edges to make the input graph a cluster graph. Remove edges to make the input graph edgeless. Remove edges to make the input graph acyclic. Remove edges to make the input graph bipartite.
Edge-Deletion or Completion problems are sometimes easier than their vertex-deletion counterparts. Add edges to make the input graph a cluster graph. Remove edges to make the input graph edgeless. Remove edges to make the input graph acyclic.
Edge-Deletion or Completion problems are sometimes easier than their vertex-deletion counterparts. Add edges to make the input graph a cluster graph. Remove edges to make the input graph edgeless. Remove edges to make the input graph acyclic. Alon, Shapira & Sudakov [2009] Given a property ᴨ such that ᴨ holds for every bipartite graph, the Minimum ᴨ-Deletion problem is NP-hard.
Edge-Deletion or Completion problems are sometimes easier than their vertex-deletion counterparts. Add edges to make the input graph a cluster graph. Remove edges to make the input graph edgeless. Remove edges to make the input graph acyclic. Alon, Shapira & Sudakov [2009] Given a property ᴨ such that ᴨ holds for every bipartite graph, the Minimum ᴨ-Deletion problem is NP-hard. (Triangle-­free)
Edge-Deletion or Completion problems are sometimes easier than their vertex-deletion counterparts. Add edges to make the input graph a cluster graph. Remove edges to make the input graph edgeless. Remove edges to make the input graph acyclic. Alon, Shapira & Sudakov [2009] Given a property ᴨ such that ᴨ holds for every bipartite graph, the Minimum ᴨ-Deletion problem is NP-hard. Colbourn & El-Mallah [1988] (Triangle-­free) Making a graph Pk-free by deleting the minimum number of edges is NP-hard for every k > 2.
edge editing edge contractions vertex deletions edge additions edge deletions Graph Class or “property”, ᴨ
edge editing edge contractions vertex deletions edge additions edge deletions Graph Class or “property”, ᴨ Perfect Trivially Perfect Cographs Permutation Graphs Bipartite Trees Weakly Chordal Chordal Strongly Chordal Split Interval Proper Interval Unit Interval Forests Caterpillars Chain Chordal Bipartite Distance Hereditary Comparability Trapezoid CoChordal Circle Planar k-Connected
Graph Modification Problems A Modern Perspective
Graph Modification Problems Why bother?
Graph Modification Algorithms
Special Graph Classes — correspond to some desirable property in real-world data.
Special Graph Classes — correspond to some desirable property in real-world data. Planar Graphs Interval Graphs Cluster Graphs k-Connected Graphs Directed Acyclic Graphs Chordal Graphs
Special Graph Classes — correspond to some desirable property in real-world data. Planar Graphs Interval Graphs Cluster Graphs k-Connected Graphs Directed Acyclic Graphs Chordal Graphs Graph Drawing Physical Mapping of DNA Correlation Clustering Reliability of Networks Deadlock Recovery, Operating Systems Gaussian Elimination over Sparse Matrices
Special Graph Classes — correspond to some desirable property in real-world data. We would like to “achieve” these properties with minimum cost, which naturally leads us to the graph modification framework. Planar Graphs Interval Graphs Cluster Graphs k-Connected Graphs Directed Acyclic Graphs Chordal Graphs Graph Drawing Physical Mapping of DNA Correlation Clustering Reliability of Networks Deadlock Recovery, Operating Systems Gaussian Elimination over Sparse Matrices
Graph Modification Problems Algorithms
Graph Class or “property”, ᴨ
Graph Class or “property”, ᴨ Forbidden Subgraph Characterization A graph G belongs to ᴨ if, and only if, G does not contain any graph from Forb(ᴨ) as an induced subgraph.
Graph Class or “property”, ᴨ Forbidden Subgraph Characterization A graph G belongs to ᴨ if, and only if, G does not contain any graph from Forb(ᴨ) as an induced subgraph. Forbidden Minor Characterization A graph G belongs to ᴨ if, and only if, G does not contain any graph from Forb(ᴨ) as a minor.
Graph Class or “property”, ᴨ Forbidden Subgraph Characterization A graph admits a forbidden subgraph characterization if, and only if, it is closed under taking induced subgraphs (hereditary), that is, the property is preserved under vertex deletions. Forbidden Minor Characterization A graph G belongs to ᴨ if, and only if, G does not contain any graph from Forb(ᴨ) as a minor.
Graph Class or “property”, ᴨ Forbidden Subgraph Characterization A graph admits a forbidden subgraph characterization if, and only if, it is closed under taking induced subgraphs (hereditary), that is, the property is preserved under vertex deletions. Forbidden Minor Characterization A graph admits a forbidden minor characterization if, and only if, it is closed under taking minors, that is, the property is preserved under vertex deletions, edge deletions and edge contractions.
Graph Class or “property”, ᴨ Forbidden Subgraph Characterization A graph admits a forbidden subgraph characterization if, and only if, it is closed under taking induced subgraphs (hereditary), that is, the property is preserved under vertex deletions. Forbidden Minor Characterization A graph admits a forbidden minor characterization if, and only if, it is closed under taking minors, that is, the property is preserved under vertex deletions, edge deletions and edge contractions.
Graph Class or “property”, ᴨ Forbidden Subgraph Characterization A graph admits a forbidden subgraph characterization if, and only if, it is closed under taking induced subgraphs (hereditary), that is, the property is preserved under vertex deletions. If there are finitely many forbidden graphs, then the corresponding graph modification problem is FPT. Forbidden Minor Characterization A graph admits a forbidden minor characterization if, and only if, it is closed under taking minors, that is, the property is preserved under vertex deletions, edge deletions and edge contractions.
Graph Class or “property”, ᴨ Forbidden Subgraph Characterization A graph admits a forbidden subgraph characterization if, and only if, it is closed under taking induced subgraphs (hereditary), that is, the property is preserved under vertex deletions. If there are finitely many forbidden graphs, then the corresponding graph modification problem is FPT. Forbidden Minor Characterization A graph admits a forbidden minor characterization if, and only if, it is closed under taking minors, that is, the property is preserved under vertex deletions, edge deletions and edge contractions. If there are finitely many forbidden minors…?
Graph Class or “property”, ᴨ Forbidden Minor Characterization A graph G belongs to ᴨ if, and only if, G does not contain any graph from Forb(ᴨ) as a minor. A graph admits a forbidden minor characterization if, and only if, it is closed under taking minors.
Graph Class or “property”, ᴨ Forbidden Minor Characterization A graph G belongs to ᴨ if, and only if, G does not contain any graph from Forb(ᴨ) as a minor. A graph admits a forbidden minor characterization if, and only if, it is closed under taking minors. In fact, there is always a finite forbidden set! (Robertson-Seymour; Graph Minors Theorem)
Graph Class or “property”, ᴨ Forbidden Minor Characterization A graph G belongs to ᴨ if, and only if, G does not contain any graph from Forb(ᴨ) as a minor. A graph admits a forbidden minor characterization if, and only if, it is closed under taking minors. In fact, there is always a finite forbidden set! (Robertson-Seymour; Graph Minors Theorem) Suppose the class ᴨ + (modifications) is closed under taking minors…
Graph Class or “property”, ᴨ Forbidden Minor Characterization A graph G belongs to ᴨ if, and only if, G does not contain any graph from Forb(ᴨ) as a minor. A graph admits a forbidden minor characterization if, and only if, it is closed under taking minors. In fact, there is always a finite forbidden set! (Robertson-Seymour; Graph Minors Theorem) Suppose the class ᴨ + (modifications) is closed under taking minors… + (modifications)
Graph Class or “property”, ᴨ Forbidden Minor Characterization A graph G belongs to ᴨ if, and only if, G does not contain any graph from Forb(ᴨ) as a minor. A graph admits a forbidden minor characterization if, and only if, it is closed under taking minors. In fact, there is always a finite forbidden set! (Robertson-Seymour; Graph Minors Theorem) Suppose the class ᴨ + (modifications) is closed under taking minors… Now: the graph modification problem has boiled down to a membership testing problem, which can be done in cubic time, given the finite forbidden set. + (modifications)
Forbidden Subgraph Characterization A graph admits a forbidden subgraph characterization if, and only if, it is closed under taking induced subgraphs (hereditary), that is, the property is preserved under vertex deletions. Forbidden Minor Characterization A graph admits a forbidden minor characterization if, and only if, it is closed under taking minors, that is, the property is preserved under vertex deletions, edge deletions and edge contractions.
Forbidden Subgraph Characterization A graph admits a forbidden subgraph characterization if, and only if, it is closed under taking induced subgraphs (hereditary), that is, the property is preserved under vertex deletions. Can the node deletion question for hereditary properties always be answered with the “graph minors hammer”? Forbidden Minor Characterization A graph admits a forbidden minor characterization if, and only if, it is closed under taking minors, that is, the property is preserved under vertex deletions, edge deletions and edge contractions.
Consider the property ᴨ of being wheel-free, that is, not having any wheel as an induced subgraph.
Consider the property ᴨ of being wheel-free, that is, not having any wheel as an induced subgraph. The property is hereditary, but not minor-closed.
Consider the property ᴨ of being wheel-free, that is, not having any wheel as an induced subgraph. Wheel-Free Deletion and Wheel-Free Vertex Deletion are W[2]-hard. Lokshtanov (2008)
Finding large induced subgraphs with property ᴨ. Khot and Raman, (2002)
Finding large induced subgraphs with property ᴨ. Khot and Raman, (2002) If ᴨ is a hereditary property that contains all independent sets and all cliques, or if ᴨ excludes some independent sets and some cliques, then the problem of finding an induced subgraph of size at least k, satisfying ᴨ, is FPT.
Finding large induced subgraphs with property ᴨ. Khot and Raman, (2002) If ᴨ is a hereditary property that contains all independent sets and all cliques, or if ᴨ excludes some independent sets and some cliques, then the problem of finding an induced subgraph of size at least k, satisfying ᴨ, is FPT. If ᴨ is a hereditary property contains all independent sets but not all cliques (or vice versa), then the problem of finding an induced subgraph of size at least k, satisfying ᴨ, is W[1]-hard.
Finding large induced subgraphs with property ᴨ. Khot and Raman, (2002) If ᴨ is a hereditary property that contains all independent sets and all cliques, or if ᴨ excludes some independent sets and some cliques, then the problem of finding an induced subgraph of size at least k, satisfying ᴨ, is FPT. If ᴨ is a hereditary property contains all independent sets but not all cliques (or vice versa), then the problem of finding an induced subgraph of size at least k, satisfying ᴨ, is W[1]-hard. Ramsey Numbers! Reduction from Independent Set
Vertex Cover Can G be made edgeless by the removal of at most k vertices? A simple randomized algorithm with an error probability of 2-k.
Vertex Cover Can G be made edgeless by the removal of at most k vertices? A simple randomized algorithm with an error probability of 2-k.
Vertex Cover Can G be made edgeless by the removal of at most k vertices? A simple randomized algorithm with an error probability of 2-k.
Feedback Vertex Set Can G be made acyclic by the removal of at most k vertices?
Feedback Vertex Set Can G be made acyclic by the removal of at most k vertices?
Feedback Vertex Set Can G be made acyclic by the removal of at most k vertices?
Feedback Vertex Set Can G be made acyclic by the removal of at most k vertices?
Feedback Vertex Set Can G be made acyclic by the removal of at most k vertices?
Feedback Vertex Set Can G be made acyclic by the removal of at most k vertices? Let’s stare at the structure of a YES-instance. #whisper: Let us also restrict ourselves to graphs of constant maximum degree, say five.
the k vertices corresponding to the FVS (n-k) vertices and at most (n-k-1) edges “outside”.
the k vertices corresponding to the FVS (n-k) vertices and at most (n-k-1) edges “outside”.
Graph Modification Algorithms
Graph Modification Algorithms
t “points of contact” between S and GS…
t “points of contact” between S and GS… at least t edges going across the two sets.
t “points of contact” between S and GS… at least t edges going across the two sets. Delete the points of contact in GS to get at most 5t “pieces”.
t “points of contact” between S and GS… at least t edges going across the two sets. Delete the points of contact in GS to get at most 5t “pieces”. Suppose each piece has constant size, say c.
t “points of contact” between S and GS… at least t edges going across the two sets. Delete the points of contact in GS to get at most 5t “pieces”. Suppose each piece has constant size, say c. The total number of edges in GS is at most 5ct.
t “points of contact” between S and GS… at least t edges going across the two sets. Delete the points of contact in GS to get at most 5t “pieces”. Suppose each piece has constant size, say c. The total number of edges in GS is at most 5ct. The number of good edges is at least t, and the number of bad edges is at most 5ct.
t “points of contact” between S and GS… at least t edges going across the two sets. Delete the points of contact in GS to get at most 5t “pieces”. Suppose each piece has constant size, say c. The total number of edges in GS is at most 5ct. The number of good edges is at least t, and the number of bad edges is at most 5ct. In this scenario, a randomly chosen edge will be good with constant probability!
Bad scenario: one of the pieces is large.
These pieces have simple structure and bounded interaction with the outer world.
These pieces have simple structure and bounded interaction with the outer world. Such pieces are called “protrusions”.
A Boundary of Constant Size Constant Treewidth
A Boundary of Constant Size Constant Treewidth
A Boundary of Constant Size Constant Treewidth
Graph Modification Algorithms
Graph Modification Algorithms
Graph Modification Algorithms
The space of t-boundaried graphs can be broken up into equivalence classes based on how they “behave” with the “other side” of the boundary.
Graph Modification Algorithms
Graph Modification Algorithms
The value of the optimal solution is the same up to a constant.
The space of t-boundaried graphs can be broken up into equivalence classes based on how they “behave” with the “other side” of the boundary.
The space of t-boundaried graphs can be broken up into equivalence classes based on how they “behave” with the “other side” of the boundary. For some problems, the number of equivalence classes is finite, allowing us to replace protrusions in graphs.
Graph Modification Algorithms
Finding protrusions? What do we replace them with?
The 퓕-Deletion Problem
The 퓕-Deletion Problem Can G be made H-minor free, for all H in퓕, by the removal of at most k vertices?
The 퓕-Deletion Problem Can G be made H-minor free, for all H in퓕, by the removal of at most k vertices? Suppose G is a YES-instance, and further, suppose퓕contains at least one planar graph.
The 퓕-Deletion Problem Can G be made H-minor free, for all H in퓕, by the removal of at most k vertices? Suppose G is a YES-instance, and further, suppose퓕contains at least one planar graph. Then GS cannot have large grids.
The 퓕-Deletion Problem Can G be made H-minor free, for all H in퓕, by the removal of at most k vertices? Suppose G is a YES-instance, and further, suppose퓕contains at least one planar graph. Then GS cannot have large grids. Thus the treewidth of GS is bounded by a constant that depends only on퓕.
The 퓕-Deletion Problem Can G be made H-minor free, for all H in퓕, by the removal of at most k vertices? Suppose G is a YES-instance, and further, suppose퓕contains at least one planar graph. Then GS cannot have large grids. Thus the treewidth of GS is bounded by a constant that depends only on퓕. The Planar 퓕-Deletion Problem
The Planar 퓕-Deletion Problem
The Planar 퓕-Deletion Problem Ensure that protrusions are removed.
The Planar 퓕-Deletion Problem Ensure that protrusions are removed. Pick an edge uniformly at random.
The Planar 퓕-Deletion Problem Ensure that protrusions are removed. Pick an edge uniformly at random. Include both endpoints.
The Planar 퓕-Deletion Problem Ensure that protrusions are removed. Pick an edge uniformly at random. Include both endpoints. Pick an endpoint u.a.r.
The Planar 퓕-Deletion Problem Ensure that protrusions are removed. Pick an edge uniformly at random. Include both endpoints. Pick an endpoint u.a.r. Repeat till G is퓕-free.
The Planar 퓕-Deletion Problem Ensure that protrusions are removed. Pick an edge uniformly at random. Include both endpoints. Pick an endpoint u.a.r. (Approximation algorithm.) Repeat till G is퓕-free.
The Planar 퓕-Deletion Problem Ensure that protrusions are removed. Pick an edge uniformly at random. Include both endpoints. Pick an endpoint u.a.r. (Approximation algorithm.) (Randomized algorithm.) Repeat till G is퓕-free.
The Planar 퓕-Deletion Problem accounts for several specific problems…
The Planar 퓕-Deletion Problem accounts for several specific problems… 퓕 K2 K3 ((k+1)-by-(k+1)) Grids K2,3, K4 K4 K3,T2 Vertex Cover Feedback Vertex Set Treewidth k Outerplanar Series-Parallel Pathwidth-One
Graph Modification Algorithms
Thanks to Graph Minors, we have a f(k)n2 algorithm for the 퓕-Deletion problem.
Thanks to Graph Minors, we have a f(k)n2 algorithm for the 퓕-Deletion problem. For Planar 퓕-Deletion, the starting point was a double-exponential algorithm. [Bodlaender, 1997]
Thanks to Graph Minors, we have a f(k)n2 algorithm for the 퓕-Deletion problem. For Planar 퓕-Deletion, the starting point was a double-exponential algorithm. [Bodlaender, 1997] Many single-exponential algorithms are known for special cases of 퓕.
Thanks to Graph Minors, we have a f(k)n2 algorithm for the 퓕-Deletion problem. For Planar 퓕-Deletion, the starting point was a double-exponential algorithm. [Bodlaender, 1997] Many single-exponential algorithms are known for special cases of 퓕. Feedback Vertex Set has Cao, Chen & Liu a O*(3.83k) algorithm. [2010]
Thanks to Graph Minors, we have a f(k)n2 algorithm for the 퓕-Deletion problem. For Planar 퓕-Deletion, the starting point was a double-exponential algorithm. [Bodlaender, 1997] Many single-exponential algorithms are known for special cases of 퓕. Cao, Chen & Liu Feedback Vertex Set has a O*(3.83k) algorithm. [2010] {K4}-Deletion has Kim, Paul & Philip a O*(2O(k)) algorithm. [2012]
Thanks to Graph Minors, we have a f(k)n2 algorithm for the 퓕-Deletion problem. For Planar 퓕-Deletion, the starting point was a double-exponential algorithm. [Bodlaender, 1997] Many single-exponential algorithms are known for special cases of 퓕. Cao, Chen & Liu Feedback Vertex Set has a O*(3.83k) algorithm. [2010] {K4}-Deletion has Kim, Paul & Philip a O*(2O(k)) algorithm. Pathwidth-1-Deletion has a O*(4.65k) algorithm. [2012] Cygan, Pilipczuk, Pilipczuk & Wojtaszczyk [2010]
Graph Modification Algorithms
Fomin, Lokshtanov, M. & Saurabh [2013] Planar퓕-Deletion admits a randomized (2O(k)n) algorithm when every graph in퓕is connected.
Fomin, Lokshtanov, M. & Saurabh [2013] Planar퓕-Deletion admits a randomized (2O(k)n) algorithm when every graph in퓕is connected. Best possible under ETH.
Fomin, Lokshtanov, M. & Saurabh [2013] Planar퓕-Deletion admits a randomized (2O(k)n) algorithm when every graph in퓕is connected. Best possible under ETH. Planar 퓕-Deletion admits a deterministic (2O(k)n log2n) algorithm when every graph in 퓕 is connected.
Fomin, Lokshtanov, M. & Saurabh [2013] Planar퓕-Deletion admits a randomized (2O(k)n) algorithm when every graph in퓕is connected. Best possible under ETH. Planar 퓕-Deletion admits a deterministic (2O(k)n log2n) algorithm when every graph in 퓕 is connected. Planar퓕-Deletion admits an O(nm) randomized algorithm that leads us to a constant-factor approximation. Can we get a deterministic constant-­factor approximation algorithm?
Fomin, Lokshtanov, M. & Saurabh [2013] Planar퓕-Deletion admits a randomized (2O(k)n) algorithm when every graph in퓕is connected. Best possible under ETH. Planar 퓕-Deletion admits a deterministic (2O(k)n log2n) algorithm when every graph in 퓕 is connected. Planar퓕-Deletion admits an O(nm) randomized algorithm that leads us to a constant-factor approximation. Can we get a deterministic constant-­factor approximation algorithm? Planar퓕-Deletion admits an deterministic algorithm that leads us to a O(log3/2(OPT)) approximation. Fomin, Lokshtanov, M., Philip & Saurabh [2013]
Fomin, Lokshtanov, M. & Saurabh [2013] Planar퓕-Deletion admits a randomized (2O(k)n) algorithm when every graph in퓕is connected. Best possible under ETH. Planar 퓕-Deletion admits a deterministic (2O(k)n log2n) algorithm when every graph in 퓕 is connected. Planar퓕-Deletion admits an O(nm) randomized algorithm that leads us to a constant-factor approximation. Can we get a deterministic constant-­factor approximation algorithm? Planar퓕-Deletion admits an deterministic algorithm that leads us to a O(log3/2(OPT)) approximation. Fomin, Lokshtanov, M., Philip & Saurabh [2013] Kim, Langer, Paul, Reidl, Rossmanith, Sau, & Sikdar [2013] Planar퓕-Deletion admits a deterministic (2O(k)n2) algorithm.
Fomin, Lokshtanov, M. & Saurabh [2013] Planar퓕-Deletion admits a randomized (2O(k)n) algorithm when every graph in퓕is connected. Best possible under ETH. Planar 퓕-Deletion admits a deterministic (2O(k)n) algorithm when every graph in 퓕 is connected. Planar퓕-Deletion admits an O(nm) randomized algorithm that leads us to a constant-factor approximation. Can we get a deterministic constant-­factor approximation algorithm? Planar퓕-Deletion admits an deterministic algorithm that leads us to a O(log3/2(OPT)) approximation. Fomin, Lokshtanov, M., Philip & Saurabh [2013] Kim, Langer, Paul, Reidl, Rossmanith, Sau, & Sikdar [2013] Planar퓕-Deletion admits a deterministic (2O(k)n2) algorithm. Fomin, Lokshtanov, M., Ramanujan& Saurabh [2015]
Graph Modification Algorithms
Many polynomial kernels are also known for special cases of 퓕.
Many polynomial kernels are also known for special cases of 퓕. Feedback Vertex Set has a Thomasse O(k2) instance kernel. [2009]
Many polynomial kernels are also known for special cases of 퓕. Thomasse Feedback Vertex Set has a O(k2) instance kernel. [2009] Vertex Cover has (Various) a O(k) vertex kernel.
Many polynomial kernels are also known for special cases of 퓕. Thomasse Feedback Vertex Set has a O(k2) instance kernel. [2009] Vertex Cover has (Various) a O(k) vertex kernel. Pathwidth-1-Deletion has a polynomial kernel Cygan, Pilipczuk, Pilipczuk & Wojtaszczyk [2010]
Many polynomial kernels are also known for special cases of 퓕. Thomasse Feedback Vertex Set has a O(k2) instance kernel. [2009] Vertex Cover has (Various) a O(k) vertex kernel. Pathwidth-1-Deletion has a polynomial kernel Cygan, Pilipczuk, Pilipczuk & Wojtaszczyk [2010] Fomin, Lokshtanov, M. & Saurabh [2013] Planar 퓕-Deletion admits a polynomial kernel.
Many polynomial kernels are also known for special cases of 퓕. Thomasse Feedback Vertex Set has a O(k2) instance kernel. [2009] Vertex Cover has (Various) a O(k) vertex kernel. Pathwidth-1-Deletion has a polynomial kernel Cygan, Pilipczuk, Pilipczuk & Wojtaszczyk [2010] Fomin, Lokshtanov, M. & Saurabh [2013] Planar 퓕-Deletion admits a polynomial kernel. Best possible under standard assumptions.
The 퓕-Deletion Problem
The 퓕-Deletion Problem Very little is known beyond the graph minors result.
The 퓕-Deletion Problem Very little is known beyond the graph minors result. The most fundamental 퓕-deletion problem that is not accounted for by Planar 퓕-Deletion is Planarization.
The 퓕-Deletion Problem Very little is known beyond the graph minors result. The most fundamental 퓕-deletion problem that is not accounted for by Planar 퓕-Deletion is Planarization. Marx and Schlotter [2012] Planarization admits an algorithm with running time O(2g(k)n2) where g(k) = kO(k3)
The 퓕-Deletion Problem Very little is known beyond the graph minors result. The most fundamental 퓕-deletion problem that is not accounted for by Planar 퓕-Deletion is Planarization. Marx and Schlotter [2012] Planarization admits an algorithm with running time O(2g(k)n2) where g(k) = kO(k3) Kawarabayashi [2009] Planarization admits an algorithm with running time O(f(k)n) where f(k) is not explicitly specified.
The 퓕-Deletion Problem Very little is known beyond the graph minors result. The most fundamental 퓕-deletion problem that is not accounted for by Planar 퓕-Deletion is Planarization. Marx and Schlotter [2012] Planarization admits an algorithm with running time O(2g(k)n2) where g(k) = kO(k3) Kawarabayashi [2009] Planarization admits an algorithm with running time O(f(k)n) where f(k) is not explicitly specified. Jansen, Lokshtanov & Saurabh [2014] Planarization admits an algorithm with running time 2O(k log k)n, achieving the best combined dependence on k and n.
The 퓕-Deletion Problem Very little is known beyond the graph minors result. The most fundamental 퓕-deletion problem that is not accounted for by Planar 퓕-Deletion is Planarization. The question of kernels is completely open. Marx and Schlotter [2012] Planarization admits an algorithm with running time O(2g(k)n2) where g(k) = kO(k3) Kawarabayashi [2009] Planarization admits an algorithm with running time O(f(k)n) where f(k) is not explicitly specified. Jansen, Lokshtanov & Saurabh [2014] Planarization admits an algorithm with running time 2O(k log k)n, achieving the best combined dependence on k and n.
Graph Modification Problems When the operation induces hardness
Contraction Problems
Contraction Problems Graph Class or “property”, ᴨ Forbidden Subgraph Characterization A graph admits a forbidden subgraph characterization if, and only if, it is closed under taking induced subgraphs (hereditary), that is, the property is preserved under vertex deletions. If there are finitely many forbidden graphs, then the corresponding graph modification problem is FPT.
Contraction Problems Graph Class or “property”, ᴨ Forbidden Subgraph Characterization A graph admits a forbidden subgraph characterization if, and only if, it is closed under taking induced subgraphs (hereditary), that is, the property is preserved under vertex deletions. If there are finitely many forbidden graphs, then the corresponding graph modification problem is FPT. Not true any more!
Contraction To C4-free graphs Can G be made C4-free by the contraction of at most k edges? W[2]-hard by a simple reduction from Hitting Set.
Contraction To C4-free graphs Can G be made C4-free by the contraction of at most k edges? W[2]-hard by a simple reduction from Hitting Set.
Graph Modification Algorithms
Ct-free Contraction isW[2]-hard if t ⩾ 4 and FPT if t⩽ 3. Pt-free Contraction isW[2]-hard if t ⩾ 5 and FPT if t ⩽ 4. Lokshtanov, M. & Saurabh [2013]
Ct-free Contraction isW[2]-hard if t ⩾ 4 and FPT if t⩽ 3. Pt-free Contraction isW[2]-hard if t ⩾ 5 and FPT if t ⩽ 4. Lokshtanov, M. & Saurabh [2013] Chordal Contraction isW[2]-hard while Clique contraction is FPT but is unlikely to admit a polynomial kernel. Cai and Guo [2013]
Ct-free Contraction isW[2]-hard if t ⩾ 4 and FPT if t⩽ 3. Pt-free Contraction isW[2]-hard if t ⩾ 5 and FPT if t ⩽ 4. Lokshtanov, M. & Saurabh [2013] Chordal Contraction isW[2]-hard while Clique contraction is FPT but is unlikely to admit a polynomial kernel. Characeterizations are open. Cai and Guo [2013]
Heggernes, van’t Hof, Lokshtanov & Paul [2011] Ct-free Contraction isW[2]-hard if t ⩾ 4 and FPT if t⩽ 3. Pt-free Contraction isW[2]-hard if t ⩾ 5 and FPT if t ⩽ 4. Chordal Contraction isW[2]-hard while Clique contraction is FPT but is unlikely to admit a polynomial kernel. Contracting to Bipartite graphs is fixed-parameter tractable with a double-exponential running time. Lokshtanov, M. & Saurabh [2013] Characeterizations are open. Cai and Guo [2013]
Heggernes, van’t Hof, Lokshtanov & Paul [2011] Ct-free Contraction isW[2]-hard if t ⩾ 4 and FPT if t⩽ 3. Pt-free Contraction isW[2]-hard if t ⩾ 5 and FPT if t ⩽ 4. Chordal Contraction isW[2]-hard while Clique contraction is FPT but is unlikely to admit a polynomial kernel. Characeterizations are open. Contracting to Bipartite graphs is fixed-parameter tractable with a double-exponential running time. Lokshtanov, M. & Saurabh [2013] Polynomial kernels are open. Cai and Guo [2013]
Heggernes, van’t Hof, Lokshtanov & Paul [2011] Ct-free Contraction isW[2]-hard if t ⩾ 4 and FPT if t⩽ 3. Pt-free Contraction isW[2]-hard if t ⩾ 5 and FPT if t ⩽ 4. Chordal Contraction isW[2]-hard while Clique contraction is FPT but is unlikely to admit a polynomial kernel. Characeterizations are open. Contracting to Bipartite graphs is fixed-parameter tractable with a double-exponential running time. Lokshtanov, M. & Saurabh [2013] Polynomial kernels are open. Heggernes, van’t Hof, Lévêque, Lokshtanov & Paul [2011] Contracting to paths is FPT and has a linear-vertex kernel; while contracting to trees is FPT but is unlikely to admit a polynomial kernel. Cai and Guo [2013]
Graph Modification Problems Completion
Chordal Completion
Yannakakis [1981] Chordal Completion The minimum fill-in problem is NP-complete (was left open in the first edition of Garey and Johnson).
Yannakakis [1981] Chordal Completion The minimum fill-in problem is NP-complete (was left open in the first edition of Garey and Johnson). Kaplan, Shamir & Tarjan [1999] Chordal Completion is fixed-parameter tractable (FPT) with a running time of O(k616k + k2mn).
Yannakakis [1981] Chordal Completion The minimum fill-in problem is NP-complete (was left open in the first edition of Garey and Johnson). Kaplan, Shamir & Tarjan [1999] Chordal Completion is fixed-parameter tractable (FPT) with a running time of O(k616k + k2mn). Bodlaender, Heggernes, & Villanger [2011] Chordal Completion is fixed-parameter tractable (FPT) with a running time of O(2.36k + k2mn).
Yannakakis [1981] Chordal Completion The minimum fill-in problem is NP-complete (was left open in the first edition of Garey and Johnson). Kaplan, Shamir & Tarjan [1999] Chordal Completion is fixed-parameter tractable (FPT) with a running time of O(k616k + k2mn). Bodlaender, Heggernes, & Villanger [2011] Chordal Completion is fixed-parameter tractable (FPT) with a running time of O(2.36k + k2mn). Fomin & Villanger [2013] Chordal Completion admits a sub exponential parameterized algorithm with a running time of O(2√k log k + k2mn).
Interval Completion
Kaplan, Shamir & Tarjan [1999] Interval Completion Chordal Completion, Strongly Chordal Completion, and Proper Interval Completion are fixed-parameter tractable (FPT).
Kaplan, Shamir & Tarjan [1999] Interval Completion Chordal Completion, Strongly Chordal Completion, and Proper Interval Completion are fixed-parameter tractable (FPT). Heggernes, Paul, Telle & Villanger [2007] Interval Completion is fixed-parameter tractable (FPT), with a running time of O(k2kn3m).
Kaplan, Shamir & Tarjan [1999] Interval Completion Chordal Completion, Strongly Chordal Completion, and Proper Interval Completion are fixed-parameter tractable (FPT). Heggernes, Paul, Telle & Villanger [2007] Interval Completion is fixed-parameter tractable (FPT), with a running time of O(k2kn3m). Cao [2007] Interval Completion has a single-exponential parameterized algorithm, with a running time of O(6k(n+m)).
Kaplan, Shamir & Tarjan [1999] Interval Completion Chordal Completion, Strongly Chordal Completion, and Proper Interval Completion are fixed-parameter tractable (FPT). Heggernes, Paul, Telle & Villanger [2007] Interval Completion is fixed-parameter tractable (FPT), with a running time of O(k2kn3m). Cao [2007] Interval Completion has a single-exponential parameterized algorithm, with a running time of O(6k(n+m)). Bliznets, Fomin, Pilipczuk & Pilipczuk [2014] Interval Completion admits a subexponential parameterized algorithm with a running time of kO(√k)nO(1).
Graph Modification Problems A Summary
Graph Modification Algorithms
Edge Deletions Kernels on Planar Graphs & the Meta-Kernel project Special Graph Classes, eg, Feedback Arc Set on Tournaments Connectivity Augmentation The descriptive complexity of graph modification Structural parameters and other objective functions Backdoor Sets!
Graph Modification Algorithms
Graph Modification Algorithms

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Graph Modification Algorithms

  • 1. Graph Modification Problems A Modern Perspective
  • 3. Setting the stage: Definitions and Preliminary Observations
  • 4. Setting the stage: Definitions and Preliminary Observations Brief excursions into specific examples
  • 5. Setting the stage: Definitions and Preliminary Observations Brief excursions into specific examples Current Trends & Future Directions
  • 6. Ingredients of a typical Graph Modification Problem
  • 9. Input Graph, G Graph Class or “property”, ᴨ
  • 10. Goal: Transform G so that it belongs to ᴨ with “minimum damages”. Input Graph, G Graph Class or “property”, ᴨ
  • 11. Goal: Transform G so that it belongs to ᴨ with “minimum damages”. Input Graph, G Graph Class or “property”, ᴨ
  • 12. Goal: Transform G so that it belongs to ᴨ with “minimum damages”. Input Graph, G Graph Class or “property”, ᴨ
  • 13. Goal: Transform G so that it belongs to ᴨ with “minimum damages”. vertex deletions Input Graph, G Graph Class or “property”, ᴨ
  • 14. Goal: Transform G so that it belongs to ᴨ with “minimum damages”. vertex deletions edge deletions Input Graph, G Graph Class or “property”, ᴨ
  • 15. Goal: Transform G so that it belongs to ᴨ with “minimum damages”. vertex deletions edge deletions edge additions Input Graph, G Graph Class or “property”, ᴨ
  • 16. Goal: Transform G so that it belongs to ᴨ with “minimum damages”. vertex deletions edge deletions edge additions edge contractions Input Graph, G Graph Class or “property”, ᴨ
  • 17. Goal: Transform G so that it belongs to ᴨ with “minimum damages”. vertex deletions edge deletions edge additions edge contractions edge editing Input Graph, G Graph Class or “property”, ᴨ
  • 18. Goal: Transform G so that it belongs to ᴨ with “minimum damages”. vertex deletions edge deletions edge additions edge contractions edge editing Input Graph, G Vertex Cover Edgeless Graphs
  • 19. Goal: Transform G so that it belongs to ᴨ with “minimum damages”. vertex deletions edge deletions edge additions edge contractions edge editing Input Graph, G Feedback Vertex Set Acyclic Graphs
  • 20. Goal: Transform G so that it belongs to ᴨ with “minimum damages”. vertex deletions edge deletions edge additions edge contractions edge editing Input Graph, G Minimum Fill-In Chordal Graphs
  • 21. Goal: Transform G so that it belongs to ᴨ with “minimum damages”. vertex deletions edge deletions edge additions edge contractions edge editing Input Graph, G Cluster Editing Cluster Graphs
  • 22. Goal: Transform G so that it belongs to ᴨ with “minimum damages”.
  • 23. Goal: Transform G so that it belongs to ᴨ with “minimum damages”. Minimum ᴨ-Completion
  • 24. Goal: Transform G so that it belongs to ᴨ with “minimum damages”. Minimum ᴨ-Completion Minimum ᴨ-Supergraph
  • 25. Goal: Transform G so that it belongs to ᴨ with “minimum damages”. Minimum ᴨ-Completion Minimum ᴨ-Deletion Minimum ᴨ-Supergraph
  • 26. Goal: Transform G so that it belongs to ᴨ with “minimum damages”. Minimum ᴨ-Completion Minimum ᴨ-Deletion Minimum ᴨ-Supergraph Maximum ᴨ-Spanning Subgraph
  • 27. Goal: Transform G so that it belongs to ᴨ with “minimum damages”. Minimum ᴨ-Completion Minimum ᴨ-Deletion Minimum ᴨ-Editing Minimum ᴨ-Supergraph Maximum ᴨ-Spanning Subgraph
  • 28. Goal: Transform G so that it belongs to ᴨ with “minimum damages”. Minimum ᴨ-Completion Minimum ᴨ-Deletion Minimum ᴨ-Editing Minimum ᴨ-Supergraph Maximum ᴨ-Spanning Subgraph Closest ᴨ-Graph
  • 29. Goal: Transform G so that it belongs to ᴨ with “minimum damages”. Minimum ᴨ-Completion Minimum ᴨ-Deletion Minimum ᴨ-Editing Minimum ᴨ-Supergraph Maximum ᴨ-Spanning Subgraph Closest ᴨ-Graph Minimum ᴨ-Vertex Deletion
  • 30. Goal: Transform G so that it belongs to ᴨ with “minimum damages”. Minimum ᴨ-Completion Minimum ᴨ-Deletion Minimum ᴨ-Editing Minimum ᴨ-Supergraph Maximum ᴨ-Spanning Subgraph Closest ᴨ-Graph Minimum ᴨ-Vertex Deletion Maximum ᴨ-Induced Subgraph
  • 31. Goal: Transform G so that it belongs to ᴨ with “minimum damages”. Minimum ᴨ-Completion Minimum ᴨ-Deletion Minimum ᴨ-Editing Minimum ᴨ-Supergraph Maximum ᴨ-Spanning Subgraph Closest ᴨ-Graph Minimum ᴨ-Vertex Deletion Maximum ᴨ-Induced Subgraph Completion to Minimum Max-Clique
  • 32. Goal: Transform G so that it belongs to ᴨ with “minimum damages”. Minimum ᴨ-Completion Minimum ᴨ-Deletion Minimum ᴨ-Editing Minimum ᴨ-Supergraph Maximum ᴨ-Spanning Subgraph Closest ᴨ-Graph Minimum ᴨ-Vertex Deletion Maximum ᴨ-Induced Subgraph Completion to Minimum Max-Clique Modification with Restrictions, eg, the Sandwich problem
  • 33. Goal: Transform G so that it belongs to ᴨ with “minimum damages”. Minimum ᴨ-Completion Minimum ᴨ-Deletion Minimum ᴨ-Editing Minimum ᴨ-Supergraph Maximum ᴨ-Spanning Subgraph Closest ᴨ-Graph Minimum ᴨ-Vertex Deletion Maximum ᴨ-Induced Subgraph Completion to Minimum Max-Clique Modification with Restrictions, eg, the Sandwich problem Restricted Classes of Input
  • 34. Graph Class or “property”, ᴨ
  • 35. Non-Trivial ᴨ admits infinitely many graphs, and also excludes infinitely many graphs. Graph Class or “property”, ᴨ
  • 36. Non-Trivial ᴨ admits infinitely many graphs, and also excludes infinitely many graphs. Monotone If G belongs to ᴨ, then all subgraphs of G also belong to ᴨ. Graph Class or “property”, ᴨ
  • 37. Non-Trivial ᴨ admits infinitely many graphs, and also excludes infinitely many graphs. Graph Class or “property”, ᴨ Monotone Hereditary If G belongs to ᴨ, then all subgraphs of G also belong to ᴨ. If G belongs to ᴨ, then all induced subgraphs of G also belong to ᴨ.
  • 38. Non-Trivial ᴨ admits infinitely many graphs, and also excludes infinitely many graphs. (Bipartite, Planar) Graph Class or “property”, ᴨ Monotone Hereditary If G belongs to ᴨ, then all subgraphs of G also belong to ᴨ. If G belongs to ᴨ, then all induced subgraphs of G also belong to ᴨ.
  • 39. Non-Trivial ᴨ admits infinitely many graphs, and also excludes infinitely many graphs. (Bipartite, Planar) Graph Class or “property”, ᴨ Monotone Hereditary If G belongs to ᴨ, then all subgraphs of G also belong to ᴨ. If G belongs to ᴨ, then all induced subgraphs of G also belong to ᴨ. (Complete graphs, Forests)
  • 40. Non-Trivial ᴨ admits infinitely many graphs, and also excludes infinitely many graphs. (Bipartite, Planar) Graph Class or “property”, ᴨ Monotone Hereditary If G belongs to ᴨ, then all subgraphs of G also belong to ᴨ. If G belongs to ᴨ, then all induced subgraphs of G also belong to ᴨ. (Complete graphs, Forests) Connectivity, Biconnectivity, Trees, Stars, Eulerian, etc.
  • 42. Lewis & Yannakakis [1980] The vertex-deletion problem is NP-complete for non-trivial, hereditary properties.
  • 43. Lewis & Yannakakis [1980] The vertex-deletion problem is NP-complete for non-trivial, hereditary properties. Yannakakis [1979] The connected vertex-deletion problem is NP-complete for non-trivial properties that hold on connected induced subgraphs.
  • 44. Lewis & Yannakakis [1980] The vertex-deletion problem is NP-complete for non-trivial, hereditary properties. Yannakakis [1979] (Edgeless, Acyclic, etc.) The connected vertex-deletion problem is NP-complete for non-trivial properties that hold on connected induced subgraphs.
  • 45. Lewis & Yannakakis [1980] The vertex-deletion problem is NP-complete for non-trivial, hereditary properties. Yannakakis [1979] (Edgeless, Acyclic, etc.) The connected vertex-deletion problem is NP-complete for non-trivial properties that hold on connected induced subgraphs.( Trees, Stars, etc.)
  • 47. Edge-Deletion or Completion problems are sometimes easier than their vertex-deletion counterparts.
  • 48. Edge-Deletion or Completion problems are sometimes easier than their vertex-deletion counterparts. Add edges to make the input graph a cluster graph.
  • 49. Edge-Deletion or Completion problems are sometimes easier than their vertex-deletion counterparts. Add edges to make the input graph a cluster graph. Remove edges to make the input graph edgeless.
  • 50. Edge-Deletion or Completion problems are sometimes easier than their vertex-deletion counterparts. Add edges to make the input graph a cluster graph. Remove edges to make the input graph edgeless. Remove edges to make the input graph acyclic.
  • 51. Edge-Deletion or Completion problems are sometimes easier than their vertex-deletion counterparts. Add edges to make the input graph a cluster graph. Remove edges to make the input graph edgeless. Remove edges to make the input graph acyclic. Remove edges to make the input graph bipartite.
  • 52. Edge-Deletion or Completion problems are sometimes easier than their vertex-deletion counterparts. Add edges to make the input graph a cluster graph. Remove edges to make the input graph edgeless. Remove edges to make the input graph acyclic.
  • 53. Edge-Deletion or Completion problems are sometimes easier than their vertex-deletion counterparts. Add edges to make the input graph a cluster graph. Remove edges to make the input graph edgeless. Remove edges to make the input graph acyclic. Alon, Shapira & Sudakov [2009] Given a property ᴨ such that ᴨ holds for every bipartite graph, the Minimum ᴨ-Deletion problem is NP-hard.
  • 54. Edge-Deletion or Completion problems are sometimes easier than their vertex-deletion counterparts. Add edges to make the input graph a cluster graph. Remove edges to make the input graph edgeless. Remove edges to make the input graph acyclic. Alon, Shapira & Sudakov [2009] Given a property ᴨ such that ᴨ holds for every bipartite graph, the Minimum ᴨ-Deletion problem is NP-hard. (Triangle-­free)
  • 55. Edge-Deletion or Completion problems are sometimes easier than their vertex-deletion counterparts. Add edges to make the input graph a cluster graph. Remove edges to make the input graph edgeless. Remove edges to make the input graph acyclic. Alon, Shapira & Sudakov [2009] Given a property ᴨ such that ᴨ holds for every bipartite graph, the Minimum ᴨ-Deletion problem is NP-hard. Colbourn & El-Mallah [1988] (Triangle-­free) Making a graph Pk-free by deleting the minimum number of edges is NP-hard for every k > 2.
  • 56. edge editing edge contractions vertex deletions edge additions edge deletions Graph Class or “property”, ᴨ
  • 57. edge editing edge contractions vertex deletions edge additions edge deletions Graph Class or “property”, ᴨ Perfect Trivially Perfect Cographs Permutation Graphs Bipartite Trees Weakly Chordal Chordal Strongly Chordal Split Interval Proper Interval Unit Interval Forests Caterpillars Chain Chordal Bipartite Distance Hereditary Comparability Trapezoid CoChordal Circle Planar k-Connected
  • 58. Graph Modification Problems A Modern Perspective
  • 61. Special Graph Classes — correspond to some desirable property in real-world data.
  • 62. Special Graph Classes — correspond to some desirable property in real-world data. Planar Graphs Interval Graphs Cluster Graphs k-Connected Graphs Directed Acyclic Graphs Chordal Graphs
  • 63. Special Graph Classes — correspond to some desirable property in real-world data. Planar Graphs Interval Graphs Cluster Graphs k-Connected Graphs Directed Acyclic Graphs Chordal Graphs Graph Drawing Physical Mapping of DNA Correlation Clustering Reliability of Networks Deadlock Recovery, Operating Systems Gaussian Elimination over Sparse Matrices
  • 64. Special Graph Classes — correspond to some desirable property in real-world data. We would like to “achieve” these properties with minimum cost, which naturally leads us to the graph modification framework. Planar Graphs Interval Graphs Cluster Graphs k-Connected Graphs Directed Acyclic Graphs Chordal Graphs Graph Drawing Physical Mapping of DNA Correlation Clustering Reliability of Networks Deadlock Recovery, Operating Systems Gaussian Elimination over Sparse Matrices
  • 66. Graph Class or “property”, ᴨ
  • 67. Graph Class or “property”, ᴨ Forbidden Subgraph Characterization A graph G belongs to ᴨ if, and only if, G does not contain any graph from Forb(ᴨ) as an induced subgraph.
  • 68. Graph Class or “property”, ᴨ Forbidden Subgraph Characterization A graph G belongs to ᴨ if, and only if, G does not contain any graph from Forb(ᴨ) as an induced subgraph. Forbidden Minor Characterization A graph G belongs to ᴨ if, and only if, G does not contain any graph from Forb(ᴨ) as a minor.
  • 69. Graph Class or “property”, ᴨ Forbidden Subgraph Characterization A graph admits a forbidden subgraph characterization if, and only if, it is closed under taking induced subgraphs (hereditary), that is, the property is preserved under vertex deletions. Forbidden Minor Characterization A graph G belongs to ᴨ if, and only if, G does not contain any graph from Forb(ᴨ) as a minor.
  • 70. Graph Class or “property”, ᴨ Forbidden Subgraph Characterization A graph admits a forbidden subgraph characterization if, and only if, it is closed under taking induced subgraphs (hereditary), that is, the property is preserved under vertex deletions. Forbidden Minor Characterization A graph admits a forbidden minor characterization if, and only if, it is closed under taking minors, that is, the property is preserved under vertex deletions, edge deletions and edge contractions.
  • 71. Graph Class or “property”, ᴨ Forbidden Subgraph Characterization A graph admits a forbidden subgraph characterization if, and only if, it is closed under taking induced subgraphs (hereditary), that is, the property is preserved under vertex deletions. Forbidden Minor Characterization A graph admits a forbidden minor characterization if, and only if, it is closed under taking minors, that is, the property is preserved under vertex deletions, edge deletions and edge contractions.
  • 72. Graph Class or “property”, ᴨ Forbidden Subgraph Characterization A graph admits a forbidden subgraph characterization if, and only if, it is closed under taking induced subgraphs (hereditary), that is, the property is preserved under vertex deletions. If there are finitely many forbidden graphs, then the corresponding graph modification problem is FPT. Forbidden Minor Characterization A graph admits a forbidden minor characterization if, and only if, it is closed under taking minors, that is, the property is preserved under vertex deletions, edge deletions and edge contractions.
  • 73. Graph Class or “property”, ᴨ Forbidden Subgraph Characterization A graph admits a forbidden subgraph characterization if, and only if, it is closed under taking induced subgraphs (hereditary), that is, the property is preserved under vertex deletions. If there are finitely many forbidden graphs, then the corresponding graph modification problem is FPT. Forbidden Minor Characterization A graph admits a forbidden minor characterization if, and only if, it is closed under taking minors, that is, the property is preserved under vertex deletions, edge deletions and edge contractions. If there are finitely many forbidden minors…?
  • 74. Graph Class or “property”, ᴨ Forbidden Minor Characterization A graph G belongs to ᴨ if, and only if, G does not contain any graph from Forb(ᴨ) as a minor. A graph admits a forbidden minor characterization if, and only if, it is closed under taking minors.
  • 75. Graph Class or “property”, ᴨ Forbidden Minor Characterization A graph G belongs to ᴨ if, and only if, G does not contain any graph from Forb(ᴨ) as a minor. A graph admits a forbidden minor characterization if, and only if, it is closed under taking minors. In fact, there is always a finite forbidden set! (Robertson-Seymour; Graph Minors Theorem)
  • 76. Graph Class or “property”, ᴨ Forbidden Minor Characterization A graph G belongs to ᴨ if, and only if, G does not contain any graph from Forb(ᴨ) as a minor. A graph admits a forbidden minor characterization if, and only if, it is closed under taking minors. In fact, there is always a finite forbidden set! (Robertson-Seymour; Graph Minors Theorem) Suppose the class ᴨ + (modifications) is closed under taking minors…
  • 77. Graph Class or “property”, ᴨ Forbidden Minor Characterization A graph G belongs to ᴨ if, and only if, G does not contain any graph from Forb(ᴨ) as a minor. A graph admits a forbidden minor characterization if, and only if, it is closed under taking minors. In fact, there is always a finite forbidden set! (Robertson-Seymour; Graph Minors Theorem) Suppose the class ᴨ + (modifications) is closed under taking minors… + (modifications)
  • 78. Graph Class or “property”, ᴨ Forbidden Minor Characterization A graph G belongs to ᴨ if, and only if, G does not contain any graph from Forb(ᴨ) as a minor. A graph admits a forbidden minor characterization if, and only if, it is closed under taking minors. In fact, there is always a finite forbidden set! (Robertson-Seymour; Graph Minors Theorem) Suppose the class ᴨ + (modifications) is closed under taking minors… Now: the graph modification problem has boiled down to a membership testing problem, which can be done in cubic time, given the finite forbidden set. + (modifications)
  • 79. Forbidden Subgraph Characterization A graph admits a forbidden subgraph characterization if, and only if, it is closed under taking induced subgraphs (hereditary), that is, the property is preserved under vertex deletions. Forbidden Minor Characterization A graph admits a forbidden minor characterization if, and only if, it is closed under taking minors, that is, the property is preserved under vertex deletions, edge deletions and edge contractions.
  • 80. Forbidden Subgraph Characterization A graph admits a forbidden subgraph characterization if, and only if, it is closed under taking induced subgraphs (hereditary), that is, the property is preserved under vertex deletions. Can the node deletion question for hereditary properties always be answered with the “graph minors hammer”? Forbidden Minor Characterization A graph admits a forbidden minor characterization if, and only if, it is closed under taking minors, that is, the property is preserved under vertex deletions, edge deletions and edge contractions.
  • 81. Consider the property ᴨ of being wheel-free, that is, not having any wheel as an induced subgraph.
  • 82. Consider the property ᴨ of being wheel-free, that is, not having any wheel as an induced subgraph. The property is hereditary, but not minor-closed.
  • 83. Consider the property ᴨ of being wheel-free, that is, not having any wheel as an induced subgraph. Wheel-Free Deletion and Wheel-Free Vertex Deletion are W[2]-hard. Lokshtanov (2008)
  • 84. Finding large induced subgraphs with property ᴨ. Khot and Raman, (2002)
  • 85. Finding large induced subgraphs with property ᴨ. Khot and Raman, (2002) If ᴨ is a hereditary property that contains all independent sets and all cliques, or if ᴨ excludes some independent sets and some cliques, then the problem of finding an induced subgraph of size at least k, satisfying ᴨ, is FPT.
  • 86. Finding large induced subgraphs with property ᴨ. Khot and Raman, (2002) If ᴨ is a hereditary property that contains all independent sets and all cliques, or if ᴨ excludes some independent sets and some cliques, then the problem of finding an induced subgraph of size at least k, satisfying ᴨ, is FPT. If ᴨ is a hereditary property contains all independent sets but not all cliques (or vice versa), then the problem of finding an induced subgraph of size at least k, satisfying ᴨ, is W[1]-hard.
  • 87. Finding large induced subgraphs with property ᴨ. Khot and Raman, (2002) If ᴨ is a hereditary property that contains all independent sets and all cliques, or if ᴨ excludes some independent sets and some cliques, then the problem of finding an induced subgraph of size at least k, satisfying ᴨ, is FPT. If ᴨ is a hereditary property contains all independent sets but not all cliques (or vice versa), then the problem of finding an induced subgraph of size at least k, satisfying ᴨ, is W[1]-hard. Ramsey Numbers! Reduction from Independent Set
  • 88. Vertex Cover Can G be made edgeless by the removal of at most k vertices? A simple randomized algorithm with an error probability of 2-k.
  • 89. Vertex Cover Can G be made edgeless by the removal of at most k vertices? A simple randomized algorithm with an error probability of 2-k.
  • 90. Vertex Cover Can G be made edgeless by the removal of at most k vertices? A simple randomized algorithm with an error probability of 2-k.
  • 91. Feedback Vertex Set Can G be made acyclic by the removal of at most k vertices?
  • 92. Feedback Vertex Set Can G be made acyclic by the removal of at most k vertices?
  • 93. Feedback Vertex Set Can G be made acyclic by the removal of at most k vertices?
  • 94. Feedback Vertex Set Can G be made acyclic by the removal of at most k vertices?
  • 95. Feedback Vertex Set Can G be made acyclic by the removal of at most k vertices?
  • 96. Feedback Vertex Set Can G be made acyclic by the removal of at most k vertices? Let’s stare at the structure of a YES-instance. #whisper: Let us also restrict ourselves to graphs of constant maximum degree, say five.
  • 97. the k vertices corresponding to the FVS (n-k) vertices and at most (n-k-1) edges “outside”.
  • 98. the k vertices corresponding to the FVS (n-k) vertices and at most (n-k-1) edges “outside”.
  • 101. t “points of contact” between S and GS…
  • 102. t “points of contact” between S and GS… at least t edges going across the two sets.
  • 103. t “points of contact” between S and GS… at least t edges going across the two sets. Delete the points of contact in GS to get at most 5t “pieces”.
  • 104. t “points of contact” between S and GS… at least t edges going across the two sets. Delete the points of contact in GS to get at most 5t “pieces”. Suppose each piece has constant size, say c.
  • 105. t “points of contact” between S and GS… at least t edges going across the two sets. Delete the points of contact in GS to get at most 5t “pieces”. Suppose each piece has constant size, say c. The total number of edges in GS is at most 5ct.
  • 106. t “points of contact” between S and GS… at least t edges going across the two sets. Delete the points of contact in GS to get at most 5t “pieces”. Suppose each piece has constant size, say c. The total number of edges in GS is at most 5ct. The number of good edges is at least t, and the number of bad edges is at most 5ct.
  • 107. t “points of contact” between S and GS… at least t edges going across the two sets. Delete the points of contact in GS to get at most 5t “pieces”. Suppose each piece has constant size, say c. The total number of edges in GS is at most 5ct. The number of good edges is at least t, and the number of bad edges is at most 5ct. In this scenario, a randomly chosen edge will be good with constant probability!
  • 108. Bad scenario: one of the pieces is large.
  • 109. These pieces have simple structure and bounded interaction with the outer world.
  • 110. These pieces have simple structure and bounded interaction with the outer world. Such pieces are called “protrusions”.
  • 111. A Boundary of Constant Size Constant Treewidth
  • 112. A Boundary of Constant Size Constant Treewidth
  • 113. A Boundary of Constant Size Constant Treewidth
  • 117. The space of t-boundaried graphs can be broken up into equivalence classes based on how they “behave” with the “other side” of the boundary.
  • 120. The value of the optimal solution is the same up to a constant.
  • 121. The space of t-boundaried graphs can be broken up into equivalence classes based on how they “behave” with the “other side” of the boundary.
  • 122. The space of t-boundaried graphs can be broken up into equivalence classes based on how they “behave” with the “other side” of the boundary. For some problems, the number of equivalence classes is finite, allowing us to replace protrusions in graphs.
  • 124. Finding protrusions? What do we replace them with?
  • 126. The 퓕-Deletion Problem Can G be made H-minor free, for all H in퓕, by the removal of at most k vertices?
  • 127. The 퓕-Deletion Problem Can G be made H-minor free, for all H in퓕, by the removal of at most k vertices? Suppose G is a YES-instance, and further, suppose퓕contains at least one planar graph.
  • 128. The 퓕-Deletion Problem Can G be made H-minor free, for all H in퓕, by the removal of at most k vertices? Suppose G is a YES-instance, and further, suppose퓕contains at least one planar graph. Then GS cannot have large grids.
  • 129. The 퓕-Deletion Problem Can G be made H-minor free, for all H in퓕, by the removal of at most k vertices? Suppose G is a YES-instance, and further, suppose퓕contains at least one planar graph. Then GS cannot have large grids. Thus the treewidth of GS is bounded by a constant that depends only on퓕.
  • 130. The 퓕-Deletion Problem Can G be made H-minor free, for all H in퓕, by the removal of at most k vertices? Suppose G is a YES-instance, and further, suppose퓕contains at least one planar graph. Then GS cannot have large grids. Thus the treewidth of GS is bounded by a constant that depends only on퓕. The Planar 퓕-Deletion Problem
  • 132. The Planar 퓕-Deletion Problem Ensure that protrusions are removed.
  • 133. The Planar 퓕-Deletion Problem Ensure that protrusions are removed. Pick an edge uniformly at random.
  • 134. The Planar 퓕-Deletion Problem Ensure that protrusions are removed. Pick an edge uniformly at random. Include both endpoints.
  • 135. The Planar 퓕-Deletion Problem Ensure that protrusions are removed. Pick an edge uniformly at random. Include both endpoints. Pick an endpoint u.a.r.
  • 136. The Planar 퓕-Deletion Problem Ensure that protrusions are removed. Pick an edge uniformly at random. Include both endpoints. Pick an endpoint u.a.r. Repeat till G is퓕-free.
  • 137. The Planar 퓕-Deletion Problem Ensure that protrusions are removed. Pick an edge uniformly at random. Include both endpoints. Pick an endpoint u.a.r. (Approximation algorithm.) Repeat till G is퓕-free.
  • 138. The Planar 퓕-Deletion Problem Ensure that protrusions are removed. Pick an edge uniformly at random. Include both endpoints. Pick an endpoint u.a.r. (Approximation algorithm.) (Randomized algorithm.) Repeat till G is퓕-free.
  • 139. The Planar 퓕-Deletion Problem accounts for several specific problems…
  • 140. The Planar 퓕-Deletion Problem accounts for several specific problems… 퓕 K2 K3 ((k+1)-by-(k+1)) Grids K2,3, K4 K4 K3,T2 Vertex Cover Feedback Vertex Set Treewidth k Outerplanar Series-Parallel Pathwidth-One
  • 142. Thanks to Graph Minors, we have a f(k)n2 algorithm for the 퓕-Deletion problem.
  • 143. Thanks to Graph Minors, we have a f(k)n2 algorithm for the 퓕-Deletion problem. For Planar 퓕-Deletion, the starting point was a double-exponential algorithm. [Bodlaender, 1997]
  • 144. Thanks to Graph Minors, we have a f(k)n2 algorithm for the 퓕-Deletion problem. For Planar 퓕-Deletion, the starting point was a double-exponential algorithm. [Bodlaender, 1997] Many single-exponential algorithms are known for special cases of 퓕.
  • 145. Thanks to Graph Minors, we have a f(k)n2 algorithm for the 퓕-Deletion problem. For Planar 퓕-Deletion, the starting point was a double-exponential algorithm. [Bodlaender, 1997] Many single-exponential algorithms are known for special cases of 퓕. Feedback Vertex Set has Cao, Chen & Liu a O*(3.83k) algorithm. [2010]
  • 146. Thanks to Graph Minors, we have a f(k)n2 algorithm for the 퓕-Deletion problem. For Planar 퓕-Deletion, the starting point was a double-exponential algorithm. [Bodlaender, 1997] Many single-exponential algorithms are known for special cases of 퓕. Cao, Chen & Liu Feedback Vertex Set has a O*(3.83k) algorithm. [2010] {K4}-Deletion has Kim, Paul & Philip a O*(2O(k)) algorithm. [2012]
  • 147. Thanks to Graph Minors, we have a f(k)n2 algorithm for the 퓕-Deletion problem. For Planar 퓕-Deletion, the starting point was a double-exponential algorithm. [Bodlaender, 1997] Many single-exponential algorithms are known for special cases of 퓕. Cao, Chen & Liu Feedback Vertex Set has a O*(3.83k) algorithm. [2010] {K4}-Deletion has Kim, Paul & Philip a O*(2O(k)) algorithm. Pathwidth-1-Deletion has a O*(4.65k) algorithm. [2012] Cygan, Pilipczuk, Pilipczuk & Wojtaszczyk [2010]
  • 149. Fomin, Lokshtanov, M. & Saurabh [2013] Planar퓕-Deletion admits a randomized (2O(k)n) algorithm when every graph in퓕is connected.
  • 150. Fomin, Lokshtanov, M. & Saurabh [2013] Planar퓕-Deletion admits a randomized (2O(k)n) algorithm when every graph in퓕is connected. Best possible under ETH.
  • 151. Fomin, Lokshtanov, M. & Saurabh [2013] Planar퓕-Deletion admits a randomized (2O(k)n) algorithm when every graph in퓕is connected. Best possible under ETH. Planar 퓕-Deletion admits a deterministic (2O(k)n log2n) algorithm when every graph in 퓕 is connected.
  • 152. Fomin, Lokshtanov, M. & Saurabh [2013] Planar퓕-Deletion admits a randomized (2O(k)n) algorithm when every graph in퓕is connected. Best possible under ETH. Planar 퓕-Deletion admits a deterministic (2O(k)n log2n) algorithm when every graph in 퓕 is connected. Planar퓕-Deletion admits an O(nm) randomized algorithm that leads us to a constant-factor approximation. Can we get a deterministic constant-­factor approximation algorithm?
  • 153. Fomin, Lokshtanov, M. & Saurabh [2013] Planar퓕-Deletion admits a randomized (2O(k)n) algorithm when every graph in퓕is connected. Best possible under ETH. Planar 퓕-Deletion admits a deterministic (2O(k)n log2n) algorithm when every graph in 퓕 is connected. Planar퓕-Deletion admits an O(nm) randomized algorithm that leads us to a constant-factor approximation. Can we get a deterministic constant-­factor approximation algorithm? Planar퓕-Deletion admits an deterministic algorithm that leads us to a O(log3/2(OPT)) approximation. Fomin, Lokshtanov, M., Philip & Saurabh [2013]
  • 154. Fomin, Lokshtanov, M. & Saurabh [2013] Planar퓕-Deletion admits a randomized (2O(k)n) algorithm when every graph in퓕is connected. Best possible under ETH. Planar 퓕-Deletion admits a deterministic (2O(k)n log2n) algorithm when every graph in 퓕 is connected. Planar퓕-Deletion admits an O(nm) randomized algorithm that leads us to a constant-factor approximation. Can we get a deterministic constant-­factor approximation algorithm? Planar퓕-Deletion admits an deterministic algorithm that leads us to a O(log3/2(OPT)) approximation. Fomin, Lokshtanov, M., Philip & Saurabh [2013] Kim, Langer, Paul, Reidl, Rossmanith, Sau, & Sikdar [2013] Planar퓕-Deletion admits a deterministic (2O(k)n2) algorithm.
  • 155. Fomin, Lokshtanov, M. & Saurabh [2013] Planar퓕-Deletion admits a randomized (2O(k)n) algorithm when every graph in퓕is connected. Best possible under ETH. Planar 퓕-Deletion admits a deterministic (2O(k)n) algorithm when every graph in 퓕 is connected. Planar퓕-Deletion admits an O(nm) randomized algorithm that leads us to a constant-factor approximation. Can we get a deterministic constant-­factor approximation algorithm? Planar퓕-Deletion admits an deterministic algorithm that leads us to a O(log3/2(OPT)) approximation. Fomin, Lokshtanov, M., Philip & Saurabh [2013] Kim, Langer, Paul, Reidl, Rossmanith, Sau, & Sikdar [2013] Planar퓕-Deletion admits a deterministic (2O(k)n2) algorithm. Fomin, Lokshtanov, M., Ramanujan& Saurabh [2015]
  • 157. Many polynomial kernels are also known for special cases of 퓕.
  • 158. Many polynomial kernels are also known for special cases of 퓕. Feedback Vertex Set has a Thomasse O(k2) instance kernel. [2009]
  • 159. Many polynomial kernels are also known for special cases of 퓕. Thomasse Feedback Vertex Set has a O(k2) instance kernel. [2009] Vertex Cover has (Various) a O(k) vertex kernel.
  • 160. Many polynomial kernels are also known for special cases of 퓕. Thomasse Feedback Vertex Set has a O(k2) instance kernel. [2009] Vertex Cover has (Various) a O(k) vertex kernel. Pathwidth-1-Deletion has a polynomial kernel Cygan, Pilipczuk, Pilipczuk & Wojtaszczyk [2010]
  • 161. Many polynomial kernels are also known for special cases of 퓕. Thomasse Feedback Vertex Set has a O(k2) instance kernel. [2009] Vertex Cover has (Various) a O(k) vertex kernel. Pathwidth-1-Deletion has a polynomial kernel Cygan, Pilipczuk, Pilipczuk & Wojtaszczyk [2010] Fomin, Lokshtanov, M. & Saurabh [2013] Planar 퓕-Deletion admits a polynomial kernel.
  • 162. Many polynomial kernels are also known for special cases of 퓕. Thomasse Feedback Vertex Set has a O(k2) instance kernel. [2009] Vertex Cover has (Various) a O(k) vertex kernel. Pathwidth-1-Deletion has a polynomial kernel Cygan, Pilipczuk, Pilipczuk & Wojtaszczyk [2010] Fomin, Lokshtanov, M. & Saurabh [2013] Planar 퓕-Deletion admits a polynomial kernel. Best possible under standard assumptions.
  • 164. The 퓕-Deletion Problem Very little is known beyond the graph minors result.
  • 165. The 퓕-Deletion Problem Very little is known beyond the graph minors result. The most fundamental 퓕-deletion problem that is not accounted for by Planar 퓕-Deletion is Planarization.
  • 166. The 퓕-Deletion Problem Very little is known beyond the graph minors result. The most fundamental 퓕-deletion problem that is not accounted for by Planar 퓕-Deletion is Planarization. Marx and Schlotter [2012] Planarization admits an algorithm with running time O(2g(k)n2) where g(k) = kO(k3)
  • 167. The 퓕-Deletion Problem Very little is known beyond the graph minors result. The most fundamental 퓕-deletion problem that is not accounted for by Planar 퓕-Deletion is Planarization. Marx and Schlotter [2012] Planarization admits an algorithm with running time O(2g(k)n2) where g(k) = kO(k3) Kawarabayashi [2009] Planarization admits an algorithm with running time O(f(k)n) where f(k) is not explicitly specified.
  • 168. The 퓕-Deletion Problem Very little is known beyond the graph minors result. The most fundamental 퓕-deletion problem that is not accounted for by Planar 퓕-Deletion is Planarization. Marx and Schlotter [2012] Planarization admits an algorithm with running time O(2g(k)n2) where g(k) = kO(k3) Kawarabayashi [2009] Planarization admits an algorithm with running time O(f(k)n) where f(k) is not explicitly specified. Jansen, Lokshtanov & Saurabh [2014] Planarization admits an algorithm with running time 2O(k log k)n, achieving the best combined dependence on k and n.
  • 169. The 퓕-Deletion Problem Very little is known beyond the graph minors result. The most fundamental 퓕-deletion problem that is not accounted for by Planar 퓕-Deletion is Planarization. The question of kernels is completely open. Marx and Schlotter [2012] Planarization admits an algorithm with running time O(2g(k)n2) where g(k) = kO(k3) Kawarabayashi [2009] Planarization admits an algorithm with running time O(f(k)n) where f(k) is not explicitly specified. Jansen, Lokshtanov & Saurabh [2014] Planarization admits an algorithm with running time 2O(k log k)n, achieving the best combined dependence on k and n.
  • 170. Graph Modification Problems When the operation induces hardness
  • 172. Contraction Problems Graph Class or “property”, ᴨ Forbidden Subgraph Characterization A graph admits a forbidden subgraph characterization if, and only if, it is closed under taking induced subgraphs (hereditary), that is, the property is preserved under vertex deletions. If there are finitely many forbidden graphs, then the corresponding graph modification problem is FPT.
  • 173. Contraction Problems Graph Class or “property”, ᴨ Forbidden Subgraph Characterization A graph admits a forbidden subgraph characterization if, and only if, it is closed under taking induced subgraphs (hereditary), that is, the property is preserved under vertex deletions. If there are finitely many forbidden graphs, then the corresponding graph modification problem is FPT. Not true any more!
  • 174. Contraction To C4-free graphs Can G be made C4-free by the contraction of at most k edges? W[2]-hard by a simple reduction from Hitting Set.
  • 175. Contraction To C4-free graphs Can G be made C4-free by the contraction of at most k edges? W[2]-hard by a simple reduction from Hitting Set.
  • 177. Ct-free Contraction isW[2]-hard if t ⩾ 4 and FPT if t⩽ 3. Pt-free Contraction isW[2]-hard if t ⩾ 5 and FPT if t ⩽ 4. Lokshtanov, M. & Saurabh [2013]
  • 178. Ct-free Contraction isW[2]-hard if t ⩾ 4 and FPT if t⩽ 3. Pt-free Contraction isW[2]-hard if t ⩾ 5 and FPT if t ⩽ 4. Lokshtanov, M. & Saurabh [2013] Chordal Contraction isW[2]-hard while Clique contraction is FPT but is unlikely to admit a polynomial kernel. Cai and Guo [2013]
  • 179. Ct-free Contraction isW[2]-hard if t ⩾ 4 and FPT if t⩽ 3. Pt-free Contraction isW[2]-hard if t ⩾ 5 and FPT if t ⩽ 4. Lokshtanov, M. & Saurabh [2013] Chordal Contraction isW[2]-hard while Clique contraction is FPT but is unlikely to admit a polynomial kernel. Characeterizations are open. Cai and Guo [2013]
  • 180. Heggernes, van’t Hof, Lokshtanov & Paul [2011] Ct-free Contraction isW[2]-hard if t ⩾ 4 and FPT if t⩽ 3. Pt-free Contraction isW[2]-hard if t ⩾ 5 and FPT if t ⩽ 4. Chordal Contraction isW[2]-hard while Clique contraction is FPT but is unlikely to admit a polynomial kernel. Contracting to Bipartite graphs is fixed-parameter tractable with a double-exponential running time. Lokshtanov, M. & Saurabh [2013] Characeterizations are open. Cai and Guo [2013]
  • 181. Heggernes, van’t Hof, Lokshtanov & Paul [2011] Ct-free Contraction isW[2]-hard if t ⩾ 4 and FPT if t⩽ 3. Pt-free Contraction isW[2]-hard if t ⩾ 5 and FPT if t ⩽ 4. Chordal Contraction isW[2]-hard while Clique contraction is FPT but is unlikely to admit a polynomial kernel. Characeterizations are open. Contracting to Bipartite graphs is fixed-parameter tractable with a double-exponential running time. Lokshtanov, M. & Saurabh [2013] Polynomial kernels are open. Cai and Guo [2013]
  • 182. Heggernes, van’t Hof, Lokshtanov & Paul [2011] Ct-free Contraction isW[2]-hard if t ⩾ 4 and FPT if t⩽ 3. Pt-free Contraction isW[2]-hard if t ⩾ 5 and FPT if t ⩽ 4. Chordal Contraction isW[2]-hard while Clique contraction is FPT but is unlikely to admit a polynomial kernel. Characeterizations are open. Contracting to Bipartite graphs is fixed-parameter tractable with a double-exponential running time. Lokshtanov, M. & Saurabh [2013] Polynomial kernels are open. Heggernes, van’t Hof, Lévêque, Lokshtanov & Paul [2011] Contracting to paths is FPT and has a linear-vertex kernel; while contracting to trees is FPT but is unlikely to admit a polynomial kernel. Cai and Guo [2013]
  • 185. Yannakakis [1981] Chordal Completion The minimum fill-in problem is NP-complete (was left open in the first edition of Garey and Johnson).
  • 186. Yannakakis [1981] Chordal Completion The minimum fill-in problem is NP-complete (was left open in the first edition of Garey and Johnson). Kaplan, Shamir & Tarjan [1999] Chordal Completion is fixed-parameter tractable (FPT) with a running time of O(k616k + k2mn).
  • 187. Yannakakis [1981] Chordal Completion The minimum fill-in problem is NP-complete (was left open in the first edition of Garey and Johnson). Kaplan, Shamir & Tarjan [1999] Chordal Completion is fixed-parameter tractable (FPT) with a running time of O(k616k + k2mn). Bodlaender, Heggernes, & Villanger [2011] Chordal Completion is fixed-parameter tractable (FPT) with a running time of O(2.36k + k2mn).
  • 188. Yannakakis [1981] Chordal Completion The minimum fill-in problem is NP-complete (was left open in the first edition of Garey and Johnson). Kaplan, Shamir & Tarjan [1999] Chordal Completion is fixed-parameter tractable (FPT) with a running time of O(k616k + k2mn). Bodlaender, Heggernes, & Villanger [2011] Chordal Completion is fixed-parameter tractable (FPT) with a running time of O(2.36k + k2mn). Fomin & Villanger [2013] Chordal Completion admits a sub exponential parameterized algorithm with a running time of O(2√k log k + k2mn).
  • 190. Kaplan, Shamir & Tarjan [1999] Interval Completion Chordal Completion, Strongly Chordal Completion, and Proper Interval Completion are fixed-parameter tractable (FPT).
  • 191. Kaplan, Shamir & Tarjan [1999] Interval Completion Chordal Completion, Strongly Chordal Completion, and Proper Interval Completion are fixed-parameter tractable (FPT). Heggernes, Paul, Telle & Villanger [2007] Interval Completion is fixed-parameter tractable (FPT), with a running time of O(k2kn3m).
  • 192. Kaplan, Shamir & Tarjan [1999] Interval Completion Chordal Completion, Strongly Chordal Completion, and Proper Interval Completion are fixed-parameter tractable (FPT). Heggernes, Paul, Telle & Villanger [2007] Interval Completion is fixed-parameter tractable (FPT), with a running time of O(k2kn3m). Cao [2007] Interval Completion has a single-exponential parameterized algorithm, with a running time of O(6k(n+m)).
  • 193. Kaplan, Shamir & Tarjan [1999] Interval Completion Chordal Completion, Strongly Chordal Completion, and Proper Interval Completion are fixed-parameter tractable (FPT). Heggernes, Paul, Telle & Villanger [2007] Interval Completion is fixed-parameter tractable (FPT), with a running time of O(k2kn3m). Cao [2007] Interval Completion has a single-exponential parameterized algorithm, with a running time of O(6k(n+m)). Bliznets, Fomin, Pilipczuk & Pilipczuk [2014] Interval Completion admits a subexponential parameterized algorithm with a running time of kO(√k)nO(1).
  • 196. Edge Deletions Kernels on Planar Graphs & the Meta-Kernel project Special Graph Classes, eg, Feedback Arc Set on Tournaments Connectivity Augmentation The descriptive complexity of graph modification Structural parameters and other objective functions Backdoor Sets!