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Polynomial Kernel for Interval Vertex Deletion

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AkankshaAgrawal55

The document presents a polynomial kernel for the interval vertex deletion problem, significantly contributing to parameterized complexity. It details the algorithm's overview, which involves computing an approximate solution and leveraging marking schemes and module components to bound sizes effectively. This work resolves a long-standing open problem in the field, with implications for computational efficiency in graph theory.

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Akanksha Agrawal*, Daniel Lokshtanov, Pranabendu Misra, Saket Saurabh, and Meirav Zehavi *Hungarian Academy of Sciences, Budapest New Horizons in Parameterized Complexity Dagstuhl Seminar, 2019 A Polynomial Kernel for Interval Vertex Deletion
Interval Vertex Deletion How far is the graph from being an interval graph? Input Graph G Integer k KERNELIZATION COMPLEXITY Parameter: Solution size (k)
Interval Vertex Deletion How far is the graph from being an interval graph? Graph G Integer k Question Is there a vertex subset S of size at most k, such that G-S is an interval graph? S interval graph KERNELIZATION COMPLEXITY Parameter: Solution size (k)
Interval Vertex Deletion: Known Results Interval Vertex Deletion admits an algorithm running in time O(10kn9). [Cao and Marx] Resolved a long standing open problem.
Interval Vertex Deletion: Known Results Resolved a long standing open problem. The current best algorithm runs in time O(8k(n+m)). [Cao] Interval Vertex Deletion admits an algorithm running in time O(10kn9). [Cao and Marx]
Interval Vertex Deletion: Known Results Interval Vertex Deletion admits an 8-approximation algorithm. [Cao] Resolved a long standing open problem. The current best algorithm runs in time O(8k(n+m)). [Cao] Interval Vertex Deletion admits an algorithm running in time O(10kn9). [Cao and Marx]
Interval Vertex Deletion: Our Result Interval Vertex Deletion admits a polynomial kernel Resolves a well-known open problem in Paramaterized Complexity
I. Overview of the Algorithm Input Graph G Integer k Compute an approximate solution interval graph
I. Overview of the Algorithm Input Graph G Integer k Compute an approximate solution Strengthen the approximate solution to obtain a redundant solution interval graph interval graph redundant solution
II. Overview of the Algorithm Input Graph G Integer k interval graph O(kd) S G-S S G-S classify connected components of G-S Modules Non-modules
III. Overview of the Algorithm G-S Modules Non-modules Bound the number of module components. Bound the size of each module component. Bound the number of non- module components. Bound the size of each non- module component. S
G-S Modules Non-modules Bound the number of module components. Bound the number of non- module components. Bound the size of each non- module component. Bounding them is easier Bound the size of each module component. III. Overview of the Algorithm S
G-S Modules Non-modules Bound the number of module components. Bound the number of non- module components. Bound the size of each module component. Bound the size of each non-module component. III. Overview of the Algorithm S
G-S Modules Non-modules Bound the number of module components. Bound the number of non- module components. Bound the size of each module component. Bound the size of each non-module component. III. Overview of the Algorithm Using Bounded Intersection Two Families Lemma (a new variant of a classic theorem of Bollobás) and some marking schemes exploiting the structure S
G-S Modules Non-modules Bound the number of module components. Bound the number of non- module components. Bound the size of each module component. Bound the size of each non-module component. III. Overview of the Algorithm Clique path decomposition S
G-S Modules Non-modules Bound the number of module components. Bound the number of non- module components. Bound the size of each module component. Bound the size of each non-module component. III. Overview of the Algorithm Clique path decomposition 1) Marking schemes to bound bag sizes in the path decomposition. S
G-S Modules Non-modules Bound the number of module components. Bound the number of non- module components. Bound the size of each module component. Bound the size of each non-module component. III. Overview of the Algorithm Clique path decomposition 1) Marking schemes to bound bag sizes in the path decomposition. 2) Many marking schemes (and Bounded Intersection Two Families Lemma) to bound the number of bags. S
Characterization of Interval Graphs (Forbidden induced subgraphs) c t b1 b2 t` trb3 c t b1 b2 t` tr b3 c t b1 bzb2 bz 1t` trb3 c2 t c1 b1 bzb2 bz 1t` trb3 C>=4 z>=2 z>=1 Interval graph if and only if none of the following are an induced subgraph.
Characterization of Interval Graphs (Forbidden induced subgraphs) c t b1 b2 t` trb3 c t b1 b2 t` tr b3 c t b1 bzb2 bz 1t` trb3 c2 t c1 b1 bzb2 bz 1t` trb3 C>=4 z>=2 z>=1 Interval graph if and only if none of the following are an induced subgraph. Obstructions!
Redundant Solution -One of the key ingredients in our algorithm
Approximate Solution Approximate solutions have been exploited to design kernels. A G-A obstruction Any obstruction contains at least one vertex from the approximate solution
Approximate Solution Approximate solutions have been exploited to design kernels. A G-A obstruction Any obstruction contains at least one vertex from the approximate solution Question: Can we strengthen the approximate solution so that any obstruction intersects it in more than c vertices? [Reasonable Size]
Approximate Solution Approximate solutions have been exploited to design kernels. A G-A obstruction Question: Can we strengthen the approximate solution so that any obstruction intersects it in more than c vertices? Any obstruction contains at least one vertex from the approximate solution Structurally Useful! [Reasonable Size]
Redundant Solution Question: Can we strengthen the approximate solution so that any obstruction intersects it in more than c vertices? Is it possible? [Reasonable Size]
Redundant Solution Question: Can we strengthen the approximate solution so that any obstruction intersects it in more than c vertices? Is it possible? Can never satisfy the condition [Reasonable Size] Obstructions with at most c vertices? A small induced cycle
Redundant Solution Question: Can we strengthen the approximate solution so that any obstruction intersects it in more than c vertices? Is it possible? Many obstructions intersect in few vertices? We need to add a lot of vertices [Reasonable Size] Many obstructions intersecting in two vertices
c-Redundant Solution W: A set of subsets of S Any solution of ``small size’’ must be a hitting set for W S G-S Question: Can we strengthen the approximate solution so that any obstruction intersects it in more than c vertices? [Reasonable Size]
c-Redundant Solution Obstruction O Any solution of ``small size’’ must be a hitting set for W S G-S W: A set of subsets of S Question: Can we strengthen the approximate solution so that any obstruction intersects it in more than c vertices? [Reasonable Size]
c-Redundant Solution Obstruction O Any solution of ``small size’’ must be a hitting set for W O contains more than c vertices from S. S G-S W: A set of subsets of S Question: Can we strengthen the approximate solution so that any obstruction intersects it in more than c vertices? [Reasonable Size]
c-Redundant Solution Obstruction O Any solution of ``small size’’ must be a hitting set for W O is taken care by the hitting set property. S G-S W: A set of subsets of S O contains more than c vertices from S. Question: Can we strengthen the approximate solution so that any obstruction intersects it in more than c vertices? [Reasonable Size]
c-Redundant Solution Obstruction O Any solution of ``small size’’ must be a hitting set for W O is taken care by the hitting set property. S G-S W: A set of subsets of S O contains more than c vertices from S. Question: Can we strengthen the approximate solution so that any obstruction intersects it in more than c vertices? [Reasonable Size] It contains one of the pink sets
c-Redundant Solution S G-S Obstruction O Any solution of ``small size’’ must be a hitting set for W Key Ingredient 1: A c-Redundant solution can be efficiently computed. W: A set of subsets of S O contains more than c vertices from S. O is taken care by the hitting set property.
Kernel with the Vertex Cover Number as Parameter
Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number) X G-X vertex cover independent set
X G-X vertex cover independent set obstruction approximate solution Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number)
Compute a c-Redundant Solution (c is a large enough constant) S X G-S independent set vertex cover Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number) k is the vertex cover number poly(k)
Bounded? Compute a c-Redundant Solution (c is a large enough constant) S G-S k is the vertex cover number Done poly(k) independent set vertex cover Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number)
Unbounded Compute a c-Redundant Solution (c is a large enough constant) S poly(k) G-S k is the vertex cover number Goal: To find an irrelevant vertex independent set vertex cover Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number)
S poly(k) G-S Consider a pair (u,v) of non-adjacent vertices in S such that {u,v} is not a set in W. u v independent set vertex cover obstruction O contains more than c vertices from S. O is taken care by the hitting set property. OR {u,v} is a set in W Useful Observations Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number)
S poly(k) G-S u v independent set vertex cover obstruction O contains more than c vertices from S. O is taken care by the hitting set property. OR {u,v} is a set in W Useful Observations Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number) Consider a pair (u,v) of non-adjacent vertices in S such that {u,v} is not a set in W.
S poly(k) G-S u v vertex cover obstruction u,v can have at most one common neighbor in G-S independent set Useful Observations Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number) Consider a pair (u,v) of non-adjacent vertices in S such that {u,v} is not a set in W.
S poly(k) G-S u v vertex cover obstruction u,v can have at most one common neighbor in G-S independent set Mark them! Useful Observations Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number) Consider a pair (u,v) of non-adjacent vertices in S such that {u,v} is not a set in W.
S poly(k) G-S vertex cover independent set poly(k) Useful Observations Consider an unmarked vertex in G-S and any two of its neighbors in S. u v Claim: (u,v) is an edge or a set in W. Why is it not marked? Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number) w
S poly(k) G-S vertex cover independent set poly(k) Useful Observations Consider an unmarked vertex in G-S and any two of its neighbors in S. u v Neighborhood is ``almost’’ a clique Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number)
S poly(k) G-S vertex cover independent set poly(k) Obstructions are ``well behaved’’ Consider an obstruction O containing an unmarked, which is ``not covered’’ by W. Obstruction O w O contains more than c vertices from S. O is taken care by the hitting set property. OR w Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number)
Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number) S poly(k) G-S vertex cover independent set poly(k) Obstructions are ``well behaved’’ Consider an obstruction O containing an unmarked, which is ``not covered’’ by W. Obstruction O w O contains more than c vertices from S. O is taken care by the hitting set property. OR w
Characterization of Interval Graphs (Forbidden induced subgraphs) c t b1 b2 t` trb3 c t b1 b2 t` tr b3 c t b1 bzb2 bz 1t` trb3 c2 t c1 b1 bzb2 bz 1t` trb3 C>=4 z>=2 z>=1 As O contains more than c vertices (c is a large constant), we have: O is one of these
Characterization of Interval Graphs (Forbidden induced subgraphs) c t b1 bzb2 bz 1t` trb3 c2 t c1 b1 bzb2 bz 1t` trb3 C>=4 z>=2 z>=1 As O contains more than c vertices (c is a large constant), we have: O is one of these
Characterization of Interval Graphs (Forbidden induced subgraphs) c t b1 bzb2 bz 1t` trb3 c2 t c1 b1 bzb2 bz 1t` trb3 C>=4 z>=2 z>=1 As O contains more than c vertices (c is a large constant), we have: Non-shallow terminals
Characterization of Interval Graphs (Forbidden induced subgraphs) c t b1 bzb2 bz 1t` trb3 c2 t c1 b1 bzb2 bz 1t` trb3 C>=4 z>=2 z>=1 As O contains more than c vertices (c is a large constant), we have: Shallow terminals
Characterization of Interval Graphs (Forbidden induced subgraphs) c t b1 bzb2 bz 1t` trb3 c2 t c1 b1 bzb2 bz 1t` trb3 C>=4 z>=2 z>=1 As O contains more than c vertices (c is a large constant), we have: Suppose that O is an induced cycle containing w w u v u and v must be in S
Consider a pair of non-adjacent vertices in S such that {u,v} is not a set in W. S poly(k) G-S u v vertex cover u,v can have at most one common neighbor in G-S independent set Mark them! Useful Observations Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number)
Characterization of Interval Graphs (Forbidden induced subgraphs) c t b1 bzb2 bz 1t` trb3 c2 t c1 b1 bzb2 bz 1t` trb3 C>=4 z>=2 z>=1 As O contains more than c vertices (c is a large constant), we have: Suppose that O is an induced cycle containing w w u v Why is it not marked? u and v must be in S
S poly(k) G-S vertex cover independent set poly(k) Obstructions are ``well behaved’’ Consider an obstruction O containing an unmarked, which is ``not covered’’ by W. Obstruction O w w is a terminal! Can be obtained using similar arguments Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number)
Polynomial Kernel for Interval Vertex Deletion (parameter: the vertex cover number) Obstructions are ``well behaved’’ w is a shallow terminal! Some Marking Schemes
Characterization of Interval Graphs (Forbidden induced subgraphs) c t b1 bzb2 bz 1t` trb3 c2 t c1 b1 bzb2 bz 1t` trb3 z>=2 z>=1 As O contains more than c vertices (c is a large constant), we have: Shallow terminals
Characterization of Interval Graphs (Forbidden induced subgraphs) c t b1 bzb2 bz 1t` trb3 c2 t c1 b1 bzb2 bz 1t` trb3 z>=2 z>=1 We will devise some marking schemes for ``induced paths’’ Shallow terminals
Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number) A long path from an obstruction not take care by the ``hitting set’’ property w (unmarked in G-S) W has at most 2 neighbors in P and its neighbors must be adjacent Pu v
Consider a pair of non-adjacent vertices in S such that {u,v} is not a set in W. S poly(k) G-S u v vertex cover obstruction u,v can have at most one common neighbor in G-S independent set Useful Observations Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number) Mark them!
Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number) A long path from an obstruction not take care by the ``hitting set’’ property w (unmarked in G-S) W has at most 2 neighbors in P and its neighbors must be adjacent Pu v
Consider u in S (we will do the steps described for every such u) S G-S u Creating Vectors Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number) Au
S G-S Creating Vectors Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number) Au Consider u in S (we will do the steps described for every such u) w x1 x2 x3 x|S| e1 e2 e3 eq 1 |S| q=|E[G[S]]| vecw
S G-S Creating Vectors Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number) Au Consider u in S (we will do the steps described for every such u) w |S| q=|E[G[S]]| x1 vecw (adjacent to x1)1 x1 x2 x3 x|S| e1 e2 e3 eq 1
S G-S Creating Vectors Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number) Au Consider u in S (we will do the steps described for every such u) w |S| q=|E[G[S]]| e1 vecw (adjacent to both endpoints) x1 x2 x3 x|S| 1 e1 e2 e3 eq 1
S G-S Creating Vectors Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number) Au Consider u in S (we will do the steps described for every such u) w |S| q=|E[G[S]]| vecw 0 at every other place x1 x2 x3 x|S| e1 e2 e3 eq 1
S G-S Creating a Set of Important Vertices Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number) Au Consider u in S (we will do the steps described for every such u) Compute basic of the vectors created (do it k+1 times) and mark the corresponding vertices Bu
S G-S Creating a Set of Important Vertices Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number) Au Consider u in S (we will do the steps described for every such u) Compute basic of the vectors created (do it k+1 times) and mark the corresponding vertices Bu Delete unmarked vertices!
Correctness Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number) Say, we deleted an unmarked vertex w from G to obtain G’
Correctness Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number) Say, we deleted an unmarked vertex w from G to obtain G’ If G-Y is an interval graph => G’-Y is an interval graph
Correctness Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number) Say, we deleted an unmarked vertex w from G to obtain G’ Suppose there is a Y of size at most k, s.t. G’-Y is an interval graph
Correctness Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number) Say, we deleted an unmarked vertex w from G to obtain G’ c t b1 bzb2 bz 1t` trb3 w u Suppose there is a Y of size at most k, s.t. G’-Y is an interval graph Obstruction in G-Y (containing w) x1 x2 x3 x| e1 e2 e3 eq 1 P
Correctness Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number) Say, we deleted an unmarked vertex w from G to obtain G’ c t b1 bzb2 bz 1t` trb3 w u Suppose there is a Y of size at most k, s.t. G’-Y is an interval graph Obstruction in G-Y (containing w) x1 x2 x3 x| e1 e2 e3 eq 1 P sum(vecw) = 1 [Restricted to P and mod 2 computation] d1 vecw1 + d2 vecw2 +…+ dr vecwr = vecw
wj Correctness Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number) Say, we deleted an unmarked vertex w from G to obtain G’ c t b1 bzb2 bz 1t` trb3 w u Suppose there is a Y of size at most k, s.t. G’-Y is an interval graph Obstruction in G-Y (containing w) x1 x2 x3 x| e1 e2 e3 eq 1 P sum(vecw) = 1 [Restricted to P and mod 2 computation] d1 vecw1 + d2 vecw2 +…+ dr vecwr = vecw There is wj with sum(vecwj) = 1
wj Correctness Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number) Say, we deleted an unmarked vertex w from G to obtain G’ c t b1 bzb2 bz 1t` trb3 w u Suppose there is a Y of size at most k, s.t. G’-Y is an interval graph Obstruction in G-Y (containing w) x1 x2 x3 x| e1 e2 e3 eq 1 P sum(vecw) = 1 [Restricted to P and mod 2 computation] d1 vecw1 + d2 vecw2 +…+ dr vecwr = vecw There is wj with sum(vecwj) = 1 Replacement for w!
wj Correctness Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number) Say, we deleted an unmarked vertex w from G to obtain G’ c t b1 bzb2 bz 1t` trb3 w u Suppose there is a Y of size at most k, s.t. G’-Y is an interval graph Obstruction in G-Y (containing w) x1 x2 x3 x| e1 e2 e3 eq 1 P sum(vecw) = 1 [Restricted to P and mod 2 computation] d1 vecw1 + d2 vecw2 +…+ dr vecwr = vecw There is wj with sum(vecwj) = 1 Replacement for w!
wj Correctness Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number) Say, we deleted an unmarked vertex w from G to obtain G’ c t b1 bzb2 bz 1t` trb3 w u Suppose there is a Y of size at most k, s.t. G’-Y is an interval graph Obstruction in G-Y (containing w) x1 x2 x3 x| e1 e2 e3 eq 1 P sum(vecw) = 1 [Restricted to P and mod 2 computation] d1 vecw1 + d2 vecw2 +…+ dr vecwr = vecw There is wj with sum(vecwj) = 1 Replacement for w!
wj Correctness Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number) Say, we deleted an unmarked vertex w from G to obtain G’ c t b1 bzb2 bz 1t` trb3 w u Suppose there is a Y of size at most k, s.t. G’-Y is an interval graph Obstruction in G-Y (containing w) x1 x2 x3 x| e1 e2 e3 eq 1 P sum(vecw) = 1 [Restricted to P and mod 2 computation] d1 vecw1 + d2 vecw2 +…+ dr vecwr = vecw There is wj with sum(vecwj) = 1 Replacement for w!
Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number) Interval Vertex Deletion admits a polynomial kernel, when parameterized by the vertex cover number S G-S
Conclusions and Open Problems Believe: Can be improved at the cost of significantly more involved arguments. The ideas of redundant solution and the linear algebra trick maybe have further algorithmic applications. Can we obtain a kernel with significantly improved size? Interval Vertex Deletion admits a polynomial kernel (parameter: solution size).
Thanks!

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Polynomial Kernel for Interval Vertex Deletion

  • 1. Akanksha Agrawal*, Daniel Lokshtanov, Pranabendu Misra, Saket Saurabh, and Meirav Zehavi *Hungarian Academy of Sciences, Budapest New Horizons in Parameterized Complexity Dagstuhl Seminar, 2019 A Polynomial Kernel for Interval Vertex Deletion
  • 2. Interval Vertex Deletion How far is the graph from being an interval graph? Input Graph G Integer k KERNELIZATION COMPLEXITY Parameter: Solution size (k)
  • 3. Interval Vertex Deletion How far is the graph from being an interval graph? Graph G Integer k Question Is there a vertex subset S of size at most k, such that G-S is an interval graph? S interval graph KERNELIZATION COMPLEXITY Parameter: Solution size (k)
  • 4. Interval Vertex Deletion: Known Results Interval Vertex Deletion admits an algorithm running in time O(10kn9). [Cao and Marx] Resolved a long standing open problem.
  • 5. Interval Vertex Deletion: Known Results Resolved a long standing open problem. The current best algorithm runs in time O(8k(n+m)). [Cao] Interval Vertex Deletion admits an algorithm running in time O(10kn9). [Cao and Marx]
  • 6. Interval Vertex Deletion: Known Results Interval Vertex Deletion admits an 8-approximation algorithm. [Cao] Resolved a long standing open problem. The current best algorithm runs in time O(8k(n+m)). [Cao] Interval Vertex Deletion admits an algorithm running in time O(10kn9). [Cao and Marx]
  • 7. Interval Vertex Deletion: Our Result Interval Vertex Deletion admits a polynomial kernel Resolves a well-known open problem in Paramaterized Complexity
  • 8. I. Overview of the Algorithm Input Graph G Integer k Compute an approximate solution interval graph
  • 9. I. Overview of the Algorithm Input Graph G Integer k Compute an approximate solution Strengthen the approximate solution to obtain a redundant solution interval graph interval graph redundant solution
  • 10. II. Overview of the Algorithm Input Graph G Integer k interval graph O(kd) S G-S S G-S classify connected components of G-S Modules Non-modules
  • 11. III. Overview of the Algorithm G-S Modules Non-modules Bound the number of module components. Bound the size of each module component. Bound the number of non- module components. Bound the size of each non- module component. S
  • 12. G-S Modules Non-modules Bound the number of module components. Bound the number of non- module components. Bound the size of each non- module component. Bounding them is easier Bound the size of each module component. III. Overview of the Algorithm S
  • 13. G-S Modules Non-modules Bound the number of module components. Bound the number of non- module components. Bound the size of each module component. Bound the size of each non-module component. III. Overview of the Algorithm S
  • 14. G-S Modules Non-modules Bound the number of module components. Bound the number of non- module components. Bound the size of each module component. Bound the size of each non-module component. III. Overview of the Algorithm Using Bounded Intersection Two Families Lemma (a new variant of a classic theorem of Bollobás) and some marking schemes exploiting the structure S
  • 15. G-S Modules Non-modules Bound the number of module components. Bound the number of non- module components. Bound the size of each module component. Bound the size of each non-module component. III. Overview of the Algorithm Clique path decomposition S
  • 16. G-S Modules Non-modules Bound the number of module components. Bound the number of non- module components. Bound the size of each module component. Bound the size of each non-module component. III. Overview of the Algorithm Clique path decomposition 1) Marking schemes to bound bag sizes in the path decomposition. S
  • 17. G-S Modules Non-modules Bound the number of module components. Bound the number of non- module components. Bound the size of each module component. Bound the size of each non-module component. III. Overview of the Algorithm Clique path decomposition 1) Marking schemes to bound bag sizes in the path decomposition. 2) Many marking schemes (and Bounded Intersection Two Families Lemma) to bound the number of bags. S
  • 18. Characterization of Interval Graphs (Forbidden induced subgraphs) c t b1 b2 t` trb3 c t b1 b2 t` tr b3 c t b1 bzb2 bz 1t` trb3 c2 t c1 b1 bzb2 bz 1t` trb3 C>=4 z>=2 z>=1 Interval graph if and only if none of the following are an induced subgraph.
  • 19. Characterization of Interval Graphs (Forbidden induced subgraphs) c t b1 b2 t` trb3 c t b1 b2 t` tr b3 c t b1 bzb2 bz 1t` trb3 c2 t c1 b1 bzb2 bz 1t` trb3 C>=4 z>=2 z>=1 Interval graph if and only if none of the following are an induced subgraph. Obstructions!
  • 20. Redundant Solution -One of the key ingredients in our algorithm
  • 21. Approximate Solution Approximate solutions have been exploited to design kernels. A G-A obstruction Any obstruction contains at least one vertex from the approximate solution
  • 22. Approximate Solution Approximate solutions have been exploited to design kernels. A G-A obstruction Any obstruction contains at least one vertex from the approximate solution Question: Can we strengthen the approximate solution so that any obstruction intersects it in more than c vertices? [Reasonable Size]
  • 23. Approximate Solution Approximate solutions have been exploited to design kernels. A G-A obstruction Question: Can we strengthen the approximate solution so that any obstruction intersects it in more than c vertices? Any obstruction contains at least one vertex from the approximate solution Structurally Useful! [Reasonable Size]
  • 24. Redundant Solution Question: Can we strengthen the approximate solution so that any obstruction intersects it in more than c vertices? Is it possible? [Reasonable Size]
  • 25. Redundant Solution Question: Can we strengthen the approximate solution so that any obstruction intersects it in more than c vertices? Is it possible? Can never satisfy the condition [Reasonable Size] Obstructions with at most c vertices? A small induced cycle
  • 26. Redundant Solution Question: Can we strengthen the approximate solution so that any obstruction intersects it in more than c vertices? Is it possible? Many obstructions intersect in few vertices? We need to add a lot of vertices [Reasonable Size] Many obstructions intersecting in two vertices
  • 27. c-Redundant Solution W: A set of subsets of S Any solution of ``small size’’ must be a hitting set for W S G-S Question: Can we strengthen the approximate solution so that any obstruction intersects it in more than c vertices? [Reasonable Size]
  • 28. c-Redundant Solution Obstruction O Any solution of ``small size’’ must be a hitting set for W S G-S W: A set of subsets of S Question: Can we strengthen the approximate solution so that any obstruction intersects it in more than c vertices? [Reasonable Size]
  • 29. c-Redundant Solution Obstruction O Any solution of ``small size’’ must be a hitting set for W O contains more than c vertices from S. S G-S W: A set of subsets of S Question: Can we strengthen the approximate solution so that any obstruction intersects it in more than c vertices? [Reasonable Size]
  • 30. c-Redundant Solution Obstruction O Any solution of ``small size’’ must be a hitting set for W O is taken care by the hitting set property. S G-S W: A set of subsets of S O contains more than c vertices from S. Question: Can we strengthen the approximate solution so that any obstruction intersects it in more than c vertices? [Reasonable Size]
  • 31. c-Redundant Solution Obstruction O Any solution of ``small size’’ must be a hitting set for W O is taken care by the hitting set property. S G-S W: A set of subsets of S O contains more than c vertices from S. Question: Can we strengthen the approximate solution so that any obstruction intersects it in more than c vertices? [Reasonable Size] It contains one of the pink sets
  • 32. c-Redundant Solution S G-S Obstruction O Any solution of ``small size’’ must be a hitting set for W Key Ingredient 1: A c-Redundant solution can be efficiently computed. W: A set of subsets of S O contains more than c vertices from S. O is taken care by the hitting set property.
  • 33. Kernel with the Vertex Cover Number as Parameter
  • 34. Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number) X G-X vertex cover independent set
  • 35. X G-X vertex cover independent set obstruction approximate solution Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number)
  • 36. Compute a c-Redundant Solution (c is a large enough constant) S X G-S independent set vertex cover Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number) k is the vertex cover number poly(k)
  • 37. Bounded? Compute a c-Redundant Solution (c is a large enough constant) S G-S k is the vertex cover number Done poly(k) independent set vertex cover Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number)
  • 38. Unbounded Compute a c-Redundant Solution (c is a large enough constant) S poly(k) G-S k is the vertex cover number Goal: To find an irrelevant vertex independent set vertex cover Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number)
  • 39. S poly(k) G-S Consider a pair (u,v) of non-adjacent vertices in S such that {u,v} is not a set in W. u v independent set vertex cover obstruction O contains more than c vertices from S. O is taken care by the hitting set property. OR {u,v} is a set in W Useful Observations Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number)
  • 40. S poly(k) G-S u v independent set vertex cover obstruction O contains more than c vertices from S. O is taken care by the hitting set property. OR {u,v} is a set in W Useful Observations Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number) Consider a pair (u,v) of non-adjacent vertices in S such that {u,v} is not a set in W.
  • 41. S poly(k) G-S u v vertex cover obstruction u,v can have at most one common neighbor in G-S independent set Useful Observations Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number) Consider a pair (u,v) of non-adjacent vertices in S such that {u,v} is not a set in W.
  • 42. S poly(k) G-S u v vertex cover obstruction u,v can have at most one common neighbor in G-S independent set Mark them! Useful Observations Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number) Consider a pair (u,v) of non-adjacent vertices in S such that {u,v} is not a set in W.
  • 43. S poly(k) G-S vertex cover independent set poly(k) Useful Observations Consider an unmarked vertex in G-S and any two of its neighbors in S. u v Claim: (u,v) is an edge or a set in W. Why is it not marked? Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number) w
  • 44. S poly(k) G-S vertex cover independent set poly(k) Useful Observations Consider an unmarked vertex in G-S and any two of its neighbors in S. u v Neighborhood is ``almost’’ a clique Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number)
  • 45. S poly(k) G-S vertex cover independent set poly(k) Obstructions are ``well behaved’’ Consider an obstruction O containing an unmarked, which is ``not covered’’ by W. Obstruction O w O contains more than c vertices from S. O is taken care by the hitting set property. OR w Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number)
  • 46. Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number) S poly(k) G-S vertex cover independent set poly(k) Obstructions are ``well behaved’’ Consider an obstruction O containing an unmarked, which is ``not covered’’ by W. Obstruction O w O contains more than c vertices from S. O is taken care by the hitting set property. OR w
  • 47. Characterization of Interval Graphs (Forbidden induced subgraphs) c t b1 b2 t` trb3 c t b1 b2 t` tr b3 c t b1 bzb2 bz 1t` trb3 c2 t c1 b1 bzb2 bz 1t` trb3 C>=4 z>=2 z>=1 As O contains more than c vertices (c is a large constant), we have: O is one of these
  • 48. Characterization of Interval Graphs (Forbidden induced subgraphs) c t b1 bzb2 bz 1t` trb3 c2 t c1 b1 bzb2 bz 1t` trb3 C>=4 z>=2 z>=1 As O contains more than c vertices (c is a large constant), we have: O is one of these
  • 49. Characterization of Interval Graphs (Forbidden induced subgraphs) c t b1 bzb2 bz 1t` trb3 c2 t c1 b1 bzb2 bz 1t` trb3 C>=4 z>=2 z>=1 As O contains more than c vertices (c is a large constant), we have: Non-shallow terminals
  • 50. Characterization of Interval Graphs (Forbidden induced subgraphs) c t b1 bzb2 bz 1t` trb3 c2 t c1 b1 bzb2 bz 1t` trb3 C>=4 z>=2 z>=1 As O contains more than c vertices (c is a large constant), we have: Shallow terminals
  • 51. Characterization of Interval Graphs (Forbidden induced subgraphs) c t b1 bzb2 bz 1t` trb3 c2 t c1 b1 bzb2 bz 1t` trb3 C>=4 z>=2 z>=1 As O contains more than c vertices (c is a large constant), we have: Suppose that O is an induced cycle containing w w u v u and v must be in S
  • 52. Consider a pair of non-adjacent vertices in S such that {u,v} is not a set in W. S poly(k) G-S u v vertex cover u,v can have at most one common neighbor in G-S independent set Mark them! Useful Observations Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number)
  • 53. Characterization of Interval Graphs (Forbidden induced subgraphs) c t b1 bzb2 bz 1t` trb3 c2 t c1 b1 bzb2 bz 1t` trb3 C>=4 z>=2 z>=1 As O contains more than c vertices (c is a large constant), we have: Suppose that O is an induced cycle containing w w u v Why is it not marked? u and v must be in S
  • 54. S poly(k) G-S vertex cover independent set poly(k) Obstructions are ``well behaved’’ Consider an obstruction O containing an unmarked, which is ``not covered’’ by W. Obstruction O w w is a terminal! Can be obtained using similar arguments Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number)
  • 55. Polynomial Kernel for Interval Vertex Deletion (parameter: the vertex cover number) Obstructions are ``well behaved’’ w is a shallow terminal! Some Marking Schemes
  • 56. Characterization of Interval Graphs (Forbidden induced subgraphs) c t b1 bzb2 bz 1t` trb3 c2 t c1 b1 bzb2 bz 1t` trb3 z>=2 z>=1 As O contains more than c vertices (c is a large constant), we have: Shallow terminals
  • 57. Characterization of Interval Graphs (Forbidden induced subgraphs) c t b1 bzb2 bz 1t` trb3 c2 t c1 b1 bzb2 bz 1t` trb3 z>=2 z>=1 We will devise some marking schemes for ``induced paths’’ Shallow terminals
  • 58. Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number) A long path from an obstruction not take care by the ``hitting set’’ property w (unmarked in G-S) W has at most 2 neighbors in P and its neighbors must be adjacent Pu v
  • 59. Consider a pair of non-adjacent vertices in S such that {u,v} is not a set in W. S poly(k) G-S u v vertex cover obstruction u,v can have at most one common neighbor in G-S independent set Useful Observations Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number) Mark them!
  • 60. Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number) A long path from an obstruction not take care by the ``hitting set’’ property w (unmarked in G-S) W has at most 2 neighbors in P and its neighbors must be adjacent Pu v
  • 61. Consider u in S (we will do the steps described for every such u) S G-S u Creating Vectors Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number) Au
  • 62. S G-S Creating Vectors Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number) Au Consider u in S (we will do the steps described for every such u) w x1 x2 x3 x|S| e1 e2 e3 eq 1 |S| q=|E[G[S]]| vecw
  • 63. S G-S Creating Vectors Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number) Au Consider u in S (we will do the steps described for every such u) w |S| q=|E[G[S]]| x1 vecw (adjacent to x1)1 x1 x2 x3 x|S| e1 e2 e3 eq 1
  • 64. S G-S Creating Vectors Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number) Au Consider u in S (we will do the steps described for every such u) w |S| q=|E[G[S]]| e1 vecw (adjacent to both endpoints) x1 x2 x3 x|S| 1 e1 e2 e3 eq 1
  • 65. S G-S Creating Vectors Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number) Au Consider u in S (we will do the steps described for every such u) w |S| q=|E[G[S]]| vecw 0 at every other place x1 x2 x3 x|S| e1 e2 e3 eq 1
  • 66. S G-S Creating a Set of Important Vertices Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number) Au Consider u in S (we will do the steps described for every such u) Compute basic of the vectors created (do it k+1 times) and mark the corresponding vertices Bu
  • 67. S G-S Creating a Set of Important Vertices Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number) Au Consider u in S (we will do the steps described for every such u) Compute basic of the vectors created (do it k+1 times) and mark the corresponding vertices Bu Delete unmarked vertices!
  • 68. Correctness Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number) Say, we deleted an unmarked vertex w from G to obtain G’
  • 69. Correctness Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number) Say, we deleted an unmarked vertex w from G to obtain G’ If G-Y is an interval graph => G’-Y is an interval graph
  • 70. Correctness Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number) Say, we deleted an unmarked vertex w from G to obtain G’ Suppose there is a Y of size at most k, s.t. G’-Y is an interval graph
  • 71. Correctness Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number) Say, we deleted an unmarked vertex w from G to obtain G’ c t b1 bzb2 bz 1t` trb3 w u Suppose there is a Y of size at most k, s.t. G’-Y is an interval graph Obstruction in G-Y (containing w) x1 x2 x3 x| e1 e2 e3 eq 1 P
  • 72. Correctness Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number) Say, we deleted an unmarked vertex w from G to obtain G’ c t b1 bzb2 bz 1t` trb3 w u Suppose there is a Y of size at most k, s.t. G’-Y is an interval graph Obstruction in G-Y (containing w) x1 x2 x3 x| e1 e2 e3 eq 1 P sum(vecw) = 1 [Restricted to P and mod 2 computation] d1 vecw1 + d2 vecw2 +…+ dr vecwr = vecw
  • 73. wj Correctness Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number) Say, we deleted an unmarked vertex w from G to obtain G’ c t b1 bzb2 bz 1t` trb3 w u Suppose there is a Y of size at most k, s.t. G’-Y is an interval graph Obstruction in G-Y (containing w) x1 x2 x3 x| e1 e2 e3 eq 1 P sum(vecw) = 1 [Restricted to P and mod 2 computation] d1 vecw1 + d2 vecw2 +…+ dr vecwr = vecw There is wj with sum(vecwj) = 1
  • 74. wj Correctness Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number) Say, we deleted an unmarked vertex w from G to obtain G’ c t b1 bzb2 bz 1t` trb3 w u Suppose there is a Y of size at most k, s.t. G’-Y is an interval graph Obstruction in G-Y (containing w) x1 x2 x3 x| e1 e2 e3 eq 1 P sum(vecw) = 1 [Restricted to P and mod 2 computation] d1 vecw1 + d2 vecw2 +…+ dr vecwr = vecw There is wj with sum(vecwj) = 1 Replacement for w!
  • 75. wj Correctness Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number) Say, we deleted an unmarked vertex w from G to obtain G’ c t b1 bzb2 bz 1t` trb3 w u Suppose there is a Y of size at most k, s.t. G’-Y is an interval graph Obstruction in G-Y (containing w) x1 x2 x3 x| e1 e2 e3 eq 1 P sum(vecw) = 1 [Restricted to P and mod 2 computation] d1 vecw1 + d2 vecw2 +…+ dr vecwr = vecw There is wj with sum(vecwj) = 1 Replacement for w!
  • 76. wj Correctness Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number) Say, we deleted an unmarked vertex w from G to obtain G’ c t b1 bzb2 bz 1t` trb3 w u Suppose there is a Y of size at most k, s.t. G’-Y is an interval graph Obstruction in G-Y (containing w) x1 x2 x3 x| e1 e2 e3 eq 1 P sum(vecw) = 1 [Restricted to P and mod 2 computation] d1 vecw1 + d2 vecw2 +…+ dr vecwr = vecw There is wj with sum(vecwj) = 1 Replacement for w!
  • 77. wj Correctness Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number) Say, we deleted an unmarked vertex w from G to obtain G’ c t b1 bzb2 bz 1t` trb3 w u Suppose there is a Y of size at most k, s.t. G’-Y is an interval graph Obstruction in G-Y (containing w) x1 x2 x3 x| e1 e2 e3 eq 1 P sum(vecw) = 1 [Restricted to P and mod 2 computation] d1 vecw1 + d2 vecw2 +…+ dr vecwr = vecw There is wj with sum(vecwj) = 1 Replacement for w!
  • 78. Polynomial Kernel for Interval Vertex Deletion (parameter: vertex cover number) Interval Vertex Deletion admits a polynomial kernel, when parameterized by the vertex cover number S G-S
  • 79. Conclusions and Open Problems Believe: Can be improved at the cost of significantly more involved arguments. The ideas of redundant solution and the linear algebra trick maybe have further algorithmic applications. Can we obtain a kernel with significantly improved size? Interval Vertex Deletion admits a polynomial kernel (parameter: solution size).