Approximation alogrithms
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The document discusses approximation algorithms in the context of NP-completeness and heuristic methods, highlighting their efficiency and quality guarantees. It covers specific problems like vertex cover and the knapsack problem, explaining how these algorithms can find solutions that are provably close to optimal. Additionally, it addresses theories on dynamic programming and scaling techniques to improve algorithm performance for NP-hard problems.
![Dynamic programming
• Q: What to remember?
• A: remember total value so far
• A[i,v]=whether v achievable with subset of first i items](https://image.slidesharecdn.com/approximationalogrithms-180315083557/85/Approximation-alogrithms-39-320.jpg)
![• A[i,v]=whether v achievable with subset of first i items
• A[i,v]= A[i-1,v] or ((v-vi) and A[i-1,v-vi])](https://image.slidesharecdn.com/approximationalogrithms-180315083557/85/Approximation-alogrithms-41-320.jpg)
![Dynamic programming
• Q: What to remember?
• A[i,v]= whether v achievable with subset of first i items
• A[i,v]=must remember size
• A[i,v]=min size achievable for subset of first i items of value v](https://image.slidesharecdn.com/approximationalogrithms-180315083557/85/Approximation-alogrithms-44-320.jpg)
Approximation alogrithms
- 1. Approximation Algorithms Seyed Mohsen Fatemi 91222096 Shahid Beheshti University
- 3. Brute-force algorithms. Develop clever enumeration strategies. Guaranteed to find optimal solution. No guarantees on running time. Heuristics. Develop intuitive algorithms. Guaranteed to run in polynomial time. No guarantees on quality of solution. Approximation algorithms. • Guaranteed to run in polynomial time. • Guaranteed to find "high quality" solution, say within 1% of optimum. Obstacle: need to prove a solution’s value is close to optimum, without even knowing what optimum value is!
- 4. Combinatorial optimization: • Scheduling classes • Planning delivery routes for trucks • … Most are NP-hard
- 5. In polynomial time, we find a solution whose value is provably within a "small" factor of the optimal solution Dealing with NP-hard problems
- 6. Approximation algorithms, vertex cover, and linear programming
- 7. Vertex Cover • Vertex cover: a subset of vertices which “covers” every edge , An edge is covered if one of its endpoint is chosen . • The Minimum Vertex Cover Problem: Find a vertex cover with minimum number of vertices.
- 18. lowerbound OPTLP
- 22. Correctness •… is it a Vertex Cover ?
- 28. Performance guarantees • An Approximation Algorithm is bounded by ρ(n) if for all input of size n , the cost c of the solution obtained by the algorithm is within a factor ρ(n) of the c* of an optimal solution
- 29. How good is that? Typical performance : within 10% of optimum
- 30. Theorem • There exists an optimal solution s.t. every coordinate is in {0, 0.5, 1} and there is a polynomial-time algorithm to construct it. • End of Vertex Cover
- 31. The Knapsack Problem • Given: capacity B knapsack , n items , item i has size si and value vi Place some items in knapsack Maximize value • It is NP-hard !
- 33. Desire: small size, high value • Naive greedy algorithm • take by order of increasing size/value
- 34. How good is Greedy ? • Greedy is bad !
- 35. Working with Special Cases : • If all items have the same size… ! Greedy is good. • If all items have the same value… ! Greedy is good. • If all items have size=value …
- 36. Greedy Algorithm For the Case size=value • A greedy algorithm for special case size=value Order items by decreasing value. • How good is that?
- 37. Knapsack special special case • All items have size = value € {1,2,…, B} • and in addition B is a "small" integer
- 39. Dynamic programming • Q: What to remember? • A: remember total value so far • A[i,v]=whether v achievable with subset of first i items
- 40. • Q: v achievable with subset of first i items iff… • A: …it depends on whether subset contains i • If not: v must be reached with first i-1 items • If yes: v-vi must be reached with first i-1 items
- 41. • A[i,v]=whether v achievable with subset of first i items • A[i,v]= A[i-1,v] or ((v-vi) and A[i-1,v-vi])
- 42. Algorithm
- 43. Less special special case • values are small integers • All values € {1,2,...,N} N : "small" integer
- 44. Dynamic programming • Q: What to remember? • A[i,v]= whether v achievable with subset of first i items • A[i,v]=must remember size • A[i,v]=min size achievable for subset of first i items of value v
- 45. • Q: v,s achievable with subset of first i items iff… • A: …it depends on whether subset contains i • If not: v,s reached with first i-1 items • If yes: v-vi , s-si reached with first i-1 items
- 49. • Q : How to modify input so that values are small integers? • A : Scale and round : **Discard items that don’t fit **1. Multiply each value by something so that values are small **2. Round values to integers Dynamic program for the scaled and rounded problem
- 51. Effect of scaling • Max scaled value = N
- 52. How do we pick N? • small enough that the runtime is polynomial • large enough that the resulting set of items has value close to the optimum
- 55. Analysis • S: output items • DP: S optimal for scaled rounded input • S*: optimal items • Scaling: S* optimal for scaled unrounded input
- 60. Lower bound OPT • Discarded items that don’t fit :
- 61. Wrapping Up
- 62. Theorem • Solution to knapsack with value at least 0.99 OPT and runtime O(poly(n))
- 64. Theorem • this gives a solution to knapsack with value at least 0.999 OPT and with runtime O(poly(n))
- 66. Theorem • knapsack has an approximation scheme
- 69. Method • 1. Simplify the input • 2. Design algorithm for “simple” inputs
- 70. Refrences : • Approximation Algorithms – Part 1 Coursera • Introduction to algorithms , CLRS , 3rd ed T.Cormen • The Design of Approximation Algorithms , David P.Williamson , David B.Shmoys • Approximation Algorithms , Vijay V. Vazirani
- 71. Good Luck ;) Teşekkürler sağ olasız