Polylogarithmic approximation algorithm for weighted F-deletion problems
- 1. Akanksha Agrawal, Daniel Lokshtanov, Pranabendu Misra, Saket Saurabh, and Meirav Zehavi Hungarian Academy of Sciences APPROX 2018, Princeton A Polylogrithmic Approximation Algorithms for Weighted F Deletion Problems
- 2. For a graph family F Input: A graph G and a weight function w: V(G) —> R. Goal: Find a minimum weight subset S V(G), such that G-S is in F. ✓ Weighted F Deletion Problem
- 3. For a graph family F Input: A graph G and a weight function w: V(G) —> R. Goal: Find a minimum weight subset S V(G), such that G-S is in F. ✓ Encompasses several NP-hard problems. Weighted F Deletion Problem
- 4. (No induced cycle on at least 4 vertices) Weighted Chordal Vertex Deletion Example: Weighted F Deletion Problem Chordal graphs
- 5. (No induced cycle on at least 4 vertices) chordal graph Chordal graphs Weighted Chordal Vertex Deletion Example: Weighted F Deletion Problem
- 6. (No induced cycle on at least 4 vertices) chordal graph not a chordal graph Chordal graphs Weighted Chordal Vertex Deletion Example: Weighted F Deletion Problem
- 7. Input: A graph G and a weight function w: V(G) —> R. Goal: Find a minimum weight subset S V(G), such that G-S is a chordal graph. ✓ Weighted Chordal Vertex Deletion Example: Weighted F Deletion Problem
- 8. Minor of a graph A graph H is a minor of G if we can obtain H from G by a sequence of the following operations: Vertex deletion.
- 9. Minor of a graph A graph H is a minor of G if we can obtain H from G by a sequence of the following operations: Vertex deletion.
- 10. Minor of a graph A graph H is a minor of G if we can obtain H from G by a sequence of the following operations: Vertex deletion. Edge deletion.
- 11. Minor of a graph A graph H is a minor of G if we can obtain H from G by a sequence of the following operations: Vertex deletion. Edge deletion.
- 12. Minor of a graph A graph H is a minor of G if we can obtain H from G by a sequence of the following operations: Vertex deletion. Edge deletion. Edge contraction.
- 13. Minor of a graph A graph H is a minor of G if we can obtain H from G by a sequence of the following operations: Vertex deletion. Edge deletion. Edge contraction.
- 14. Minor of a graph A graph H is a minor of G if we can obtain H from G by a sequence of the following operations: Vertex deletion. Edge deletion. Edge contraction.
- 15. Weighted Planar H-Minor-Free Deletion Example: Weighted F Deletion Problem H: A finite family of graphs containing at least one planar graph.
- 16. Weighted Planar H-Minor-Free Deletion Example: Weighted F Deletion Problem H: A finite family of graphs containing at least one planar graph. G(H): The family of graphs excluding each graph in H as a minor.
- 17. Weighted Planar H-Minor-Free Deletion Example: Weighted F Deletion Problem G(H): The family of graphs excluding each graph in H as a minor. Planar H-Minor-Free Deletion = Weighted F= G(H) Deletion Problem H: A finite family of graphs containing at least one planar graph.
- 18. Weighted Planar H-Minor-Free Deletion Example: Weighted F Deletion Problem Encompasses several NP-hard problems like: Weighted Vertex Cover Weighted Feedback Vertex Set Treewidth η-Deletion
- 19. Weighted Distance Hereditary Deletion Example: Weighted F Deletion Problem (Every connected induced subgraph preserves distances) Distance hereditary graphs
- 20. Weighted Distance Hereditary Deletion Example: Weighted F Deletion Problem (Every connected induced subgraph preserves distances) Distance hereditary graphs u v not a distance hereditary graph
- 21. Weighted Distance Hereditary Deletion Example: Weighted F Deletion Problem (Every connected induced subgraph preserves distances) Distance hereditary graphs u v a distance hereditary graph
- 22. Our Results Weighted Planar H-Minor-Free Deletion: • Randomized O(log1.5 n)-approximation algorithm. • Deterministic O(log2 n)-approximation algorithm. Weighted Chordal Vertex Deletion: O(log2 n)-approximation algorithm. • Weighted Multicut: constant factor approximation. Weighted Distance Hereditary Deletion: O(log3 n)- approximation algorithm. Unified framework
- 25. Our Methodology For its typical application, the solution forms a balanced separator. Add balanced separator to the approximate solution and recurse. A strengthening of the approach of Leighton and Rao
- 26. Our Methodology For its typical application, the solution forms a balanced separator. Add balanced separator to the approximate solution and recurse. A strengthening of the approach of Leighton and Rao Not enough for our purpose!
- 27. Our Methodology S* (optimal solution) X (well structured set, possibly very large) (G-S*) - X
- 28. Our Methodology S* (optimal solution) X (well structured set, possibly very large) (G-S*) - X Balanced separator V1 V2
- 29. Our Methodology X G-X Step 1. (well structured set)
- 30. Our Methodology X (well structured set) G-X W* (inexpensive balanced separator) Step 1.
- 31. Our Methodology X (well structured set) G-X W Step 2: Compute a balanced separator of G-X using the known approximation algorithm. V1 V2 (inexpensive balanced separator)
- 32. Our Methodology X (well structured set) G-X W V1 V2 Add to the solution (inexpensive balanced separator) Step 2: Compute a balanced separator of G-X using the known approximation algorithm.
- 33. Our Methodology X (well structured set) W Step 3: Recursive step. V1 V2 (inexpensive balanced separator)
- 34. Our Methodology X (well structured set) W S1 S2 V1 V2 (inexpensive balanced separator) Step 3: Recursive step.
- 35. Our Methodology W Belongs to FV1-S1 X (well structured set) V2-S2 (inexpensive balanced separator) Step 3: Recursive step.
- 36. Our Methodology X (well structured set) W Special instance V1-S1 V2-S2 Step 4: Solve the special instance (using a special algorithm). (inexpensive balanced separator)
- 37. Our Methodology X (well structured set) W Special instance V1-S1 V2-S2 S3 (inexpensive balanced separator) Step 4: Solve the special instance (using a special algorithm).
- 38. Our Methodology X (well structured set) W Special instance V1-S1 V2-S2 S3 S1 S2 (inexpensive balanced separator) Step 4: Solve special instance (using special algorithm).
- 39. Z Weighted Planar H-Minor-Free Deletion Example: Weighted F Deletion Problem
- 40. x y b a c w p q s A B C D a,b,c a,b,w b,c,p,q a,c,s x,y X Graph G Tree decomposition (X,T) ZTree Decomposition
- 41. x y b a c w p q s A B C D a,b,c a,b,w b,c,p,q a,c,s x,y X Graph G Every vertex is contained in contained in at least one bag. ZTree Decomposition Tree decomposition (X,T)
- 42. x y b a c w p q s A B C D a,b,c a,b,w b,c,p,q a,c,s x,y X Graph G For each (u,v) E(G), there is a bag containing u and v. 2 ZTree Decomposition Tree decomposition (X,T)
- 43. x y b a c w p q s A B C D a,b,c a,b,w b,c,p,q a,c,s x,y X Graph G For each v V(G), set of bags containing v is a sub-tree of T. 2 ZTree Decomposition Tree decomposition (X,T)
- 44. Z Weighted Planar H-Minor-Free Deletion Example: Weighted F Deletion Problem Treewidth property A graph excluding each graph in H as a minor has treewidth at most c=c(H). Recall that H contains at least one planar graph!
- 45. Our Methodology X (well structured set) W (cheap balanced separator) Special instance V1-S1 V2-S2 S3 S1 S2 Step 4: Solve special instance (using special algorithm).
- 46. Weighted Planar H-Minor-Free Deletion Example: Weighted F Deletion Problem X (well structured set: size at most c+1) G-X H-minor-free Special instance
- 47. Z Weighted Planar H-Minor-Free Deletion Example: Weighted F Deletion Problem Treewidth property A graph excluding each graph in H as a minor has treewidth at most c=c(H). Recall that H contains at least one planar graph!
- 48. Weighted Planar H-Minor-Free Deletion Example: Weighted F Deletion Problem X (well structured set: size at most c+1) G-X Treewidth at most c H-minor-free Resolving Special instance
- 49. Weighted Planar H-Minor-Free Deletion Example: Weighted F Deletion Problem X G-X Compute tree decomposition (width at most c) Resolving Special instance
- 50. Weighted Planar H-Minor-Free Deletion Example: Weighted F Deletion Problem X G-X Compute tree decomposition (width at most c) Resolving Special instance
- 51. Weighted Planar H-Minor-Free Deletion Example: Weighted F Deletion Problem X G-X Compute tree decomposition of G (width at most 2c+1) Resolving Special instance
- 52. Weighted Planar H-Minor-Free Deletion Example: Weighted F Deletion Problem Tree decomposition of G (width at most 2c+1) Resolving Special instance
- 53. Weighted Planar H-Minor-Free Deletion Example: Weighted F Deletion Problem Resolving Special instance Dynamic programming over tree decomposition! Tree decomposition of G (width at most 2c+1)
- 54. Weighted Planar H-Minor-Free Deletion Algorithm S* (optimal solution) G-S*
- 55. Weighted Planar H-Minor-Free Deletion Algorithm S* (optimal solution) G-S* Treewidth at most c
- 56. Weighted Planar H-Minor-Free Deletion Algorithm S* (optimal solution) G-S* Small Balanced separator
- 57. • For each vertex subset X of size at most c+1, test if G-X is H-minor-free. Yes: Resolve using the algorithm for special cases. No: Recurse. Weighted Planar H-Minor-Free Deletion Algorithm
- 58. Weighted Planar H-Minor-Free Deletion Recursive step G X W Compute a balanced separator W of G-X using the known approximation algorithm. At least one of them is good!
- 59. X (well structured set) G-X W Step 2: We have a balanced separator of G-X using the known approximation algorithm. V1 V2 Add to the solution Weighted Planar H-Minor-Free Deletion Recursive step (inexpensive balanced separator)
- 60. X (well structured set) W V1 V2 Weighted Planar H-Minor-Free Deletion Recursive step (inexpensive balanced separator) Step 3: Recursive step.
- 61. X (well structured set) W S1 S2 V1 V2 Weighted Planar H-Minor-Free Deletion Recursive step (inexpensive balanced separator) Step 3: Recursive step.
- 62. W (cheap balanced separator) Step 3: Recursive step. Belongs to FV1-S1 X (well structured set) V2-S2 Weighted Planar H-Minor-Free Deletion Recursive step
- 63. X (well structured set) W (cheap balanced separator) Special instance V1-S1 V2-S2 Step 4: Solve the special instance. Weighted Planar H-Minor-Free Deletion Recursive step
- 64. X (well structured set) W (cheap balanced separator) Special instance V1-S1 V2-S2 Step 4: Solve special instance (using special algorithm). S3 Weighted Planar H-Minor-Free Deletion Recursive step
- 65. X (well structured set) W (cheap balanced separator) Special instance V1-S1 V2-S2 Step 4: Solve special instance (using special algorithm). S3 S1 S2 Weighted Planar H-Minor-Free Deletion Recursive step
- 66. Does Weighted Planar H-Minor-Free Deletion admit a constant-factor approximation algorithm? What about Weighted H-Minor-Free Deletion? Does Weighted Chordal Vertex Deletion admit a constant-factor approximation algorithm? Does Weighted Rankwidth-η Vertex Deletion admit a O(logO(1)n)-factor approximation algorithm? Open Questions
- 67. Thanks