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Iterative Compression

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- 1. Iterative Compression

2. InputA graph G = (V,E) with n vertices, m edges, and k.Is there a vertex cover of size at most k?Vertex CoverQuestion 3. InputA graph G = (V,E) with n vertices, m edges, and k.Is there a vertex cover of size at most k?Vertex CoverQuestion 4. InputA graph G = (V,E) with n vertices, m edges, and k.Is there a vertex cover of size at most k?A vertex cover of size (k+1).Vertex CoverQuestion 5. InputA graph G with a vertex cover of size k+1.Is there a vertex cover of size at most k?A vertex cover of size (k+1).Vertex CoverQuestion 6. InputA graph G with a vertex cover of size k+1.Is there a vertex cover of size at most k?A vertex cover of size (k+1).Vertex CoverQuestion 7. InputA graph G with a vertex cover of size k+1.Is there a vertex cover of size at most k?A vertex cover of size (k+1). Let us guess how a vertex cover of size at most k interacts with this one.Vertex CoverQuestion 8. InputA graph G with a vertex cover of size k+1.Is there a vertex cover of size at most k?A vertex cover of size (k+1). Let us guess how a vertex cover of size at most k interacts with this one.Vertex CoverQuestion 9. InputA graph G with a vertex cover of size k+1.Is there a vertex cover of size at most k?A vertex cover of size (k+1). Let us guess how a vertex cover of size at most k interacts with this one.Vertex CoverQuestion 10. InputA graph G with a vertex cover of size k+1.Is there a vertex cover of size at most k?A vertex cover of size (k+1).Vertex CoverQuestion 11. InputA graph G with a vertex cover of size k+1.Is there a vertex cover of size at most k?A vertex cover of size (k+1). Can there be an edge between two red vertices?Vertex CoverQuestion 12. InputA graph G with a vertex cover of size k+1.Is there a vertex cover of size at most k?A vertex cover of size (k+1).Vertex CoverQuestion 13. InputA graph G with a vertex cover of size k+1.Is there a vertex cover of size at most k?A vertex cover of size (k+1). Now we need to make up for the work that the red vertices were doing.Vertex CoverQuestion 14. InputA graph G with a vertex cover of size k+1.Is there a vertex cover of size at most k?Vertex CoverQuestion 15. InputA graph G with a vertex cover of size k+1.Is there a vertex cover of size at most k?Vertex CoverQuestion 16. InputA graph G = (V,E) with n vertices, m edges, and k.Is there a vertex cover of size at most k?Vertex CoverQuestion 17. InputA graph G = (V,E) with n vertices, m edges, and k.Is there a vertex cover of size at most k?Vertex CoverQuestion 18. InputA graph G = (V,E) with n vertices, m edges, and k.Is there a vertex cover of size at most k?Vertex CoverQuestion 19. InputA graph G = (V,E) with n vertices, m edges, and k.Is there a vertex cover of size at most k?Vertex CoverQuestion 20. InputA graph G = (V,E) with n vertices, m edges, and k.Is there a vertex cover of size at most k?The first k+2 vertices in G.Vertex CoverQuestion 21. InputA graph G = (V,E) with n vertices, m edges, and k.Is there a vertex cover of size at most k?QuestionThe first k+2 vertices in G. It has an easy vertex cover of size k+1.Vertex Cover 22. InputA graph G = (V,E) with n vertices, m edges, and k.Is there a vertex cover of size at most k?QuestionThe first k+2 vertices in G. It has an easy vertex cover of size k+1.Vertex Cover 23. InputA graph G = (V,E) with n vertices, m edges, and k.Is there a vertex cover of size at most k?QuestionThe first k+2 vertices in G. It has an easy vertex cover of size k+1.Vertex Cover 24. InputA graph G = (V,E) with n vertices, m edges, and k.Is there a vertex cover of size at most k?QuestionThe first k+2 vertices in G. It has an easy vertex cover of size k+1.Vertex Cover 25. InputA graph G = (V,E) with n vertices, m edges, and k.Is there a vertex cover of size at most k?QuestionThe first k+2 vertices in G. It has an easy vertex cover of size k+1.Vertex Cover 26. InputA graph G = (V,E) with n vertices, m edges, and k.Is there a vertex cover of size at most k?QuestionThe first k+2 vertices in G. It has an easy vertex cover of size k+1. Compress again rinse, repeat.Vertex Cover 27. InputA graph G = (V,E) with n vertices, m edges, and k.Is there a subset of vertices S of size at most k that intersects all the edges?QuestionWe have a O*(2k) algorithm for Vertex Cover.Vertex Cover 28. InputA graph G = (V,E) with n vertices, m edges, and k.Is there a subset of vertices S of size at most k that intersects all the edges?QuestionWe have a O*(2k) algorithm for Vertex Cover.Vertex Cover 29. InputA graph on n vertices, m edges and an integer k.Is there a subset of at most k vertices whose removal makes the graph acyclic?A Feedback Vertex Set of size (k+1). Let us guess how a FVS of size at most k interacts with this one.Feedback Vertex SetQuestion 30. InputA graph on n vertices, m edges and an integer k.Is there a subset of at most k vertices whose removal makes the graph acyclic?Feedback Vertex SetQuestion 31. InputA graph on n vertices, m edges and an integer k.Is there a subset of at most k vertices whose removal makes the graph acyclic?Feedback Vertex SetQuestion 32. InputA graph on n vertices, m edges and an integer k.Is there a subset of at most k vertices whose removal makes the graph acyclic?Feedback Vertex SetQuestion 33. InputA graph on n vertices, m edges and an integer k.Is there a subset of at most k vertices whose removal makes the graph acyclic?Feedback Vertex SetQuestion 34. InputA graph on n vertices, m edges and an integer k.Is there a subset of at most k vertices whose removal makes the graph acyclic?QuestionA vertex with two neighbors in the same component is forced.Feedback Vertex Set 35. InputA graph on n vertices, m edges and an integer k.Is there a subset of at most k vertices whose removal makes the graph acyclic?Feedback Vertex SetQuestion 36. InputA graph on n vertices, m edges and an integer k.Is there a subset of at most k vertices whose removal makes the graph acyclic?QuestionA leaf that has exactly one neighbor above can be preprocessed.Feedback Vertex Set 37. InputA graph on n vertices, m edges and an integer k.Is there a subset of at most k vertices whose removal makes the graph acyclic?Feedback Vertex SetQuestion 38. InputA graph on n vertices, m edges and an integer k.Is there a subset of at most k vertices whose removal makes the graph acyclic?Questiona leaf with at least two neighbors in different components.Feedback Vertex Set 39. InputA graph on n vertices, m edges and an integer k.Is there a subset of at most k vertices whose removal makes the graph acyclic?Feedback Vertex SetQuestion 40. InputA graph on n vertices, m edges and an integer k.Is there a subset of at most k vertices whose removal makes the graph acyclic?QuestionThe leaf merges two components when we dont include it.Feedback Vertex Set 41. InputA graph on n vertices, m edges and an integer k.Is there a subset of at most k vertices whose removal makes the graph acyclic?Feedback Vertex SetQuestion 42. InputA graph on n vertices, m edges and an integer k.Is there a subset of at most k vertices whose removal makes the graph acyclic?QuestionThe number of components on top decreases.Feedback Vertex Set 43. InputA graph on n vertices, m edges and an integer k.Is there a subset of at most k vertices whose removal makes the graph acyclic?QuestionStart with a leaf. !Two neighbors in one component: forced. At most one neighbor above: preprocess. At least two neighbors, all in different components above: branch.Feedback Vertex Set 44. InputA graph on n vertices, m edges and an integer k.Is there a subset of at most k vertices whose removal makes the graph acyclic?Feedback Vertex SetQuestion 45. InputA graph on n vertices, m edges and an integer k.Is there a subset of at most k vertices whose removal makes the graph acyclic?QuestionLet t denote the number of components among the red vertices. Let w = (k+t).Feedback Vertex Set 46. InputA graph on n vertices, m edges and an integer k.Is there a subset of at most k vertices whose removal makes the graph acyclic?QuestionLet t denote the number of components among the red vertices. Let w = (k+t). Include v k drops by 1 Exclude v t drops by at least 1Feedback Vertex Set 47. InputA graph on n vertices, m edges and an integer k.Is there a subset of at most k vertices whose removal makes the graph acyclic?QuestionLet t denote the number of components among the red vertices. Let w = (k+t). Include v k drops by 1 Exclude v t drops by at least 1 Either way, w drops by at least one.Feedback Vertex Set 48. InputA graph on n vertices, m edges and an integer k.Is there a subset of at most k vertices whose removal makes the graph acyclic? Either way, w drops by at least one.Feedback Vertex SetQuestion 49. InputA graph on n vertices, m edges and an integer k.Is there a subset of at most k vertices whose removal makes the graph acyclic? Either way, w drops by at least one. Running time: 2w = 2(k+t)Feedback Vertex SetQuestion 50. InputA graph on n vertices, m edges and an integer k.Is there a subset of at most k vertices whose removal makes the graph acyclic? Either way, w drops by at least one. Running time: 2w = 2(k+t) 2k+k 4kFeedback Vertex SetQuestion 51. InputA graph on n vertices, m edges and an integer k.Is there a subset of at most k vertices whose removal makes the graph acyclic? Either way, w drops by at least one. Running time: 2w = 2(k+t) 2k+k 4k Overall Running TimeFeedback Vertex SetQuestion 52. InputA graph on n vertices, m edges and an integer k.Is there a subset of at most k vertices whose removal makes the graph acyclic? Either way, w drops by at least one. Running time: 2w = 2(k+t) 2k+k 4k Overall Running Time ki=1k k k 4 =5 iFeedback Vertex SetQuestion 53. InputA tournament T on n vertices and an integer k.Is there a subset of k arcs that can be reversed to make the tournament acyclic? This implies an O(5k) algorithm for FVS.Feedback Vertex SetQuestion 54. InputA tournament T on n vertices and an integer k.Is there a subset of k arcs that can be reversed to make the tournament acyclic? This implies an O(5k) algorithm for FVS.Feedback Vertex SetQuestion 55. InputA graph on n vertices, m edges and an integer k.Is there a subset of at most k vertices whose removal makes the graph bipartite?QuestionA Odd Cycle Transversal of size (k+1). Let us guess how a OCT of size at most k interacts with this one.Odd Cycle Transversal (OCT) 56. InputA graph on n vertices, m edges and an integer k.Is there a subset of at most k vertices whose removal makes the graph bipartite?QuestionOdd Cycle Transversal (OCT) 57. InputA graph on n vertices, m edges and an integer k.Is there a subset of at most k vertices whose removal makes the graph bipartite?QuestionOdd Cycle Transversal (OCT) 58. InputA graph on n vertices, m edges and an integer k.Is there a subset of at most k vertices whose removal makes the graph bipartite?QuestionOdd Cycle Transversal (OCT) 59. InputA graph on n vertices, m edges and an integer k.Is there a subset of at most k vertices whose removal makes the graph bipartite?QuestionOdd Cycle Transversal (OCT) 60. InputA graph on n vertices, m edges and an integer k.Is there a subset of at most k vertices whose removal makes the graph bipartite?QuestionOdd Cycle Transversal (OCT) 61. InputA graph on n vertices, m edges and an integer k.Is there a subset of at most k vertices whose removal makes the graph bipartite?QuestionOdd Cycle Transversal (OCT) 62. InputA graph on n vertices, m edges and an integer k.Is there a subset of at most k vertices whose removal makes the graph bipartite?QuestionOdd Cycle Transversal (OCT) 63. InputA graph on n vertices, m edges and an integer k.Is there a subset of at most k vertices whose removal makes the graph bipartite?QuestionOdd Cycle Transversal (OCT) 64. InputA graph on n vertices, m edges and an integer k.Is there a subset of at most k vertices whose removal makes the graph bipartite?QuestionOdd Cycle Transversal (OCT) 65. InputA graph on n vertices, m edges and an integer k.Is there a subset of at most k vertices whose removal makes the graph bipartite?QuestionOdd Cycle Transversal (OCT) 66. InputA graph on n vertices, m edges and an integer k.Is there a subset of at most k vertices whose removal makes the graph bipartite?QuestionCan we remove at most b vertices so that the resulting graph has a bipartition extending this pre-coloring?Odd Cycle Transversal (OCT) 67. InputA graph on n vertices, m edges and an integer k.Is there a subset of at most k vertices whose removal makes the graph bipartite? GREENQuestionYELLOWCan we remove at most b vertices so that the resulting graph has a bipartition extending this pre-coloring?Odd Cycle Transversal (OCT) 68. InputA graph on n vertices, m edges and an integer k.Is there a subset of at most k vertices whose removal makes the graph bipartite? GREENQuestionYELLOWCan we remove at most b vertices so that the resulting graph has a bipartition extending this pre-coloring?Odd Cycle Transversal (OCT) 69. InputA tournament T on n vertices and an integer k.Is there a subset of k arcs that can be reversed to make the tournament acyclic?QuestionThis implies an O*(3k) algorithm for OCT.Odd Cycle Transversal (OCT) 70. InputA tournament T on n vertices and an integer k.Is there a subset of k arcs that can be reversed to make the tournament acyclic?QuestionThis implies an O*(3k) algorithm for OCT.Odd Cycle Transversal (OCT) 71. Take Away 72. Show that the problem has a hereditary property Solve the compression question, or even a Disjoint version. 73. Show that the problem has a hereditary property Solve the compression question, or even a Disjoint version.Is a framework that can be setup quite easily. 74. Show that the problem has a hereditary property Solve the compression question, or even a Disjoint version.Is a framework that can be setup quite easily.The compression algorithm can be fairly non-trivial.