expansions for reductions

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  • 1. Expansions for Reductions
  • 2. Expansions for Reductions Joint work with Fedor Fomin, Daniel Lokshtanov, Geevarghese Philip, and Saket Saurabh University of Bergen University of California, San Diego The Institute of Mathematical Sciences
  • 3. Talk Outline
  • 4. Talk Outline |N(S)| q|S| q matchings, mutually disjoint in B
  • 5. Talk Outline q-Expansion Lemma |N(S)| q|S| q matchings, mutually disjoint in B
  • 6. Talk Outline q-Expansion Lemma with applications to Vertex Cover, Feedback Vertex set, and beyond.
  • 7. Part 1A Tale of q Matchings
  • 8. B AConsider a bipartite graph one of whose parts (say B) is at least twice as big as the other (call this A).
  • 9. B AAssume that there are no isolated vertices in B. bleh
  • 10. Suppose, further, that for every subset S in A, N(S) is at least twice as large as |S|.
  • 11. N(S) SSuppose, further, that for every subset S in A, N(S) is at least twice as large as |S|.
  • 12. B Aen there exist two matchings saturating A, bleh
  • 13. B Aen there exist two matchings saturating A, bleh
  • 14. B Aen there exist two matchings saturating A, and disjoint in B.
  • 15. Claim:If N(A) q|A|, then there exists a subset S of A such that: there q matchings saturating the subset S that are vertex-disjoint in B.provided B does not have any isolated vertices.
  • 16. Claim:If N(A) q|A|, then there exists a subset S of A such that: there q matchings saturating the subset S that are vertex-disjoint in B.provided B does not have any isolated vertices.
  • 17. Claim:If |B| q|A|, then there exists a subset S of A such that: there q matchings saturating the subset S that are vertex-disjoint in B,provided B does not have any isolated vertices.
  • 18. Crucially: it turns out that the endpoints of the matchingsin B (the larger set) do not have neighbors outside S.
  • 19. B A
  • 20. Part 2Two-Expansions and FVS
  • 21. Ingredients
  • 22. a high-degree vertex, v Ingredients
  • 23. a small hitting set, sans va high-degree vertex, v Ingredients
  • 24. Given a high-degree vertex v, nding a small hitting set thatdoes not contain v.Any hitting set whose size is a polynomial function of k.When the largest collection of vertex disjoint paths fromN(v) to N(v) is small.
  • 25. Given a high-degree vertex v, nding a small hitting set thatdoes not contain v.A subset whose removal makes the graph acyclic.When the largest collection of vertex disjoint paths fromN(v) to N(v) is small.
  • 26. Given a high-degree vertex v, nding a small hitting set thatdoes not contain v.A polynomial function of k.When the largest collection of vertex disjoint paths fromN(v) to N(v) is small.
  • 27. Given a high-degree vertex v, nding a small hitting set thatdoes not contain v.At least twice the size of the hitting set.When the largest collection of vertex disjoint paths fromN(v) to N(v) is small.
  • 28. Given a high-degree vertex v, nding a small hitting set thatdoes not contain v.Find an approximate hitting set T .When the largest collection of vertex disjoint paths fromN(v) to N(v) is small.
  • 29. Given a high-degree vertex v, nding a small hitting set thatdoes not contain v.If T does not contain v, we are done.When the largest collection of vertex disjoint paths fromN(v) to N(v) is small.
  • 30. Given a high-degree vertex v, nding a small hitting set thatdoes not contain v.Else: v T . Delete T v from G.When the largest collection of vertex disjoint paths fromN(v) to N(v) is small.
  • 31. Given a high-degree vertex v, nding a small hitting set thatdoes not contain v.e only remaining cycles pass through v.When the largest collection of vertex disjoint paths fromN(v) to N(v) is small.
  • 32. Given a high-degree vertex v, nding a small hitting set thatdoes not contain v.Find an optimal cut set for paths from N(v) to N(v).When the largest collection of vertex disjoint paths fromN(v) to N(v) is small.
  • 33. Given a high-degree vertex v, nding a small hitting set thatdoes not contain v.Why is this cut set small enough?When the largest collection of vertex disjoint paths fromN(v) to N(v) is small.
  • 34. Given a high-degree vertex v, nding a small hitting set thatdoes not contain v.When is this cut set small enough?When the largest collection of vertex disjoint paths fromN(v) to N(v) is small.
  • 35. Given a high-degree vertex v, nding a small hitting set thatdoes not contain v.When is this cut set small enough?When the largest collectio