1. Matroids & Representative Sets Daniel Lokshtanov

2. Alice vs Bob F = {{a,b,c}, {a,c,d}, {b,c,e}} {b, e} {a,c,d} {a, c}

3. Rules of the game Board: universe of size n All Alice’s sets have size p Bob a picks set B of size q Alice wins if she has a set disjoint from B

4. Lazy Alice Alice does not like remembering all those sets. Alice hates losing to Bob. Can she forget a set A from F, and be sure this will not make the difference between winning and losing?

5. (Ir)relevant Sets Alice may forget exactly the irrelevant sets

6. Only relevant sets? B1B1 B2B2 B3B3 BmBm F = { A2A2 A1A1 A3A3 AmAm } …

7. Bollobás’ Lemma [1966] No dependence on universe size n at all!

8. Bollobás’ helps Alice yay!

9. Proof of Bollobás’ Lemma AiAi BiBi BjBj AjAj

10. Representative Sets

11. Computational Problem

12. Computing Representative Sets But first – an easy application

13. d-Hitting Set Easy branching in time d k Next: kernel with O(k d ) sets and elements

14. d-Hitting Set as a Game F = {{a,b,c}, {a,c,d}, {b,c,e}} Is {b, e} a hitting set? No, since {a,c,d}

15. Kernel for d-Hitting Set

16. Why is the kernel correct? May not change a YES instance into a NO instance. Can a NO instance change into a YES instance? NO instance = Alice always wins YES instance = Bob can win We did not forget any sets that made the difference between Alice winning and losing!

17. Playing on a matroid

18. Alice vs Bob on a matroid F = {{a,b,c}, {a,c,d}, {b,c,e}} Do you have a set that fits {b, e} ? &%¤&!!

19. Representative Sets

20. Computing Representative Sets

21. Playing on a matroid p=4, q=2 304958029038502923840293850230905801012095830321520385 292302302310958042093582023039528303202335020322022582 2202302350203209802104+4267429810983502239582820320502 340958683040938323035802092309532029385308209821522998 208357298739829872398253982359823987235239729019380205 230958203958293958203958203958203958522938572938575292 M = p+q F ? ! 232401 018920 320110 848338 053002

22. Fit vs Determinant If Alices set A and Bob’s set B overlap, then the same column is used twice  determinant is 0! Determinant is nonzero if and only if A fits B.

23. Matrix game a b c d p+q p q c

24. Generalized Laplace Expansion almost correct MBMB MAMA p+q p q To compute Det dot product! *

25. Giant Vector game a bc d c

26. Basis If Alice keeps vectors v 1,v 2,v 3 and v 3 = v 1 + v 2 and v 3 fits Bob’s vector v B Then either v 1 or v 2 fits v B Alice only needs to keep linearly independent vectors!

27. Wrap up

28. Computing Representative Sets

29. Application - Treewidth DP Have seen several approaches for single exponential algorithms for connectivity problems parameterized by treewidth. Representative sets gives yet another one

30. Hamiltonian Path

31. Representative Sets for Matroid Classes Is it possible to compute representative sets for uniform matroids, graphic matroids or transversal matroids faster than for linear matroids in general? For uniform matroids, the answer is yes (but proof is sort of complicated)

32. Application – k-Path

33. k-Path

34. Extend all paths that can be extended by v

35. B u w v Size q+1

36. k-Path

37. Input: (directed) graph G, integer k. Question: Is there a simple directed cycle on at least k vertices? Theorem: 8 k poly(n) algorithm.

38. In a shortest cycle C on at least k vertices, we can replace any subpath on k vertices by any other path on k vertices, which is disjoint from the k vertices after it on C.

39. P uv P vw P wv v u w

40. Guess a vertex u that a shortest cycle C of length at least k passes through. For every vertex v and integer p, define P[u, p] to be the set of (vertex sets of) paths on exactly p vertices from u to v. For every vertex v compute a set P’[v] that k- represents P[v,k] using the method from the k-path algorithm.

42. Speeding up

43. Exercises Book: 12.9, 12.11, 12.13, 5.9