Matroids & Representative Sets Daniel Lokshtanov

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

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

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?

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

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

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

Bollobás’ helps Alice yay!

Proof of Bollobás’ Lemma AiAi BiBi BjBj AjAj

Representative Sets

Computational Problem

Computing Representative Sets But first – an easy application

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

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}

Kernel for d-Hitting Set

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!

Playing on a matroid

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} ? &%¤&!!

Representative Sets

Computing Representative Sets

Playing on a matroid p=4, q= M = p+q F ? !

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.

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

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

Giant Vector game a bc d c

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!

Wrap up

Computing Representative Sets

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

Hamiltonian Path

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)

Application – k-Path

k-Path

Extend all paths that can be extended by v

B u w v Size q+1

k-Path

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

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.

P uv P vw P wv v u w

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.

Speeding up

Exercises Book: 12.9, 12.11, 12.13, 5.9