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 nAll Alice’s sets have size pBob a picks set B of size qAlice 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

A F is irrelevant if: for every set B of size q such that A B = , there is a set A’ F such that A’ B = .

Alice may forget exactly the irrelevant sets

Only relevant sets?

B1 B2 B3 Bm

F = { A2A1 A3 Am }…

Bollobás’ Lemma [1966]

Let A1, A2, … , Am be sets of size p and

B1, B2, … , Bm be sets of size q s.t:

then .

No dependence on universe size n at all!

Bollobás’ helps Alice

Bollobás’ lemma immediately implies that Alice only needs to remember at most sets.

yay!

Proof of Bollobás’ Lemma

Consider a random permutation of the universe.

The events «all of Ai before all of Bi» and «All of Aj before all of Bj» are disjoint!

So and hence .

P[all of Ai is before all of Bi] = .Ai BiBj Aj

Representative Sets

Let F be a family of p-sets. Then q-represents F if for every B of size q such that there exists an with there exists an with .

Corollary of Bollobás: For every F there is an of size at most that q-represents F.

Computational Problem

Given a family F of p-sets and an integer q, compute a family of size at most that q-represents F.

Computing Representative Sets

Will show: we can compute representative sets in time essentially where is the matrix multiplication constant < 2.38.

But first – an easy application

d-Hitting Set

Input: Family F = {S1,…,Sm} of sets of size d over universe U = {v1, …, vn}, integer k.

Question: Does there exist a set X U of size at most k such that for every Si F, Si X ?

Easy branching in time dk

Next: kernel with O(kd) 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

Compute a k-representative subfamily F’ F of size at most .

Remove all elements not in F’ (at most dkd)

Output the instance F’, k.

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

Suppose now that the universe is the edge set of a matroid.

A set A fits a set B if - A and B are disjoint and- A B is independent in the 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}?

&%¤&!!

Note: this game on a uniform matroid of rank p+q is exactly the old game.

Representative Sets

Let F be a family of p-sets (in a matroid M). Then q-represents F if:

for every B of size q such thatthere exists an that fits B there exists an that also fits B.

Note: representation in a uniform matroid of rank p+q is exactly the old representation.

Computing Representative Sets

Input: Family F of p-sets over a matroid, integer q, matrix M representing the matroid.

Task: Compute a q-representative subfamily of size at most .

Playing on a matroidp=4, q=2

3049580290385029238402938502309058010120958303215203852923023023109580420935820230395283032023350203220225822202302350203209802104+4267429810983502239582820320502340958683040938323035802092309532029385308209821522998208357298739829872398253982359823987235239729019380205230958203958293958203958203958203958522938572938575292

M = p+q

F ?!

232401018920320110848338053002

Det

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

ab

cd p+q

p

p+q

q

c

Generalized Laplace Expansionalmost correct

MBMA p+q

p q

To compute Det

Compute the determinants of all

p p submatrices of MA

Compute the determinants of all

q q submatrices of MB

dimensional vector vA

dimensional vector vB

dot product!*

Giant Vector game

a (𝑝+𝑞𝑝 )b c

d

c

(𝑝+𝑞𝑝 )

⋅ 0 ?

Basis

If Alice keeps vectors v1,v2,v3 and v3 = v1 + v2

and v3 fits Bob’s vector vB

Then either v1 or v2 fits vB Alice only needs to keep linearly independent vectors!

At most of them, sincevectors are - dimensional

Finding the basis takes time.

Wrap up

Alice has a family of p-sets, family of p (p+q) matricesfamily of - dimensonal vectors.

Keep linearly independent vectors, keep the corresponding sets!

Computing Representative Sets

Theorem: we can compute representative sets of size in time essentially where is the matrix multiplication constant < 2.38.

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

Input: (directed) graph G, integer k.Question: Is there a simple directed path on k vertices?

Theorem: There is a deterministic time algorithm for k-Path.

k-Path

Fix a source vertex u.

For vertex v and integer p, define P[v,p] to be the set of (vertex sets of) paths on exactly p vertices from u to v.

Goal: for every v and p k compute a set P’[v,p] that (k-p)-represents P[v,p].

k-PathGoal: for every v, p k compute a set P’[v,p] that (k-p)-represents P[v,p].

X [v , p ]=¿wv∈ E(G)P′ [w , p−1 ] ⋅ \{v }P′ [u ,1 ]= {{u } }

P′ [v , p ]=reduce(X [v , p ] , k − p)

Need to prove: that (k-p)-represents P[v,p] assuming (k-p+1)-represents P[w,p-1]

Extend all paths that can beextended by v

Need to prove: that (k-p)-represents P[v,p] assuming (k-p+1)-represents P[w,p-1]

BBu

wv

Size q+1

In

In

k-Path

Size of family to reduce:

Time to reduce:

Total time:

Also works for weighted k-Path.

( 𝑘𝑝−1)2𝑜(𝑘) ⋅n

Application – k-Cycle

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

Theorem: 8kpoly(n) algorithm.

k-Cycle – main lemma

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.

k-Cycle – main lemma proof

Puv

Pvw

Pwv

v

u

w

k-Cycle – algorithm

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.

Size:

k-Cycle – algorithm

Guess the k’th vertex v on the cycle C.

Main lemma there is a k-cycle containing a path from P’[v] as a subpath!

Check whether there is a path Q in P’[v] such that there is a path back from v to u in G\Q.

Time 8kpoly(n).

Speeding up

Using similar methods, but trading off space for time one can speed up k-Path to 2.619k and k-Cycle to 6.75k.

Exercises

Book: 12.9, 12.11, 12.13, 5.9

Thank You!