Algebraic Structures and Algorithms for Matching and Matroid Problems

Algebraic Structures and Algorithmsfor Matching and Matroid Problems Nick Harvey

Perfect Matching • Given graph G=(V,E) • Perfect matching is M⊆E such that each vertex incident with exactlyone edge in M

Matching History Dense Graphs m=n2 O(n4) • All are non-trivial • Mucha-Sankowski uses sophisticateddynamic connectivity data structures (Holm et al. ’01)to maintain “canonical partition” (Cheriyan ’97) O(n2.5) O(n2.38) (n = # vertices, m = # edges) • All are non-trivial! • Mucha ’05: “[Our] algorithm is quite complicated and heavily relies on graph theoretic results and techniques. It would be nice to have a strictly algebraic, and possibly simpler, matching algorithm”. (<2.38 is exponent for matrix multiplication)

Matching Algorithms Dense Graphs m=n2 • Conceptually simple • Implemented in ~200 lines of MATLAB * * * Randomized, but Las Vegas via [Cheriyan ’97]

(Linear) Matroid Intersection • Find largest set of columns S such thatAS and BS are both linearly independent AS A = BS B =

Matroid Intersection • Find largest set of columns S such thatAS and BS are both linearly independent • Why? • Many problems are a special case: • Bipartite matching • Arboresence (directed spanning tree), … • Powerful tool: • Bounded-Degree Sp. Tree [Goemans ’06]

Matroid Intersection History (for matrices of size n x m)

Matroid Intersection History • Essentially optimal: • Best known alg to compute rank takes O(mn-1) • Computing rank reduces to matroid intersection * * * † O(mn1.38) * Assumes matroids are linear † Randomized, and assumes matroids can berepresented over same field

Generic Matching Algorithm For each e E If e is contained in a perfect matching Add e to solution Delete endpoints of e Else Delete edge e

Generic Matching Algorithm Key step For each e E If e is contained in a perfect matching Add e to solution Delete endpoints of e Else Delete edge e • How can we test this? • Randomization and linear algebra play key role

Matching Outline • Implementing Generic Algorithm • Tutte Matrix & Properties • Rabin-Vazirani Algorithm • Rank-1 Updates • Rabin-Vazirani with Rank-1 Updates • Our Recursive Algorithm (overview) • Our Recursive Algorithm (fast updates)

Matching & Tutte Matrix • Let G=(V,E) be a graph • Define variable x{u,v} {u,v}∈E • Define a skew-symmetric matrix T s.t. ± x{u,v} if {u,v}∈E Tu,v = 0 otherwise a b c

Properties of Tutte Matrix Lemma [Tutte’47]: G has a perfect matchingiff T is non-singular. Lemma [RV’89]: G[V\{u,v}] has a perfect matching iff (T-1)u,v 0.

Properties of Tutte Matrix Lemma [Tutte’47]: G has a perfect matchingiff T is non-singular. Lemma [RV’89]: G[V\{u,v}] has a perfect matching iff (T-1)u,v 0. u (T-1)u,v 0 v

Properties of Tutte Matrix Lemma [Tutte’47]: G has a perfect matchingiff T is non-singular. Lemma [RV’89]: G[V\{u,v}] has a perfect matching iff (T-1)u,v 0. G[V\{u,v}] has perfectmatching (T-1)u,v 0

Properties of Tutte Matrix Lemma [Tutte’47]: G has a perfect matchingiff T is non-singular. Lemma [RV’89]: G[V\{u,v}] has a perfect matching iff (T-1)u,v 0. u G[V\{u,v}] has perfectmatching (T-1)u,v 0 v

Properties of Tutte Matrix Lemma [Tutte’47]: G has a perfect matchingiff T is non-singular. Lemma [RV’89]: G[V\{u,v}] has a perfect matching iff (T-1)u,v 0. Computing T-1 very slow: Contains variables! Lemma [Lovász’79]: These results hold w.h.p. if we randomly choose values for x{u,v}’s.

Matching Algorithm Rabin-Vazirani ’89 (Takes O(n) time) Compute T-1 For each {u,v}  E If T-1u,v 0 Add {u,v} to matching Recurse on G[V\{u,v}] • Total time: O(n+1) time • Natural question: Can we recomputeT-1 quickly in each iteration?

Matching Outline • Tutte Matrix & Properties • Rabin-Vazirani Algorithm • Rank-1 Updates • Rabin-Vazirani with Rank-1 Updates • Our Recursive Algorithm (overview) • Our Recursive Algorithm (fast updates)

Rank-1 Updates • Let T be a n x n matrix • T + uvT is called a rank-1 update of T + ∙ T u vT

Rank-1 Updates • Let T be a n x n matrix • T + uvT is called a rank-1 update of T • Computing rank-1 update takes O(n2) time + T uvT

Rank-1 Updates • Let T be a n x n matrix • T + uvT is called a rank-1 update of T • Claim: If you modify one entry of T,then T-1 is modified by a rank-1 update. T = T-1 =

Rank-1 Updates • Let T be a n x n matrix • T + uvT is called a rank-1 update of T • Claim: If you modify one entry of T,then T-1 is modified by a rank-1 update. • Claim: If you delete O(1) rows and columns of T then T-1 is affected by O(1) rank-1 updates. T-1 can be updated in O(n2) time

RV Algorithm + Rank-1 Updates Mucha-Sankowski ’04 Compute T-1 For each {u,v}  E If T-1u,v 0 Add {u,v} to matching Update T-1(via rank-1 updates) Recurse on G[ V\{u,v} ] • Total runtime: O(n3) time • Can updates use Fast Matrix Multiplication? • Bipartite graphs: Yes • Non-bip. graphs: Yes, but non-trivial [MS’04]

RV Algorithm + Rank-1 Updates Mucha-Sankowski ’04 Compute T-1 For each {u,v}  E If T-1u,v 0 Add {u,v} to matching Update T-1(via rank-1 updates) Recurse on G[ V\{u,v} ] • Total runtime: O(n3) time • Can updates use Fast Matrix Multiplication? New approach: An unusual recursion!

New Recursive Approach (Here c=4 parts) • Partition into c parts {V1,…,Vc} (arbitrarily) • For each pair of parts {Va,Vb} (arbitrary order) • Recurse on G[Va ⋃Vb]

New Recursive Approach • Partition into c parts {V1,…,Vc} (arbitrarily) • For each pair of parts {Va,Vb} (arbitrary order) • Recurse on G[Va ⋃Vb]

New Recursive Approach • Partition into c parts {V1,…,Vc} (arbitrarily) • For each pair of parts {Va,Vb} (arbitrary order) • Recurse on G[Va ⋃Vb] • Base case: 2 vertices

New Recursive Approach • Partition into c parts {V1,…,Vc} (arbitrarily) • For each pair of parts {Va,Vb} (arbitrary order) • Recurse on G[Va ⋃Vb] • Base case: 2 vertices {u,v} • If T-1u,v0, add {u,v} to matching, update T-1

Recursion F.A.Q. • Why not just recurse on G[ Va ]? • Edges between parts would be missed • Claim: Our recursion examines every pair of vertices  examines every edge • Why does algorithm work? • It implements Rabin-Vazirani Algorithm! • Isn’t this horribly slow? • No: we’ll see recurrence next • Partition into c parts {V1,…,Vc} (arbitrarily) • For each pair of parts {Va,Vb} (arbitrary order) • Recurse on G[Va ⋃Vb] • Base case: 2 vertices {u,v} • If T-1u,v0, add {u,v} to matching, update T-1

Final Matching Algorithm If base case with 2 vertices {u,v} If T-1u,v 0, add {u,v} to matching Else Partition into c parts {V1,…,Vc} For each pair {Va,Vb} Recurse on G[Va ⋃Vb] Apply updates to current subproblem s = size of subproblem

Final Matching Algorithm If base case with 2 vertices {u,v} If T-1u,v 0, add {u,v} to matching Else Partition into c parts {V1,…,Vc} For each pair {Va,Vb} Recurse on G[Va ⋃Vb] Apply updates to current subproblem Assume: O(s) time s = size of subproblem

Time Analysis • Basic Divide-and-Conquer • If then • Since ,just choose c large enough! c = 13 is large enough if  = 2.38

Handling Updates T =

Handling Updates v v u u u u v v T = T-1 = • Delete vertices u and v

Handling Updates (Naively) v v u u u u v v T = T-1 = • Delete vertices u and v clear rows / columns • Causes rank-1 updates to T-1 • Algorithm still takes (n3) time

Postponed Just-in-time Updates v u u v T-1 = • Don’t update entire matrix! • Just update parent in recursion tree • Updates outside of parent are postponed

Invert Multiply Postponed Updates v u u v T-1 = • Accumulate batches of updates • Claim: New updates can be applied with matrix multiplication and inversion

Final Matching Algorithm If base case with 2 vertices {u,v} If T-1u,v 0, add {u,v} to matching Else Partition into c parts {V1,…,Vc} For each pair {Va,Vb} Recurse on G[Va ⋃Vb] Apply updates to current subproblem Invariant: Before / after child subproblem,parent’s submatrix is completely updated  in every base case, T-1u,v is up-to-date!

Graph G children Just apply postponed updates All edges chosen in base cases Only take edges that can be extended to a perfect matching in the whole graph. This decision is possible because invariant ensures that T-1u,v is up-to-date.

Matching Summary • Can compute a perfect matching inO(n) = O(n2.38) time • Algorithm uses only simple randomization, linear algebra and divide-and-conquer • Easy to implement • Extensions for: (by existing techniques) • maximummatchings • Las Vegas

J A Matroid Intersection O(mn-1) algorithm J x3 x4 BT x5 x6 x7 x8 More Extensions • Same philosophy applies to manymatching-like problems • Design matrix capturing problem • Design algorithm to test if element in OPT • Design method to efficiently do updates • Design “algebraic structure” capturing problem

More Extensions • Same philosophy applies to manymatching-like problems • Design matrix capturing problem • Design “algebraic structure” capturing problem Bipartite Matroid Matching & Basic Path-Matching O(n) algorithm

Open Problems • Fast, deterministic methods to choosevalues for indeterminates? • Implications for Matching in NC • Scaling algorithms for weighted problems? • O(n2) algorithms for matching bycompletely different techniques? • e.g., via Karger-Levine randomized max flow

Algebraic Structures and Algorithms for Matching and Matroid Problems