On Approximate Fully-Dynamic Matching and Online Matrix-Vector Multiplication

TL;DR

This paper explores the complexity of maintaining approximate maximum bipartite matchings and vertex covers in dynamic graphs, establishing connections with the OMv conjecture and providing faster algorithms for these problems.

Contribution

It establishes equivalences between dynamic matching/vertex cover algorithms and the OMv conjecture, and introduces faster algorithms for approximate solutions in both online and offline settings.

Findings

Proves that sublinear update times for dynamic problems imply the falsehood of the OMv conjecture.

Provides a randomized algorithm for approximate matching and vertex cover with sublinear amortized update time.

Offers improved offline algorithms for approximate maximum matching and vertex cover using fast matrix multiplication.

Abstract

We study connections between the problem of fully dynamic $(1-\epsilon)$-approximate maximum bipartite matching, and the dual $(1+\epsilon)$-approximate vertex cover problem, with the online matrix-vector ($\mathsf{OMv}$) conjecture which has recently been used in several fine-grained hardness reductions. We prove that there is an online algorithm that maintains a $(1+\epsilon)$-approximate vertex cover in amortized $n^{1-c}\epsilon^{-C}$ time for constants $c, C > 0$ for fully dynamic updates if and only if the $\mathsf{OMv}$ conjecture is false. Similarly, we prove that there is an online algorithm that maintains a $(1-\epsilon)$-approximate maximum matching in amortized $n^{1-c}\epsilon^{-C}$ time if and only if there is a nontrivial algorithm for another dynamic problem, which we call dynamic approximate $\mathsf{OMv}$, that has seemingly no matching structure. This provides some evidence against achieving amortized sublinear update times for approximate fully dynamic matching and vertex cover. Leveraging these connections, we obtain faster algorithms for approximate fully dynamic matching in both the online and offline settings. 1. We give a randomized algorithm that with high probability maintains a $(1-\epsilon)$-approximate bipartite matching and $(1+\epsilon)$-approximate vertex cover in fully dynamic graphs, in amortized $O(\epsilon^{-O(1)} \frac{n}{2^{\Omega(\sqrt{\log n})}})$ update time. Our algorithm leverages fast algorithms for $\mathsf{OMv}$ due to Larsen-Williams [SODA 2017]. 2. We give a randomized offline algorithm for $(1-\epsilon)$-approximate maximum matching with amortized runtime $O(n^{.58}\epsilon^{-O(1)})$ by using fast matrix multiplication, significantly improving over the runtimes achieved via online algorithms. We also give an offline algorithm that maintains a $(1+\epsilon)$-approximate vertex cover in amortized $O(n^{.723}\epsilon^{-O(1)})$ time.