Regularized Box-Simplex Games and Dynamic Decremental Bipartite Matching

TL;DR

This paper introduces efficient algorithms for solving regularized box-simplex games, which are then used to develop faster dynamic algorithms for decremental bipartite matching, improving previous runtimes significantly.

Contribution

The paper presents a novel reduction framework connecting regularized box-simplex game solvers to dynamic bipartite matching algorithms, achieving near-linear time solutions with improved runtimes.

Findings

New near-linear time solvers for regularized box-simplex games.

A reduction framework translating game solutions into dynamic matching algorithms.

Improved total runtime for dynamic decremental bipartite matching to $ ilde{O}(m imes ext{poly}(rac{1}{ ext{epsilon}}))$.

Abstract

Box-simplex games are a family of bilinear minimax objectives which encapsulate graph-structured problems such as maximum flow [She17], optimal transport [JST19], and bipartite matching [AJJ+22]. We develop efficient near-linear time, high-accuracy solvers for regularized variants of these games. Beyond the immediate applications of such solvers for computing Sinkhorn distances, a prominent tool in machine learning, we show that these solvers can be used to obtain improved running times for maintaining a (fractional) $\epsilon$-approximate maximum matching in a dynamic decremental bipartite graph against an adaptive adversary. We give a generic framework which reduces this dynamic matching problem to solving regularized graph-structured optimization problems to high accuracy. Through our reduction framework, our regularized box-simplex game solver implies a new algorithm for dynamic decremental bipartite matching in total time $\tilde{O}(m \cdot \epsilon^{-3})$, from an initial graph with $m$ edges and $n$ nodes. We further show how to use recent advances in flow optimization [CKL+22] to improve our runtime to $m^{1 + o(1)} \cdot \epsilon^{-2}$, thereby demonstrating the versatility of our reduction-based approach. These results improve upon the previous best runtime of $\tilde{O}(m \cdot \epsilon^{-4})$ [BGS20] and illustrate the utility of using regularized optimization problem solvers for designing dynamic algorithms.