Regularized Box-Simplex Games and Dynamic Decremental Bipartite Matching
Arun Jambulapati, Yujia Jin, Aaron Sidford, Kevin Tian

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) -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…
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