Online Dependent Rounding Schemes for Bipartite Matchings, with Applications

TL;DR

This paper introduces the first online dependent rounding schemes for bipartite matchings that surpass the traditional 1-1/e ratio, enabling improved online algorithms in various applications.

Contribution

It presents the first generic online dependent rounding schemes for bipartite matchings with no restrictions on fractional inputs, achieving ratios above 0.646 and 0.652.

Findings

Achieved rounding ratios of 0.646 for b-matchings and 0.652 for simple matchings.

Broken the 1-1/e barrier in online matching problems.

Enabled improved algorithms for online edge coloring and stochastic optimization.

Abstract

We introduce the abstract problem of rounding an unknown fractional bipartite $b$-matching $\bf{x}$ revealed online (e.g., output by an online fractional algorithm), exposed node-by-node on~one~side. The objective is to maximize the \emph{rounding ratio} of the output matching $M$, which is the minimum over all fractional $b$-matchings $\bf{x}$, and edges $e$, of the ratio $\Pr[e\in M]/x_e$. In analogy with the highly influential offline dependent rounding schemes of Gandhi et al.~(FOCS'02, JACM'06), we refer to such algorithms as \emph{online dependent rounding schemes} (ODRSes). This problem, with additional restrictions on the possible inputs $\bf{x}$, has played a key role in recent developments in online computing. We provide the first generic $b$-matching ODRSes that impose no restrictions on $\bf{x}$. Specifically, we provide ODRSes with rounding ratios of $0.646$ and $0.652$ for $b$-matchings and simple matchings, respectively. This breaks the natural barrier of $1-1/e$, prevalent for online matching problems, and numerous online problems more broadly. Using our ODRSes, we provide a number of algorithms with similar better-than-$(1-1/e)$ ratios for several problems in online edge coloring, stochastic optimization, and more. Our techniques, which have already found applications in several follow-up works (Patel and Wajc SODA'24, Blikstad et al.~SODA'25, Braverman et al.~SODA'25, and Aouad et al.~2024), include periodic use of \emph{offline} contention resolution schemes (in online algorithm design), grouping nodes, and a new scaling method which we call \emph{group discount and individual markup}.