Online Dependent Rounding Schemes for Bipartite Matchings, with Applications

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}.