Maximum Weight b-Matchings in Random-Order Streams

TL;DR

This paper introduces a new streaming algorithm for maximum weight b-matching in random order streams, achieving a better approximation ratio than previous methods, especially for small weights, using a novel edge-degree constrained subgraph technique.

Contribution

The paper presents the first algorithm to beat a 2-approximation for weighted b-matching in semi-streaming, generalizing prior work and employing a generalized edge-degree constrained subgraph approach.

Findings

Achieves a $2 - rac{1}{2W} + ext{epsilon}$ approximation ratio.

Uses memory proportional to the size of the optimal matching and polynomial factors.

Improves upon previous approximation ratios for small weights.

Abstract

We consider the maximum weight $b$-matching problem in the random-order semi-streaming model. Assuming all weights are small integers drawn from $[1,W]$, we present a $2 - \frac{1}{2W} + \varepsilon$ approximation algorithm, using a memory of $O(\max(|M_G|, n) \cdot poly(\log(m),W,1/\varepsilon))$, where $|M_G|$ denotes the cardinality of the optimal matching. Our result generalizes that of Bernstein [Bernstein, 2015], which achieves a $3/2 + \varepsilon$ approximation for the maximum cardinality simple matching. When $W$ is small, our result also improves upon that of Gamlath et al. [Gamlath et al., 2019], which obtains a $2 - \delta$ approximation (for some small constant $\delta \sim 10^{-17}$) for the maximum weight simple matching. In particular, for the weighted $b$-matching problem, ours is the first result beating the approximation ratio of $2$. Our technique hinges on a generalized weighted version of edge-degree constrained subgraphs, originally developed by Bernstein and Stein [Bernstein and Stein, 2015]. Such a subgraph has bounded vertex degree (hence uses only a small number of edges), and can be easily computed. The fact that it contains a $2 - \frac{1}{2W} + \varepsilon$ approximation of the maximum weight matching is proved using the classical K\H{o}nig-Egerv\'ary's duality theorem.