Optimal Rounding for Two-Stage Bipartite Matching

TL;DR

This paper develops polynomial-time approximation algorithms for two-stage bipartite matching problems, achieving near-optimal guarantees by leveraging negative association properties and fractional rounding techniques.

Contribution

It introduces new approximation algorithms with guarantees matching upper bounds, improving previous results for two-stage bipartite matching with known distributions.

Findings

Achieves 7/8 approximation for vertex-weighted graphs.

Achieves approximately 0.828 approximation for edge-weighted graphs.

Extends results to settings with only sample access to distributions.

Abstract

We study two-stage bipartite matching, in which the edges of a bipartite graph on vertices $(B_1 \cup B_2, I)$ are revealed in two batches. In stage one, a matching must be selected from among revealed edges $E \subseteq B_1 \times I$. In stage two, edges $E^\theta \subseteq B_2 \times I$ are sampled from a known distribution, and a second matching must be selected between $B_2$ and unmatched vertices in $I$. The objective is to maximize the total weight of the combined matching. We design polynomial-time approximations to the optimum online algorithm, achieving guarantees of $7/8$ for vertex-weighted graphs and $2\sqrt{2}-2 \approx 0.828$ for edge-weighted graphs under arbitrary distributions. Both approximation ratios match known upper bounds on the integrality gap of the natural fractional relaxation, improving upon the best-known approximation of 0.767 by Feng, Niazadeh, and Saberi for unweighted graphs whose second batch consists of independently arriving nodes. Our results are obtained via an algorithm that rounds a fractional matching revealed in two stages, aiming to match offline nodes (respectively, edges) with probability proportional to their fractional weights, up to a constant-factor loss. We leverage negative association (NA) among offline node availabilities -- a property induced by dependent rounding -- to derive new lower bounds on the expected size of the maximum weight matching in random graphs where one side is realized via NA binary random variables. Moreover, we extend these results to settings where we have only sample access to the distribution. In particular, $\text{poly}(n,\epsilon^{-1})$ samples suffice to obtain an additive loss of $\epsilon$ in the approximation ratio for the vertex-weighted problem; a similar bound holds for the edge-weighted problem with an additional (unavoidable) dependence on the scale of edge weights.