Optimal Rounding for Two-Stage Bipartite Matching

2025-10-22

AI Summary

This paper studies the two-stage online bipartite matching problem: edges arrive in two batches—first, a set of known edges arrives, and a matching is selected; second, new edges are sampled from a known weight distribution, and another matching is chosen—aiming to maximize the total expected weight across both stages. We propose an optimal rounding algorithm based on the natural fractional relaxation. For vertex-weighted and edge-weighted variants, it achieves tight approximation ratios of $7/8$ and $2sqrt{2}-2 approx 0.828$, respectively—the first to match the corresponding integrality gap upper bounds. Key technical contributions include: (i) a two-stage rounding scheme guided by the fractional solution; (ii) negative association (NA) analysis to control dependency; (iii) a novel lower bound on the expected maximum-weight matching in random bipartite graphs; and (iv) a distribution-approximation method requiring only $mathrm{poly}(n,varepsilon^{-1})$ samples. The algorithm runs in polynomial time and improves significantly over the prior best ratio of $0.767$.

Technology Category

• Machine Learning: Online Learning & Bandits
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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 $E subseteq B_1 imes I$. In stage two, edges $E^θsubseteq B_2 imes I$ 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 $2sqrt{2}-2 approx 0.828$ 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, $ ext{poly}(n,ε^{-1})$ samples suffice to obtain an additive loss of $ε$ 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.

Problem

• Optimizing two-stage bipartite matching with sequentially revealed edges
• Designing polynomial-time approximations for maximum weight matching
• Developing rounding algorithms for fractional matching under uncertainty

Innovation

• Polynomial-time approximations for two-stage bipartite matching
• Dependent rounding with negative association properties
• Sample-based distribution approximation with additive loss guarantees