Edge-weighted Matching in the Dark

2025-07-25

AI Summary

This paper studies the oblivious matching problem on edge-weighted bipartite graphs without distributional assumptions, aiming to break the long-standing $1-1/e approx 0.632$ competitive ratio barrier. We propose the first algorithmic framework based on parameterized quadratic ranking, where dual function parameterization and distribution-free analysis reduce algorithm design to a tractable quadratic programming problem. Theoretically, we prove that our framework satisfies natural monotonicity and feasibility constraints; optimal parameters are obtained via numerical optimization. Experiments demonstrate a competitive ratio of $0.659$, surpassing the previous best $0.641$—the first substantive improvement over the classical bound in the distribution-free setting. This advances the theoretical frontier of query-commit matching.

Technology Category

• Search and Optimization: Distributed Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Learning Preferences or Rankings

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We present a $0.659$-competitive Quadratic Ranking algorithm for the Oblivious Bipartite Matching problem, a distribution-free version of Query-Commit Matching. This result breaks the $1-frac{1}{e}$ barrier, addressing an open question raised by Tang, Wu, and Zhang (JACM 2023). Moreover, the competitive ratio of this distribution-free algorithm improves the best existing $0.641$ ratio for Query-Commit Matching achieved by the distribution-dependent algorithm of Chen, Huang, Li, and Tang (SODA 2025). Quadratic Ranking is a novel variant of the classic Ranking algorithm. We parameterize the algorithm with two functions, and let two key expressions in the definition and analysis of the algorithm be quadratic forms of the two functions. We show that the quadratic forms are the unique choices that satisfy a set of natural properties. Further, they allow us to optimize the choice of the two functions using powerful quadratic programming solvers.

Problem

• Develops a competitive algorithm for Oblivious Bipartite Matching
• Breaks the 1-1/e barrier in competitive ratio
• Improves upon existing distribution-dependent algorithm performance

Innovation

• Quadratic Ranking algorithm for matching
• Breaks 1-1/e barrier competitively
• Uses quadratic programming solvers