Edge-weighted Matching in the Dark

TL;DR

This paper introduces a new quadratic ranking algorithm for Oblivious Bipartite Matching that surpasses previous competitive ratios, breaking longstanding barriers and improving distribution-free matching performance.

Contribution

The paper develops a novel quadratic ranking algorithm for Oblivious Bipartite Matching, achieving a new competitive ratio of 0.659 and addressing an open problem in the field.

Findings

Achieved a 0.659-competitive ratio, surpassing the 1-1/e barrier.

Improved the best known distribution-free algorithm ratio from 0.641.

Introduced quadratic forms as a key component for optimizing the algorithm.

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.