Tight Pair Query Lower Bounds for Matching and Earth Mover's Distance

2025-10-18

AI Summary

This paper addresses two fundamental problems: (1) the query complexity of estimating maximum matching size in an $n$-vertex graph via adjacency matrix queries, and (2) lower bounds on the computational complexity of Earth Mover’s Distance (EMD) estimation. Technically, the authors establish the first superlinear lower bound $Omega(n^{2-delta})$ for maximum matching estimation in the adjacency matrix query model, for any $delta > 0$, resolving a central open problem posed at STOC’24 and closing a long-standing gap between prior upper and lower bounds. Their proof integrates adversarial query complexity analysis, intricate graph-theoretic constructions, and a novel reduction from EMD estimation under the $(1,2)$-metric space assumption. As a consequence, the FOCS’23 algorithm for maximum matching estimation is shown to be asymptotically optimal, and the STOC’24 EMD estimation algorithm is proven to admit no substantial improvement in this setting.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Learning to Search
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

How many adjacency matrix queries (also known as pair queries) are required to estimate the size of a maximum matching in an $n$-vertex graph $G$? We study this fundamental question in this paper. On the upper bound side, an algorithm of Bhattacharya, Kiss, and Saranurak [FOCS'23] gives an estimate that is within $εn$ of the right bound with $n^{2-Ω_ε(1)}$ queries, which is subquadratic in $n$ (and thus sublinear in the matrix size) for any fixed $ε> 0$. On the lower bound side, while there has been a lot of progress in the adjacency list model, no non-trivial lower bound has been established for algorithms with adjacency matrix query access. In particular, the only known lower bound is a folklore bound of $Ω(n)$, leaving a huge gap. In this paper, we present the first superlinear in $n$ lower bound for this problem. In fact, we close the gap mentioned above entirely by showing that the algorithm of [BKS'23] is optimal. Formally, we prove that for any fixed $δ> 0$, there is a fixed $ε> 0$ such that an estimate that is within $εn$ of the true bound requires $Ω(n^{2-δ})$ adjacency matrix queries. Our lower bound also has strong implications for estimating the earth mover's distance between distributions. For this problem, Beretta and Rubinstein [STOC'24] gave an $n^{2-Ω_ε(1)}$ time algorithm that obtains an additive $ε$-approximation and works for any distance function. Whether this can be improved generally, or even for metric spaces, had remained open. Our lower bound rules out the possibility of any improvements over this bound, even under the strong assumption that the underlying distances are in a (1, 2)-metric.

Problem

• Establishing superlinear query lower bounds for maximum matching estimation
• Determining optimal adjacency matrix queries for graph matching algorithms
• Proving tight lower bounds for Earth Mover's Distance approximation

Innovation

• Established first superlinear query lower bound
• Proved optimality of existing subquadratic algorithm
• Extended lower bounds to Earth Mover's Distance