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

TL;DR

This work resolves a long-standing gap in the dense-graph (adjacency-matrix) model for estimating the maximum matching size: it proves a near-tight lower bound of $\Omega(n^{2-\delta})$ queries for any fixed $\delta>0$ to achieve an additive error of $\varepsilon n$, effectively matching the best known subquadratic upper bound by Bhattacharya, Kiss, and Saranurak. The authors develop a novel hierarchical multigraph construction with ground, labelled-ground, and pseudo edges, plus a parallel pseudo-edge mechanism that ensures non-edges carry limited information, enabling a strong coupling argument that cannot distinguish YES/NO instances with sublinear queries. They also translate the max-matching lower bound into a lower bound for additive $\varepsilon$-approximation of Earth Mover's Distance under a $(1,2)$-metric, showing that subquadratic-time improvements in this setting are unlikely. The results have broader implications for sublinear graph problems and dynamic algorithms, and they close the gap between upper and lower bounds in this fundamental model by establishing optimality of the subquadratic rate. Overall, the paper introduces a sophisticated probabilistic construction and coupling framework to prove hardness in the dense adjacency-matrix model, with concrete consequences for EMD and related sublinear problems.

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.

Figures (2)

• Figure 1: An illustration of ground, pseudo, and real edges. As shown in the figure, pseudo edges form a subset of ground edges, while real edges are a subset of pseudo edges.
• Figure 2: An illustration of the levelled construction for a gadget on level $L-1$. The ground edges and pseudo edges do not depend on the vertex labels. Level-$L$ pseudo edges are a subset of the ground edges, and level-$\ell$ pseudo edges are a subset of the Level-$(\ell+1)$ pseudo edges. The level-$\ell$ Laballed-ground edges are determined based on the vertex labels and the level-$\ell$ gadgets between them, here the $X$-$Y$ gadget on level $L-1$. The level-$\ell$ real edges are the intersection of level-$\ell$ pseudo edges and level-$\ell$ labelled-ground edges.

Theorems & Definitions (153)

• Theorem 1
• Proposition 3.1: erdHos1964randomerdHos1966existencefriezepittel
• Proposition 3.2: Yao's Lemma. Theorem 3 of Yao77
• Proposition 3.3: Chernoff Bound
• Definition 4.1: EMD
• Theorem 2
• Remark 1
• Claim 4.2
• proof
• Lemma 5.1