The Matching Number Characterizes the Query Complexity of Edge Counting and Degree Moment Estimation

Abstract

Consider an unknown graph $G = (V,E)$ with vertex degrees $d_v$, accessed only by sampling uniformly random edges and querying the degrees of their endpoints. We study the query complexity of estimating the number of edges $m$ and the degree moments $M_s = \sum_{v \in V} (d_v)^s$ for fixed integers $s \ge 2$. In this model, [Beretta and Tětek, TALG 2024] estimate $m$ within relative error $\varepsilon$ using $O(\log n/\varepsilon^2+\sqrt n/\varepsilon)$ expected queries without knowing the number of vertices $n$, and [Bhattacharya and Pradhan, CIAC 2025] estimate $M_s$ using $O(n^{1-1/s}/\varepsilon^2)$ queries when $n$ is given. We show that the query complexity of both problems is governed by the matching number $ν$ of $G$ (the maximum number of pairwise disjoint edges) rather than the number of vertices $n$, and that the dependence on $\varepsilon$ for degree moments can be polynomially improved. Our algorithms rest on a simple observation: the endpoints of a maximum matching cover every edge, so a random endpoint of a random edge falls on this set of $2ν$ vertices with probability at least $1/2$. For every $\varepsilon\in(0,1]$, we give advice-free algorithms that estimate $m$ and $M_s$ within relative error $\varepsilon$ with probability at least $2/3$, without knowledge of $n$, $ν$, or other graph parameters. Specifically, our algorithms use $O(\min\{ν/\varepsilon, 1/\varepsilon^2+\sqrtν/\varepsilon\})$ expected queries for $m$ and $O(\min\{ν^{1-1/s}/\varepsilon^{2/s}, ν^{1-1/(2s-1)}/\varepsilon^{2/(2s-1)}\})$ for $M_s$. Finally, our lower bounds show that for every $k \ge 2$ and sufficiently small $\varepsilon$, these query complexities are optimal up to constant factors over graphs with matching number at most $k$.