We study local computation algorithms (LCA) for maximum matching. An LCA does not return its output entirely, but reveals parts of it upon query. For matchings, each query is a vertex $v$; the LCA should return whether $v$ is matched -- and if so to which neighbor -- while spending a small time per query. In this paper, we prove that any LCA that computes a matching that is at most an additive of $\epsilon n$ smaller than the maximum matching in $n$-vertex graphs of maximum degree $\Delta$ must take at least $\Delta^{\Omega(1/\varepsilon)}$ time. This comes close to the existing upper bounds that take $(\Delta/\epsilon)^{O(1/\epsilon^2)} polylog(n)$ time. In terms of sublinear time algorithms, our techniques imply that any algorithm that estimates the size of maximum matching up to an additive error of $\epsilon n$ must take $\Delta^{\Omega(1/\epsilon)}$ time. This negatively resolves a decade old open problem of the area (see Open Problem 39 of sublinear.info) on whether such estimates can be achieved in $poly(\Delta/\epsilon)$ time.

翻译：我们研究最大匹配的局部计算算法（LCA）。LCA不一次性返回完整输出，而是根据查询逐步揭示部分内容。对于匹配问题，每个查询对应一个顶点$v$；LCA需返回该顶点是否被匹配——若匹配，还需指出其匹配的邻居——同时保证每次查询的耗时极小。本文证明：对于最大度为$\Delta$的$n$顶点图，任何能够计算与最大匹配相差至多$\epsilon n$（加法误差）的匹配的LCA，其单次查询时间至少为$\Delta^{\Omega(1/\varepsilon)}$。这一结果逼近现有上界$(\Delta/\epsilon)^{O(1/\epsilon^2)} polylog(n)$。在亚线性时间算法方面，我们的技术表明：任何能够以加法误差$\epsilon n$估计最大匹配大小的算法，其运行时间至少为$\Delta^{\Omega(1/\epsilon)}$。这否定了该领域一个悬而未决十余年的公开问题（见sublinear.info的开放问题39），即是否可以在$poly(\Delta/\epsilon)$时间内完成此类估计。