Local Computation Algorithms for Maximum Matching: New Lower Bounds

TL;DR

This paper establishes lower bounds on the time complexity of local computation algorithms for maximum matching, showing that achieving near-optimal solutions or estimates requires super-polynomial time in certain parameters, thus resolving a long-standing open problem.

Contribution

It proves new lower bounds on the time complexity of LCAs for maximum matching, demonstrating inherent computational hardness for approximate solutions.

Findings

Any LCA approximating maximum matching within additive   n requires super-polynomial time.

Estimating maximum matching size within additive   n also requires super-polynomial time.

Negatively resolves a decade-old open problem on sublinear algorithms for maximum matching estimation.

Abstract

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.