Beating Greedy Matching in Sublinear Time

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 εn smaller than the maximum matching in n-vertex graphs of maximum degree ∆ must take at least ∆ Ω(1/ε) time. This comes close to the existing upper bounds that takeIn terms of sublinear time algorithms, our techniques imply that any algorithm that estimates the size of maximum matching up to an additive error of εn must take ∆ Ω(1/ε) 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(∆/ε) time.

Sublinear Algorithms for TSP via Path Covers

“…This is a constant size program that can be easily solved; the solution is 19/10. 4 This completes the proof.…”

“…This remains an important open problem in the study of sublinear time maximum matching algorithms. See in particular [4]. This implies that short of a major result in the study maximum matchings in the sublinear time model, which have received significant attention in the literature (see [27,2,4,3,6] and references therein), our path cover algorithm has an optimal approximation ratio.…”

Dynamic Matching with Better-than-2 Approximation in Polylogarithmic Update Time

“…it has been a key open problem [BRRS23] whether non-trivial (1 + ǫ)-approximate matching algorithms in dense graphs exist in the sublinear model. Hence, it remains unclear whether an improved dynamic (1 + ǫ)-approximation algorithm is possible via this new connection or even possible at all.…”

“…Behnezhad et al [BRRS23] posted an open question about sublinear matching algorithms as follows "ruling out say a 1.01-approximation in n 2−Ω(1) time would also be extremely interesting." 4 .…”

“…To compare with Theorem 1.3, let us discuss only algorithms that use the adjacency matrix. Behnezhad et al [BRRS23] first broke the 2-approximation barrier by computing a (2 − Ω γ (1), o(n))-approximate matching in Õ(n 1+γ ) time. Then (3/2, ǫn)-approximation algorithms with n 2−Θ(ǫ 2 ) time were shown independently in [BKS23,BRR23].…”