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.

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Motivated by applications in the gig economy, we study approximation algorithms for a sequential pricing problem. The input is a bipartite graph G = (I, J, E) between individuals I and jobs J. The platform has a value of v j for matching job j to an individual worker. In order to find a matching, the platform can consider the edges (ij) ∈ E in any order and make a one-time take-it-or-leave-it offer of a price π ij = w of its choosing to i for completing j. The worker accepts the offer with a known probability p ijw ; in this case the job and the worker are irrevocably matched. What is the best way to make offers to maximize revenue and/or social welfare?The optimal algorithm is known to be NP-hard to compute (even if there is only a single job). With this in mind, we design efficient approximations to the optimal policy via a new Random-Order Online Contention Resolution Scheme (RO-OCRS) for matching. Our main result is a 0.456-balanced RO-OCRS in bipartite graphs and a 0.45-balanced RO-OCRS in general graphs. These algorithms improve on the recent bound of 1 2 (1 − e −2 ) ≈ 0.432 of [BGMS21], and improve on the best known lower bounds for the correlation gap of matching, despite applying to a significantly more restrictive setting. As a consequence of our OCRS results, we obtain a 0.456approximate algorithm for the sequential pricing problem. We further extend our results to settings where workers can only be contacted a limited number of times, and show how to achieve improved results for this problem, via improved algorithms for the well-studied stochastic probing problem.