Online Matching with Set and Concave Delays

TL;DR

This paper introduces the first non-clairvoyant algorithms for online min-cost perfect matching with set delay, focusing on size-based delays, and establishes fundamental lower bounds for algorithm competitiveness.

Contribution

It presents the first non-clairvoyant algorithms for MPMD-Set with size-based delay and provides lower bounds, advancing understanding of online matching with complex delay functions.

Findings

First non-clairvoyant algorithms for MPMD-Set with size-based delay.

Lower bounds of Ω(n) for deterministic and Ω(log n) for randomized algorithms.

An m-competitive deterministic algorithm for uniform concave delays in the clairvoyant setting.

Abstract

We initiate the study of online problems with set delay, where the delay cost at any given time is an arbitrary function of the set of pending requests. In particular, we study the online min-cost perfect matching with set delay (MPMD-Set) problem, which generalises the online min-cost perfect matching with delay (MPMD) problem introduced by Emek et al. (STOC 2016). In MPMD, $m$ requests arrive over time in a metric space of $n$ points. When a request arrives the algorithm must choose to either match or delay the request. The goal is to create a perfect matching of all requests while minimising the sum of distances between matched requests, and the total delay costs incurred by each of the requests. In contrast to previous work we study MPMD-Set in the non-clairvoyant setting, where the algorithm does not know the future delay costs. We first show no algorithm is competitive in $n$ or $m$. We then study the natural special case of size-based delay where the delay is a non-decreasing function of the number of unmatched requests. Our main result is the first non-clairvoyant algorithms for online min-cost perfect matching with size-based delay that are competitive in terms of $m$. In fact, these are the first non-clairvoyant algorithms for any variant of MPMD. Furthermore, we prove a lower bound of $\Omega(n)$ for any deterministic algorithm and $\Omega(\log n)$ for any randomised algorithm. These lower bounds also hold for clairvoyant algorithms. Finally, we also give an $m$-competititve deterministic algorithm for uniform concave delays in the clairvoyant setting.