The Min-Cost Matching with Concave Delays Problem

TL;DR

This paper introduces competitive online algorithms for min-cost perfect matching with concave delays, extending from single location to metric spaces and bichromatic cases, achieving constant and logarithmic competitive ratios.

Contribution

It provides the first $O(1)$-competitive algorithm for single location and extends it to $O( ext{log} n)$ competitiveness in metric spaces with concave delays, including bichromatic matching.

Findings

Single location algorithm is $O(1)$-competitive.

Extended algorithms achieve $O( ext{log} n)$ competitiveness in metric spaces.

Algorithms adapted for bichromatic matching with similar competitiveness.

Abstract

We consider the problem of online min-cost perfect matching with concave delays. We begin with the single location variant. Specifically, requests arrive in an online fashion at a single location. The algorithm must then choose between matching a pair of requests or delaying them to be matched later on. The cost is defined by a concave function on the delay. Given linear or even convex delay functions, matching any two available requests is trivially optimal. However, this does not extend to concave delays. We solve this by providing an $O(1)$-competitive algorithm that is defined through a series of delay counters. Thereafter we consider the problem given an underlying $n$-points metric. The cost of a matching is then defined as the connection cost (as defined by the metric) plus the delay cost. Given linear delays, this problem was introduced by Emek et al. and dubbed the Min-cost perfect matching with linear delays (MPMD) problem. Liu et al. considered convex delays and subsequently asked whether there exists a solution with small competitive ratio given concave delays. We show this to be true by extending our single location algorithm and proving $O(\log n)$ competitiveness. Finally, we turn our focus to the bichromatic case, wherein requests have polarities and only opposite polarities may be matched. We show how to alter our former algorithms to again achieve $O(1)$ and $O(\log n)$ competitiveness for the single location and for the metric case.