TY - GEN

T1 - The min-cost matching with concave delays problem

AU - Azar, Yossi

AU - Ren, Runtian

AU - Vainstein, Danny

N1 - Publisher Copyright:
Copyright © 2021 by SIAM

PY - 2021

Y1 - 2021

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85105359297&partnerID=8YFLogxK

U2 - 10.1137/1.9781611976465.20

DO - 10.1137/1.9781611976465.20

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AN - SCOPUS:85105359297

T3 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

SP - 301

EP - 320

BT - Proceedings of the Thirty-Second Annual ACM-SIAM Symposium on Discrete Algorithms

A2 - Marx, Daniel

PB - Society for Industrial and Applied Mathematics (SIAM)

Y2 - 10 January 2021 through 13 January 2021

ER -