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)
T2 - 32nd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2021
Y2 - 10 January 2021 through 13 January 2021
ER -