Metrical service systems with transformations

Sébastien Bubeck, Niv Buchbinder, Christian Coester, Mark Sellke

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

2 Scopus citations


We consider a generalization of the fundamental online metrical service systems (MSS) problem where the feasible region can be transformed between requests. In this problem, which we call T-MSS, an algorithm maintains a point in a metric space and has to serve a sequence of requests. Each request is a map (transformation) ft : At → Bt between subsets At and Bt of the metric space. To serve it, the algorithm has to go to a point at ∈ At, paying the distance from its previous position. Then, the transformation is applied, modifying the algorithm’s state to ft(at). Such transformations can model, e.g., changes to the environment that are outside of an algorithm’s control, and we therefore do not charge any additional cost to the algorithm when the transformation is applied. The transformations also allow to model requests occurring in the k-taxi problem. We show that for α-Lipschitz transformations, the competitive ratio is Θ(α)n-2 on n-point metrics. Here, the upper bound is achieved by a deterministic algorithm and the lower bound holds even for randomized algorithms. For the k-taxi problem, we prove a competitive ratio of Õ((nlog k)2). For chasing convex bodies, we show that even with contracting transformations no competitive algorithm exists. The problem T-MSS has a striking connection to the following deep mathematical question: Given a finite metric space M, what is the required cardinality of an extension M ⊇ M where each partial isometry on M extends to an automorphism? We give partial answers for special cases.

Original languageEnglish
Title of host publication12th Innovations in Theoretical Computer Science Conference, ITCS 2021
EditorsJames R. Lee
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959771771
StatePublished - 1 Feb 2021
Event12th Innovations in Theoretical Computer Science Conference, ITCS 2021 - Virtual, Online
Duration: 6 Jan 20218 Jan 2021

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
ISSN (Print)1868-8969


Conference12th Innovations in Theoretical Computer Science Conference, ITCS 2021
CityVirtual, Online


FundersFunder number
National Science Foundation
Microsoft Research
United States - Israel Binational Science Foundation2018352
Nederlandse Organisatie voor Wetenschappelijk Onderzoek639.023.812
Israel Science Foundation2233/19


    • Competitive analysis
    • K-taxi
    • Metrical task systems
    • Online algorithms


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