TY - GEN

T1 - Metrical service systems with transformations

AU - Bubeck, Sébastien

AU - Buchbinder, Niv

AU - Coester, Christian

AU - Sellke, Mark

N1 - Publisher Copyright:
© Sébastien Bubeck, Niv Buchbinder, Christian Coester, and Mark Sellke.

PY - 2021/2/1

Y1 - 2021/2/1

N2 - 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.

AB - 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.

KW - Competitive analysis

KW - K-taxi

KW - Metrical task systems

KW - Online algorithms

UR - http://www.scopus.com/inward/record.url?scp=85115201977&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.ITCS.2021.21

DO - 10.4230/LIPIcs.ITCS.2021.21

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AN - SCOPUS:85115201977

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 12th Innovations in Theoretical Computer Science Conference, ITCS 2021

A2 - Lee, James R.

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

Y2 - 6 January 2021 through 8 January 2021

ER -