Efficient approximations for the online dispersion problem

Jing Chen, Bo Li, Yingkai Li

Research output: Chapter in book / Conference proceedingConference article published in proceeding or bookAcademic researchpeer-review

Abstract

The dispersion problem has been widely studied in computational geometry and facility location, and is closely related to the packing problem. The goal is to locate n points (e.g., facilities or persons) in a κ-dimensional polytope, so that they are far away from each other and from the boundary of the polytope. In many real-world scenarios however, the points arrive and depart at different times, and decisions must be made without knowing future events. Therefore we study, for the first time in the literature, the online dispersion problem in Euclidean space. There are two natural objectives when time is involved: the all-time worst-case (ATWC) problem tries to maximize the minimum distance that ever appears at any time; and the cumulative distance (CD) problem tries to maximize the integral of the minimum distance throughout the whole time interval. Interestingly, the online problems are highly non-trivial even on a segment. For cumulative distance, this remains the case even when the problem is time-dependent but offline, with all the arriving and departure times given in advance. For the online ATWC problem on a segment, we construct a deterministic polynomial-time algorithm which is (2 ln 2+ϵ)-competitive, where ϵ > 0 can be arbitrarily small and the algorithm's running time is polynomial in 1/ϵ. We show this algorithm is actually optimal. For the same problem in a square, we provide a 1.591-competitive algorithm and a 1.183 lower-bound. Furthermore, for arbitrary κ-dimensional polytopes with κ ≥ 2, we provide a 2/1-ϵ-competitive algorithm and a 7/6 lower-bound. All our lower-bounds come from the structure of the online problems and hold even when computational complexity is not a concern. Interestingly, for the offline CD problem in arbitrary κ-dimensional polytopes, we provide a polynomial-time black-box reduction to the online ATWC problem, and the resulting competitive ratio increases by a factor of at most 2. Our techniques also apply to online dispersion problems with different boundary conditions.

Original languageEnglish
Title of host publication44th International Colloquium on Automata, Languages, and Programming, ICALP 2017
EditorsAnca Muscholl, Piotr Indyk, Fabian Kuhn, Ioannis Chatzigiannakis
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959770415
DOIs
Publication statusPublished - 1 Jul 2017
Externally publishedYes
Event44th International Colloquium on Automata, Languages, and Programming, ICALP 2017 - Warsaw, Poland
Duration: 10 Jul 201714 Jul 2017

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume80
ISSN (Print)1868-8969

Conference

Conference44th International Colloquium on Automata, Languages, and Programming, ICALP 2017
Country/TerritoryPoland
CityWarsaw
Period10/07/1714/07/17

Keywords

  • Competitive algorithms
  • Dispersion
  • Geometric optimization
  • Online algorithms
  • Packing

ASJC Scopus subject areas

  • Software

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