An Improved Approximation Algorithm for Dynamic Minimum Linear Arrangement

Marcin Bienkowski*, Guy Even*

*Corresponding author for this work

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

Abstract

The dynamic offline linear arrangement problem deals with reordering n elements subject to a sequence of edge requests. The input consists of a sequence of m edges (i.e., unordered pairs of elements). The output is a sequence of permutations (i.e., bijective mapping of the elements to n equidistant points). In step t, the order of the elements is changed to the t-th permutation, and then the t-th request is served. The cost of the output consists of two parts per step: request cost and rearrangement cost. The former is the current distance between the endpoints of the request, while the latter is proportional to the number of adjacent element swaps required to move from one permutation to the consecutive permutation. The goal is to find a minimum cost solution. We present a deterministic O(log n log log n)-approximation algorithm for this problem, improving over a randomized O(log2 n)-approximation by Olver et al. [22]. Our algorithm is based on first solving spreading-metric LP relaxation on a time-expanded graph, applying a tree decomposition on the basis of the LP solution, and finally converting the tree decomposition to a sequence of permutations. The techniques we employ are general and have the potential to be useful for other dynamic graph optimization problems.

Original languageEnglish
Title of host publication41st International Symposium on Theoretical Aspects of Computer Science, STACS 2024
EditorsOlaf Beyersdorff, Mamadou Moustapha Kante, Orna Kupferman, Daniel Lokshtanov
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959773119
DOIs
StatePublished - Mar 2024
Event41st International Symposium on Theoretical Aspects of Computer Science, STACS 2024 - Clermont-Ferrand, France
Duration: 12 Mar 202414 Mar 2024

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume289
ISSN (Print)1868-8969

Conference

Conference41st International Symposium on Theoretical Aspects of Computer Science, STACS 2024
Country/TerritoryFrance
CityClermont-Ferrand
Period12/03/2414/03/24

Keywords

  • Graph Problems
  • Minimum Linear Arrangement
  • Optimization Problems
  • approximation Algorithms
  • dynamic Variant

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