TY - GEN

T1 - An Improved Approximation Algorithm for Dynamic Minimum Linear Arrangement

AU - Bienkowski, Marcin

AU - Even, Guy

N1 - Publisher Copyright:
© Marcin Bienkowski and Guy Even; licensed under Creative Commons License CC-BY 4.0.

PY - 2024/3

Y1 - 2024/3

N2 - 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.

AB - 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.

KW - Graph Problems

KW - Minimum Linear Arrangement

KW - Optimization Problems

KW - approximation Algorithms

KW - dynamic Variant

UR - http://www.scopus.com/inward/record.url?scp=85187777246&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.STACS.2024.15

DO - 10.4230/LIPIcs.STACS.2024.15

M3 - ???researchoutput.researchoutputtypes.contributiontobookanthology.conference???

AN - SCOPUS:85187777246

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 41st International Symposium on Theoretical Aspects of Computer Science, STACS 2024

A2 - Beyersdorff, Olaf

A2 - Kante, Mamadou Moustapha

A2 - Kupferman, Orna

A2 - Lokshtanov, Daniel

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

T2 - 41st International Symposium on Theoretical Aspects of Computer Science, STACS 2024

Y2 - 12 March 2024 through 14 March 2024

ER -