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
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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 -