Abstract
A kswap W for a vertex cover S of a graph G is a vertex set of size at most k such that S' = (S โงต W) โช (W โงต S), the symmetric difference of S and W, is a vertex cover of G. If S' < S, then W is improving. In LSVertex Cover, one is given a vertex cover S of a graph G and wants to know if there is an improving kswap for S in G. In applications of LSVertex Cover, k is a very small parameter that can be set by a user to determine the tradeoff between running time and solution quality. Consequently, k can be considered to be a constant. Motivated by this and the fact that LSVertex Cover is W[1]hard with respect to k, we aim for algorithms with running time ๐^f(k) โ
n^๐ช(1) where ๐ is a structural graph parameter upperbounded by n. We say that such a running time grows mildly with respect to ๐ and strongly with respect to k. We obtain algorithms with such a running time for ๐ being the hindex of G, the treewidth of G, or the modularwidth of G. In addition, we consider a novel parameter, the maximum degree over all quotient graphs in a modular decomposition of G. Moreover, we adapt these algorithms to the more general problem where each vertex is assigned a weight and where we want to find a dimproving kswap, that is, a kswap which decreases the weight of the vertex cover by at least d.
BibTeX  Entry
@InProceedings{komusiewicz_et_al:LIPIcs.IPEC.2022.20,
author = {Komusiewicz, Christian and Morawietz, Nils},
title = {{Parameterized Local Search for Vertex Cover: When Only the Search Radius Is Crucial}},
booktitle = {17th International Symposium on Parameterized and Exact Computation (IPEC 2022)},
pages = {20:120:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959772600},
ISSN = {18688969},
year = {2022},
volume = {249},
editor = {Dell, Holger and Nederlof, Jesper},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2022/17376},
URN = {urn:nbn:de:0030drops173764},
doi = {10.4230/LIPIcs.IPEC.2022.20},
annote = {Keywords: Local Search, Structural parameterization, Fixedparameter tractability}
}
Keywords: 

Local Search, Structural parameterization, Fixedparameter tractability 
Collection: 

17th International Symposium on Parameterized and Exact Computation (IPEC 2022) 
Issue Date: 

2022 
Date of publication: 

14.12.2022 