Abstract
Capacitated versions of Vertex Cover and Dominating Set have been studied intensively in terms of polynomial time approximation algorithms. Although the problems Dominating Set and Vertex Cover have been subjected to considerable scrutiny in the parameterized complexity world, this is not true for their capacitated versions. Here we make an attempt to understand the behavior of the problems Capacitated Dominating Set and Capacitated Vertex Cover from the perspective of parameterized complexity.
The original, uncapacitated versions of these problems, Vertex Cover and Dominating Set, are known to be fixed parameter tractable when parameterized by a structure of the graph called the treewidth (tw). In this paper we show that the capacitated versions of these problems behave differently. Our results are:
– Capacitated Dominating Set is W1-hard when parameterized by treewidth. In fact, Capacitated Dominating Set is W1-hard when parameterized by both treewidth and solution size k of the capacitated dominating set.
– Capacitated Vertex Cover is W1-hard when parameterized by treewidth.
– Capacitated Vertex Cover can be solved in time 2O(tw logk) n O(1) where tw is the treewidth of the input graph and k is the solution size. As a corollary, we show that the weighted version of Capacitated Vertex Cover in general graphs can be solved in time 2O(k logk) n O(1). This improves the earlier algorithm of Guo et al. [15] running in time \(O(1.2^{k^2}+n^2)\). Capacitated Vertex Cover is, therefore, to our knowledge the first known “subset problem” which has turned out to be fixed parameter tractable when parameterized by solution size but W1-hard when parameterized by treewidth.
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Dom, M., Lokshtanov, D., Saurabh, S., Villanger, Y. (2008). Capacitated Domination and Covering: A Parameterized Perspective. In: Grohe, M., Niedermeier, R. (eds) Parameterized and Exact Computation. IWPEC 2008. Lecture Notes in Computer Science, vol 5018. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-79723-4_9
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