We present improved parameterized algorithms for the feedback vertex set problem on both unweighted and weighted graphs. Both algorithms run in time O(5 k kn 2 ). The algorithms construct a feedback vertex set of size at most k (in the weighted case this set is of minimum weight among the feedback vertex sets of size at most k) in a given graph G of n vertices, or report that no such feedback vertex set exists in G.
We show that the treewidth and the minimum fill-in of an n-vertex graph can be computed in time O(1.8899 n ). Our results are based on combinatorial proofs that an n-vertex graph has O(1.7087 n ) minimal separators and O(1.8135 n ) potential maximal cliques. We also show that for the class of AT-free graphs the running time of our algorithms can be reduced to O(1.4142 n ).
We obtain an algorithmic meta-theorem for the following optimization problem. Let ϕ be a Counting Monadic Second Order Logic (CMSO) formula and t ≥ 0 be an integer. For a given graph G = (V, E), the task is to maximize |X| subject to the following: there is a set
Capacitated versions of Dominating Set and Vertex Cover 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 the 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 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 W[1]-hard when parameterized by treewidth. In fact, Capacitated Dominating Set is W[1]-hard when parameterized by both treewidth and solution size k of the capacitated dominating set.• Capacitated Vertex Cover is W[1]-hard when parameterized by treewidth.• Capacitated Vertex Cover can be solved in time 2 O(tw log k) 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 2 O(k log k) n O(1) . This improves the earlier algorithm of Guo et al.[15] running in time O(1.2We would also like to point out that our W[1]-hardness result for Capacitated Vertex Cover, when parameterized by treewidth, makes it (to the best of our knowledge) the first known "subset problem" which has turned out to be fixed parameter tractable when parameterized by solution size but W[1]-hard when parameterized by treewidth.
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