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Parameterized algorithms for feedback set problems and their duals in tournaments. (English) Zbl 1086.68105
Summary: The parameterized feedback vertex (arc) set problem is to find whether there are $$k$$ vertices (arcs) in a given graph whose removal makes the graph acyclic. The parameterized complexity of this problem in general directed graphs is a long standing open problem. We investigate the problems on tournaments, a well studied class of directed graphs. We consider both weighted and unweighted versions.
We also address the parametric dual problems which are also natural optimization problems. We show that they are fixed parameter tractable not just in tournaments but in oriented directed graphs (where there is at most one directed arc between a pair of vertices). More specifically, the dual problem we show fixed parameter tractable are: Given an oriented directed graph, is there a subset of $$k$$ vertices (arcs) that forms an acyclic directed subgraph of the graph?
Our main results include:
$$\bullet$$ an $$O((2.4143)^{k}n^{\omega})^{1}$$ algorithm for weighted feedback vertex set problem, and an $$O((2.415)^{k}n^{\omega })$$ algorithm for weighted feedback arc set problem in tournaments;
$$\bullet$$ an $$O((e2^{k}/k)^{k}k^{2}+\min \{m \mathrm{lg} n,n^{2}\})$$ algorithm for the dual of feedback vertex set problem (maximum vertex induced acyclic graph) in oriented directed graphs, and an $$O(4^{k}k+m)$$ algorithm for the dual of feedback arc set problem (maximum arc induced acyclic graph) in general directed graphs.
We also show that the dual of feedback vertex set is $$W[1]$$-hard in general directed graphs and the feedback arc set problem is fixed parameter tractable in dense directed graphs. Our results are the first non-trivial results for these problems.

MSC:
 68R10 Graph theory (including graph drawing) in computer science 05C20 Directed graphs (digraphs), tournaments 05C85 Graph algorithms (graph-theoretic aspects) 68Q25 Analysis of algorithms and problem complexity
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References:
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