FPT Algorithms to Enumerate and Count Acyclic and Totally Cyclic Orientations

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Abstract

In this paper, we deal with counting and enumerating problems for two types of graph orientations: acyclic and totally cyclic orientations. Counting is known to be #P-hard for both of them. To circumvent this issue, we propose Fixed Parameter Tractable (FPT) algorithms. For the enumeration task, we construct a Binary Decision Diagram (BDD) to represent all orientations of the two kinds, instead of explicitly enumerating them. We prove that the running time of this construction is bounded by O*(2pw2/4+o(pw2)) with respect to the pathwidth pw. We then develop faster FPT algorithms to count acyclic and totally acyclic orientations, running in O*(2bw2/2+o(bw2)) time, where bw denotes the branch-width of the given graph. These counting algorithms are obtained by applying the observations in our enumerating algorithm to branch decomposition.

Keywords

Acyclic Orientations
Totally Cyclic Orientations
Parameterized Algorithms
FPT Algorithms
Path-width
Branch-width
Binary Decision Diagram

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