An Exact Algorithm for Subset Feedback Vertex Set on Chordal Graphs
Given a graph G = (V,E) and a set S ⊆ V, a set U ⊆ V is a subset feedback vertex set of (G,S) if no cycle in G[V ∖ U] contains a vertex of S. The Subset Feedback Vertex Set problem takes as input G, S, and an integer k, and the question is whether (G,S) has a subset feedback vertex set of cardinality or weight at most k. Both the weighted and the unweighted versions of this problem are NP-complete on chordal graphs, even on their subclass split graphs. We give an algorithm with running time O(1.6708 n ) that enumerates all minimal subset feedback vertex sets on chordal graphs with n vertices. As a consequence, Subset Feedback Vertex Set can be solved in time O(1.6708 n ) on chordal graphs, both in the weighted and in the unweighted case. On arbitrary graphs, the fastest known algorithm for the problems has O(1.8638 n ) running time.
KeywordsExact Algorithm Chordal Graph Reduction Rule Graph Class Split Graph
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