Breaking the Barrier 2^k for Subset Feedback Vertex Set in Chordal Graphs
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The Subset Feedback Vertex Set problem (SFVS), to delete k vertices from a given graph such that any vertex in a vertex subset (called a terminal set) is not in a cycle in the remaining graph, generalizes the famous Feedback Vertex Set problem and Multiway Cut problem. SFVS remains NP-hard even in split and chordal graphs, and SFVS in Chordal Graphs can be considered as a special case of the 3-Hitting Set problem. However, it is not easy to solve SFVS in Chordal Graphs faster than 3-Hitting Set. In 2019, Philip, Rajan, Saurabh, and Tale (Algorithmica 2019) proved that SFVS in Chordal Graphs can be solved in 2^k n^𝒪(1), slightly improving the best result 2.076^k n^𝒪(1) for 3-Hitting Set. In this paper, we break the "2^k-barrier" for SFVS in Chordal Graphs by giving a 1.619^k n^𝒪(1)-time algorithm. Our algorithm uses reduction and branching rules based on the Dulmage-Mendelsohn decomposition and a divide-and-conquer method.
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