Fixed Parameter Enumeration Algorithm for Feedback Vertex Set in Weighted Undirected Graphs
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Feedback vertex set (FVS) problem is a classical NP-complete problem. There are important applications of the problem in many areas. Many studies have been done on the FVS problem, but there is no effective algorithm to enumerate the FVS in weighted undirected graphs. In this paper, a fixed-parameter enumeration algorithm based on the branch-and-search technique is given by analyzing the structure of the FVS problem in weighted undirected graphs. In the algorithm, the feedback vertex set problem is transformed to feedback edge set (FES) problem and the z minimum-weight FES of size k are enumerated by enumerating z maximum-weight forests, then the z minimum-weight FVS of size k can be found. The fixed-parameter enumeration algorithm for FVS runs in time O(5kn2 (logn+k)+3kz(n2logn+z)).