We present an 8-approximation algorithm for the problem of finding a minimum weight subset feedback vertex set. The input in this problem consists of an undirected graph G=(V,E) with vertex weights w(v) and a subset of vertices S called special vertices. A cycle is called interesting if it contains at least one special vertex. A subset of vertices is called a subset feedback vertex set with respect to S if it intersects every interesting cycle The goal is to find a minimum weight subset feedback vertex set. The best pervious algorithm for the general case provided only a logarithmic approximation factor. The minimum weight subset feedback vertex set problem generalizes two NP-Complete problems: the minimum weight feedback vertex set problem in undirected graphs and the minimum weight multiway vertex cut problem. The main tool that we use in our algorithm and its analysis is a new version of multi-commodity flow which we call relaxed multi-commodity flow. Relaxed multi-commodity flow is a hybrid of multi-commodity flow and multi-terminal flow.
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