Approximating Minimum Subset Feedback Sets in Undirected Graphs with Applications

Let G=(V,E) be a weighted undirected graph where all weights are at least one. We consider the following generalization of feedback set problems. Let $S \subset V$ be a subset of the vertices. A cycle is called interesting if it intersects the set S. A subset feedback edge (vertex) set is a subset of the edges (vertices) that intersects all interesting cycles. In minimum subset feedback problems the goal is to find such sets of minimum weight. This problem has a variety of applications, among them genetic linkage analysis and circuit testing. The case in which S consists of a single vertex is equivalent to the multiway cut problem, in which the goal is to separate a given set of terminals. Hence, the subset feedback problem is NP-complete and also generalizes the multiway cut problem. We provide a polynomial time algorithm for approximating the subset feedback edge set problem that achieves an approximation factor of two. This implies a $\Delta$-approximation algorithm for the subset feedback vertex set problem, where $\Delta$ is the maximum degree in G. We also consider the multicut problem and show how to achieve an $O(\log \tau^*)$ approximation factor for this problem, where $\tau^*$ is the value of the optimal fractional solution. To achieve the $O(\log \tau^*)$ factor we employ a bootstrapping technique.