Constant factor approximation for subset feedback set problems via a new LP relaxation

We consider subset feedback edge and vertex set problems in undirected graphs. The input to these problems is an undirected graph G = (V, E) and a set S = {s₁, S₂,.·, S_k} c V of k terminals. A cycle in G is interesting if it contains a terminal. In the Subset Feedback Edge Set problem (Subset-FES) the input graph is edge-weighted and the goal is to remove a minimum weight set of edges such that no interesting cycle remains. In the Subset Feedback Vertex Set problem (subset-FVS) the input graph is node-weighted and the goal is to remove a minimum weight set of nodes such that no interesting cycle remains. A 2-approximation is known for subset-FES [12] and a 8-approximation is known for SuBSET-FVS [13]. The algorithm and analysis for SuBSET-FVS is complicated. One reason for the difficulty in addressing feedback set problems in undirected graphs has been the lack of LP relaxations with constant factor integrality gaps; the natural LP has an integrality gap of θ(logn). In this paper, we introduce new LP relaxations for subset-FES and Subset-FVS and show that their integrality gap is at most 13. Our LP formulation and rounding are simple although the analysis is non-obvious.