The Approximability of Partial Vertex Covers in Trees.
Motivated by applications in risk management of computational systems, we focus our attention on a special case of the partial vertex cover problem, where the underlying graph is assumed to be a tree. Here, we consider four possible versions of this setting, depending on whether vertices and edges are weighted or not. Two of these versions, where edges are assumed to be unweighted, are known to be polynomial-time solvable (Gandhi, Khuller, and Srinivasan, 2004). However, the computational complexity of this problem with weighted edges, and possibly with weighted vertices, has not been determined yet. The main contribution of this paper is to resolve these questions, by fully characterizing which variants of partial vertex cover remain intractable in trees, and which can be efficiently solved. In particular, we propose a pseudo-polynomial DP-based algorithm for the most general case of having weights on both edges and vertices, which is proven to be NPhard. This algorithm provides a polynomial-time solution method when weights are limited to edges, and combined with additional scaling ideas, leads to an FPTAS for the general case. A secondary contribution of this work is to propose a novel way of using centroid decompositions in trees, which could be useful in other settings as well.