Partial multicuts in trees

Let T = (V, E) be an undirected tree, in which each edge is associated with a non-negative cost, and let { s₁, t₁ }, ..., { s_k, t_k } be a collection of k distinct pairs of vertices. Given a requirement parameter t ≤ k, the partial multicut on a tree problem asks to find a minimum cost set of edges whose removal from T disconnects at least t out of these k pairs. This problem generalizes the well-known multicut on a tree problem, in which we are required to disconnect all given pairs. The main contribution of this paper is an (frac(8, 3) + ε{lunate})-approximation algorithm for partial multicut on a tree, whose run time is strongly polynomial for any fixed ε{lunate} > 0. This result is achieved by introducing problem-specific insight to the general framework of using the Lagrangian relaxation technique in approximation algorithms. Our algorithm utilizes a heuristic for the closely related prize-collecting variant, in which we are not required to disconnect all pairs, but rather incur penalties for failing to do so. We provide a Lagrangian multiplier preserving algorithm for the latter problem, with an approximation factor of 2. Finally, we present a new 2-approximation algorithm for multicut on a tree, based on LP-rounding.