A Deterministic Algorithm for the COST-DISTANCE Problem

The COST-DISTANCE network design problem is the following. We are given an undirected graph G = (V,E), a designated root vertex r ∈ V, and a set of terminals S ⊂ of V. We are also given two non-negative real valued functions defined on E, namely, a cost function c and a length function l, and a non-negative weight function w on the set S. The goal is to find a tree T that connects the terminals in S to the root r and minimizes σ e ∈ Tc(e) + σ t ∈ Sw(t)lT(r,t), where lT(r,t) is the length of the path in T from t to r. We give a deterministic O(log k) approximation algorithm for the COST-DISTANCE network design problem, in a sense derandomizing the algorithm given in [4]. Our algorithm is based on a natural linear programming relaxation of the problem and in the process we show that its integrality gap is O(log k).