Improved approximation algorithm for steiner κ-Forest with nearly uniform weights

In the Steiner κ-Forest problem we are given an edge weighted graph, a collection D of node pairs, and an integer κ ≤ |D|. The goal is to find a minimum cost subgraph that connects at least κ pairs. The best known ratio for this problem is min{O( √n),O(√κ)} [8]. In [8] it is also shown that ratio ρ for Steiner κ-Forest implies ratio O(ρ · log2 n) for the Dial-a-Ride problem: given an edge weighted graph and a set of items with a source and a destination each, find a minimum length tour to move each object from its source to destination, but carrying at most κ objects at a time. The only other algorithm known for Dial-a-Ride, besides the one resulting from [8], has ratio O( √n) [4]. We obtain ratio n_0.448 for Steiner κ-Forest and Dial-a-Ride with unit weights, breaking the O(√n) ratio barrier for this natural special case. We also show that if the maximum weight of an edge is O(n_ε), then one can achieve ratio O(n_{(1+ε)·0.448}), which is less than √n if ε is small enough. To prove our main result we consider the following generalization of the Minimum κ-Edge Subgraph (Mk-ES) problem, which we call Min-Cost ℓ-Edge-Profit Subgraph (MCℓ-EPS): Given a graph G = (V,E) with edge-profits p = {p_e : e ∈ E} and node-costs c = {c_v : v ∈ V }, and a lower profit bound ℓ, find a minimum node-cost subgraph of G of edge profit at least ℓ. The Mk-ES problem is a special case of MCℓ-EPS with unit node costs and unit edge profits. The currently best known ratio for Mk-ES is n_3-2√2+ε (note that 3-2√2 < 0.1716) [5]. We extend this ratio to MCℓ-EPS for arbitrary node weights and edge profits that are polynomial in n, which may be of independent interest.