Sharing the cost more efficiently: improved approximation for multicommodity rent-or-buy

In the multicommodity rent-or-buy (MROB) network design problem we are given a network together with a set of k terminal pairs (s1,t 1), . . ., (sk, tk). The goal is to provision the network so that a given amount of ow can be shipped between si and ti for all 1 ≤ i ≤ k simultaneously. In order to provision the network one can either rent capacity on edges at some cost per unit of ow, or buy them at some larger fixed cost. Bought edges have no incremental, ow-dependent cost. The overall objective is to minimize the total provisioning cost. Recently, Gupta et al. presented a 12-approximation for the MROB problem. Their algorithm chooses a subset of the terminal pairs in the graph at random and then buys the edges of an approximate Steiner forest for these pairs. This technique has previously been introduced for the single sink rent-or-buy network design problem. In this paper we give a 6.828-approximation for the MROB problem by refining the algorithm of Gupta et al. and simplifying their analysis. The improvement in our paper is based on a more careful adaptation and simplified analysis of the primal-dual algorithm for the Steiner forest problem due to Agrawal, Klein and Ravi. Our result significantly reduces the gap between the single-sink and multi-sink case.