Improved approximation algorithms for network design problems

1 Introduction We consider a class of network design problems in which one needs to find a minimum-cost network satisfying cer-tam connectivity requirements. For example, in the snr-vivable network design problem, the requirements specify that there should be at least r(v, UJ) edge-disjoint paths between each pair of vertices v and w. We present an approximation algorithm with a performance guarantee of 2TH(fmax) = 2(1 + $ + g +. .. + &) where fmax is the maximum requirement. This improves upon the best previously known performance guarantee of 2fmax. We also show that our analysis is tight, up to a constant factor. In addition , we present approximation algorithms for two natural variations of the problem involving capacities. Our algorithms are based on a primal-dual approximation method for approximating a class of integer linear programs. Network design problems have a wide range of practical applications, ranging from telecommunications to transportation problems [lo]. In this paper, we will investigate approximation algorithms for several basic network design problems. In a network design problem, the input consists of an undirected graph G = (V, E), where each edge e E E has a nonneg-ative cost c(e), and we wish to select a minimum-cost subgraph that satisfies certain specified connec-tivity requirements. A general way to specify these constraints is to require, for each S c V, that the subgraph must contain at least f(S) edges in the cut 6(S) = {(v, w) E E : v E S, w $! S}, where we assume that there is a polynomial-time subroutine to compute f. Young Investigator Grant CC%8858097 with matching funds from AT&T, DEC, and 3M, and a grant from Powell Foundation. There has been a great deal of recent attention on the design and analysis of approximation algorithms for network design problems. There are two primary variants of these problems: one in which a solution may include multiple copies of each edge, and one in which this is forbidden. We first focus on the latter case. Extending work of Klein & Ravi [9] for proper function with range {0,2}, Williamson, Goemans, Mihail, & Vazirani [14] gave a polynomial-time 2fmax-approximation algorithm when the function f is proper, where fmax = maxscv f(S) and a p-approximation algorithm is an algorithm that always delivers a solution of cost at most p times the optimum. We shall defer the definition of proper functions , but we note …