Generalized Steiner Network

Abstract

The generalized Steiner network problem asks to find a solution E 0 of minimum cost P e2E 0 c(e). This problem generalizes several classical network design problems. Some examples include minimum spanning tree, Steiner tree and Steiner forest. The most general special case for which a 2-approximation was previously known is the Steiner forest problem [1,4]. Williamson et al. [8] were the first to show a non-trivial approximation algorithm for the generalized Steiner network problem, achieving a 2k-approximation, where k = max i; j2V fr i; j g. This result was improved to O(log k)-approximation by Goemans et al. [3]. Key Results The main result of [6] is a factor-2 approximation algorithm for the generalized Steiner network problem. The techniques used in the design and the analysis of the algorithm seem to be of independent interest. The 2-approximation is achieved for a more general problem, defined as follows. The input is a multigraph G = (V ; E) with costs c() on edges, and connectivity requirement function f : 2 V ! Z. Function f is weakly sub-modular, i. e., it has the following properties: 1. f (V) = 0. 2. For all A; B Â V, at least one of the following two conditions holds: f (A) + f (B) Ä f (A n B) + f (B n A). f (A) + f (B) Ä f (A \ B) + f (A [ B). For any subset S Â V of vertices, let ı(S) denote the set of edges with exactly one endpoint in S. The goal is to find a minimum-cost subset of edges E 0 Â E, such that for every subset S Â V of vertices, jı(S) \ E 0 j j f (S). This problem can be equivalently expressed as an integer program. For each edge e 2 E, let x e be the indicator variable of whether e belongs to the solution. (IP) min X e2E c(e)x e subject to: X e2ı(S) x e f (S) 8S Â V (1) x e 2 f0; 1g 8 e 2 E (2) It is easy to see that the generalized Steiner network problem is a special case of (IP), where for each S Â V, f (S) = max i2S; j6 2S fr i; j g. Techniques The approximation algorithm uses the LP-rounding technique. The initial linear program (LP) is obtained from (IP) by replacing the integrality constraint (2) with: 0 Ä x e Ä 1 8e 2 E (3) It is assumed that there is a separation oracle for (LP). It is easy to see that such an oracle exists if (LP) is obtained from the generalized Steiner network problem. The key result used in the design and the analysis of the algorithm is summarized in the following theorem. Theorem 1 In any basic solution of (LP), there is at least one edge e 2 E with x e 1/2. The approximation algorithm works by iterative LP-rounding. Given a basic optimal solution of (LP), let E Â E be the subset of edges e with x e 1/2. The edges of E are removed from the graph (and are eventually added to the solution), and the problem is then solved recursively on the residual graph, by solving (LP) on G = (V; E n E), where for each subset S Â V, the new requirement is f (S) jı(S) \ E j. The main observation that leads to factor-2 approximation is the following: if E 0 is a 2-approximation for the residual problem, then E 0 [ E is a 2-approximation for the original problem. Given any solution to (LP), set S Â V is called tight iff constraint (1) holds with equality for S. The proof of Theorem 1 involves constructing a large laminar family of tight sets (a family where for every pair of sets, either one set contains the other, or the two sets are disjoint). After that a clever accounting scheme that charges edges to the sets of the laminar family is used to show that there is at least one edge e 2 E with x e 1/2. Applications Generalized Steiner network is a very basic and natural network design problem that has many applications in different areas, including the design of communication networks , VLSI design and vehicle routing. One example is the design of survivable communication networks, which remain functional even after the failure of some network components (see [5] for more details). Open Problems The 2-approximation algorithm of Jain [6] for generalized Steiner network is based on LP-rounding, and it has high running time. It would be interesting to design a combina-torial approximation algorithm for this problem. It is not known whether a better approximation is possible for generalized Steiner network. Very few hardness of approximation results are known for this type of problems. The best current hardness factor stands on 1:01063 [2],