CS261 - Optimization Paradigms Lecture Notes for 2009-2010 Academic Year

Given a connected graph G = (V, E) with non-negative edge costs, and a set of " special " nodes S ⊂ V , a subgraph of G is a Steiner tree, if it is a tree that spans (connects) all the (" special ") nodes in S. The Steiner Tree problem is to find a Steiner Tree of minimum weight (cost). Steiner Tree is an important NP-hard problem that is often encountered in practice. Examples include design of multicast trees, design of signal distribution networks, etc. Since it is NP-hard, we cannot expect to design a polynomial-time algorithm that solves it to optimality. In this lecture we will describe a simple polynomial time algorithm that produces an answer that is within a factor 2 of optimum. Since a minimum cost spanning tree (MST) of the given graph is a Steiner tree, the intuitively easiest approximation is to find the MST of the given graph. Unfortunately, there are graphs for which this is a bad choice, as illustrated in fig. (1). n n n nodes 1 1 2 1 1 1 1 2 1 1 (a) (c) (b) 1 1 2 1 1 Figure 1: An example showing that MST is not a good heuristic for the minimum Steiner tree problem. (a) The original graph. Circled nodes are the special nodes. (b) The minimum cost spanning tree. (c) The minimum cost Steiner tree. In this graph, the MST has weight (n − 1). Also, in this case, the MST is minimal in the Steiner tree sense, that is, removing any edge from it will cause it to not be a Steiner tree. The minimum cost Steiner tree, however, has weight 2. The approximation is therefore only good to a factor of (n − 1)/2, which is quite bad. Next we will describe a better approximation algorithm, which is guaranteed to produce a Steiner tree no worse than 2 times the optimal solution. The algorithm to find an approximate optimal Steiner tree for a graph G = (V, E) and special nodes S is as follows: 5 1. Compute the complete graph of distances between all of the special nodes. The distance between two nodes is the length of the shortest path between them. We construct a complete graph G ′ = (V ′ , E ′), where V ′ corresponds to the special nodes S, and the weight of each …