We introduce the following notion of compressing an undirected graph G with edge-lengths and terminal vertices @math . A distance-preserving minor is a minor G' (of G) with possibly different edge-lengths, such that @math and the shortest-path distance between every pair of terminals is exactly the same in G and in G'. What is the smallest f*(k) such that every graph G with k=|R| terminals admits a distance-preserving minor G' with at most f*(k) vertices? Simple analysis shows that @math . Our main result proves that @math , significantly improving over the trivial @math . Our lower bound holds even for planar graphs G, in contrast to graphs G of constant treewidth, for which we prove that O(k) vertices suffice.
Coppersmith and Elkin @cite_11 studied a problem similar to ours, except that they seek subgraphs with few edges (rather than minors). Among other things, they prove that for every weighted graph @math and every set of @math terminals (sources), there exists a weighted subgraph @math , called , that preserves terminal distances exactly and has @math edges. They also show a nearly-matching lower bound on @math . Dor, Halperin and Zwick @cite_9 similarly asked for a graph with few edges, though not necessarily a subgraph or a minor, that preserves all distances. Woodruff @cite_5 combined their notion of with Coppersmith and Elkin's , and studied the size of arbitrary graphs preserving only distances between given sets of terminals (sources) in the given graph @math .
@cite_5 @cite_9 @cite_11
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Given a graph @math and a subset @math of terminals, a of @math is a tree that spans @math . In the vertex-weighted Steiner tree (VST) problem, each vertex is assigned a non-negative weight, and the goal is to compute a minimum weight Steiner tree of @math . We study a natural generalization of the VST problem motivated by multi-level graph construction, the (V-GSST), which can be stated as follows: given a graph @math and terminals @math , where each terminal @math requires a facility of a minimum grade of service @math , compute a Steiner tree @math by installing facilities on a subset of vertices, such that any two vertices requiring a certain grade of service are connected by a path in @math with the minimum grade of service or better. Facilities of higher grade are more costly than facilities of lower grade. Multi-level variants such as this one can be useful in network design problems where vertices may require facilities of varying priority. While similar problems have been studied in the edge-weighted case, they have not been studied as well in the more general vertex-weighted case. We first describe a simple heuristic for the V-GSST problem whose approximation ratio depends on @math , the number of grades of service. We then generalize the greedy algorithm of [Klein & Ravi, 1995] to show that the V-GSST problem admits a @math -approximation, where @math is the set of terminals requiring some facility. This result is surprising, as it shows that the (seemingly harder) multi-grade problem can be approximated as well as the VST problem, and that the approximation ratio does not depend on the number of grades of service. In the 0-extension problem, we are given a weighted graph with some nodes marked as terminals and a semimetric on the set of terminals. Our goal is to assign the rest of the nodes to terminals so as to minimize the sum, over all edges, of the product of the edge's weight and the distance between the terminals to which its endpoints are assigned. This problem generalizes the multiway cut problem of [SIAM J. Comput. , 23 (1994), pp. 864--894] and is closely related to the metric labeling problem introduced by Kleinberg and Tardos [Proceedings of the 40th IEEE Annual Symposium on Foundations of Computer Science, New York, 1999, pp. 14--23]. We present approximation algorithms for 0-Extension . In arbitrary graphs, we present a O(log k)-approximation algorithm, k being the number of terminals. We also give O(1)-approximation guarantees for weighted planar graphs. Our results are based on a natural metric relaxation of the problem previously considered by Karzanov [European J. Combin., 19 (1998), pp. 71--101]. It is similar in flavor to the linear programming relaxation of Garg, Vazirani, and Yannakakis [SIAM J. Comput. , 25 (1996), pp. 235--251] for the multicut problem, and similar to relaxations for other graph partitioning problems. We prove that the integrality ratio of the metric relaxation is at least @math for a positive c for infinitely many k. Our results improve some of the results of Kleinberg and Tardos, and they further our understanding on how to use metric relaxations. We introduce the following notion of compressing an undirected graph @math with (nonnegative) edge-lengths and terminal vertices @math . A distance-preserving minor is a minor @math (of @math ) with possibly different edge-lengths, such that @math and the shortest-path distance between every pair of terminals is exactly the same in @math and in @math . We ask: what is the smallest @math such that every graph @math with @math terminals admits a distance-preserving minor @math with at most @math vertices? Simple analysis shows that @math . Our main result proves that @math , significantly improving on the trivial @math . Our lower bound holds even for planar graphs @math , in contrast to graphs @math of constant treewidth, for which we prove that @math vertices suffice. We introduce the following notion of compressing an undirected graph G with (nonnegative) edge-lengths and terminal vertices R⊆V(G). A distance-preserving minor is a minor G′ (of G) with possibly different edge-lengths, such that R⊆V(G′) and the shortest-path distance between every pair of terminals is exactly the same in G and in G′. We ask: what is the smallest f*(k) such that every graph G with k=|R| terminals admits a distance-preserving minor G′ with at most f*(k) vertices? Simple analysis shows that f*(k)≤O(k4). Our main result proves that f*(k)≥Ω(k2), significantly improving over the trivial f*(k)≥k. Our lower bound holds even for planar graphs G, in contrast to graphs G of constant treewidth, for which we prove that O(k) vertices suffice.