Approximating k-Spanner Problems for k > 2

Abstract

Given a graph G = (V,E), a subgraph G′ = (V,H), H ⊆ E is a k-spanner (respectively, k-DSS) of G if for any pair of vertices u,w ∈ V it satisfies d _H(u,w) ≤ k · d _G(u,w) (resp., d _H(u,w) ≤ k). The basic k- spanner (resp., k-DSS ) problem is to find a k-spanner (resp., k-DSS) of a given graph G with the smallest possible number of edges.

This paper considers approximation algorithms for these and some related problems for k > 2. Both problems are known to be $\Omega (2^{log^{1 - \in } n} ) -$ inapproximable [11,13]. The basic k-spanner problem over undirected graphs with k > 2 has been given a sublinear ratio approximation algorithm (with ratio roughly $O(n^{\frac{2} {{k + 1}}} ))$ ), but no such algorithms were known for other variants of the problem, including the directed and the client- server variants, as well as for the k-DSS problem. We present the first approximation algorithms for these problems with sublinear approximation ratio.

The second contribution of this paper is in characterizing some wide families of graphs on which the problems do admit a logarithmic and a polylogarithmic approximation ratios. These families are characterized as containing graphs that have optimal or “near-optimal” spanners with certain desirable properties, such as being a tree, having low arboricity or having low girth. All our results generalize to the directed and the client-server variants of the problems. As a simple corollary, we present an algorithm that given a graph G builds a subgraph with Õ(n) edges and stretch bounded by the tree-stretch of G, namely the minimum maximal stretch of a spanning tree for G.

The analysis of our algorithms involves the novel notion of edge-dominating systems developed in the paper. The technique introduced in the paper enables us to reduce the studied algorithmic questions of approximability of the k-spanner and k-DSS problems to purely graph-theoretical questions concerning the existence of certain combinatorial objects in families of graphs.

Supported in part by grants from the Israel Science Foundation and from the Israel Ministry of Science and Art.