A k-spanner of a connected graph G = (V, E) is a subgraph Gʹ consisting of all the vertices of V and a subset of the edges, with the additional property that the distance between any two vertices in Gʹ is larger than that distance in G by no more than a factor of k. This paper concerns the hardness of finding spanners with the number of edges close to the optimum. It is proved that for every fixed k approximating the spanner problem is at least as hard as approximating the set cover problem We also consider a weighted version of the spanner problem. We prove that in the case k = 2 the problem admits an O(log n)-ratio approximation, and in the case k ≥ 5, there is no 2log1-ϵ n-ratio approximation, for any ϵ > 0, unless NP ⊆ DTIME(npolylog n).
|Title of host publication||Approximation Algorithms for Combinatorial Optimization - International Workshop, APPROX 1998, Proceedings|
|Editors||José Rolim, Klaus Jansen|
|Number of pages||12|
|ISBN (Print)||3540647368, 9783540647362|
|State||Published - 1998|
|Event||International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 1998 - Aalborg, Denmark|
Duration: 18 Jul 1998 → 19 Jul 1998
|Name||Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)|
|Conference||International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 1998|
|Period||18/07/98 → 19/07/98|
Bibliographical notePublisher Copyright:
© Springer-Verlag Berlin Heidelberg 1998.