New sparseness results on graph spanners
SCG '92 Proceedings of the eighth annual symposium on Computational geometry
On sparse spanners of weighted graphs
Discrete & Computational Geometry
Journal of Algorithms
STOC '95 Proceedings of the twenty-seventh annual ACM symposium on Theory of computing
Approximation algorithms for NP-hard problems
Approximation algorithms for NP-hard problems
Design networks with bounded pairwise distance
STOC '99 Proceedings of the thirty-first annual ACM symposium on Theory of computing
On the red-blue set cover problem
SODA '00 Proceedings of the eleventh annual ACM-SIAM symposium on Discrete algorithms
Distributed computing: a locality-sensitive approach
Distributed computing: a locality-sensitive approach
On the Hardness of Approximation Spanners
APPROX '98 Proceedings of the International Workshop on Approximation Algorithms for Combinatorial Optimization
The Client-Server 2-Spanner Problem and Applications to Network Design
The Client-Server 2-Spanner Problem and Applications to Network Design
A PTAS for the Sparsest Spanners Problem on Apex-Minor-Free Graphs
MFCS '08 Proceedings of the 33rd international symposium on Mathematical Foundations of Computer Science
Approximating Shortest Paths in Graphs
WALCOM '09 Proceedings of the 3rd International Workshop on Algorithms and Computation
Transitive-closure spanners: a survey
Property testing
Transitive-closure spanners: a survey
Property testing
Improved approximation for the directed spanner problem
ICALP'11 Proceedings of the 38th international colloquim conference on Automata, languages and programming - Volume Part I
Label cover instances with large girth and the hardness of approximating basic k-spanner
ICALP'12 Proceedings of the 39th international colloquium conference on Automata, Languages, and Programming - Volume Part I
Approximation algorithms for spanner problems and Directed Steiner Forest
Information and Computation
| Hi-index | 0.00 |
This paper studies the approximability of the sparse k-spanner problem. An O(log n)-ratio approximation algorithm is known for the problem for k = 2. For larger values of k, the problem admits only a weaker O(n1/⌊k⌋)-approximation ratio algorithm [14]. On the negative side, it is known that the k-spanner problem is weakly inapproximable, namely, it is NP-hard to approximate the problem with ratio O(log n), for every k ≥ 2 [11]. This lower bound is tight for k = 2 but leaves a considerable gap for small constants k 2. This paper considerably narrows the gap by presenting a strong (or Class III [10]) inapproximability result for the problem for any constant k 2, namely, showing that the problem is inapproximable within a ratio of O(2logƐ n), for any fixed 0 NP ⊆ DTIME(npolylog n). Hence the k-spanner problem exhibits a "jump" in its inapproximability once the required stretch is increased from k = 2 to k = 2+δ. This hardness result extends into a result of O(2logƐ n)-inapproximability for the k-spanner problem for k = logµ n and 0 O(2log1-µ n)-approximation ratio for the problem, implied by the algorithm of [14] for the case k = logµ n. To the best of our knowledge, this is the first example for a set of Class III problems for which the upper and lower bounds "converge" in this sense. Our main result implies also the same hardness for some other variants of the problem whose strong inapproximability was not known before, such as the uniform k-spanner problem, the unit-weight k-spanner problem, the 3-spanner augmentation problem and the "all-server" k-spanner problem for any constant k.