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SIAM Journal on Computing
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SWAT '90 Proceedings of the second Scandinavian workshop on Algorithm theory
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STOC '92 Proceedings of the twenty-fourth annual ACM symposium on Theory of computing
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SCG '92 Proceedings of the eighth annual symposium on Computational geometry
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STOC '99 Proceedings of the thirty-first annual ACM symposium on Theory of computing
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APPROX '98 Proceedings of the International Workshop on Approximation Algorithms for Combinatorial Optimization
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Extremal Graph Theory
Delaunay graphs are almost as good as complete graphs
SFCS '87 Proceedings of the 28th Annual Symposium on Foundations of Computer Science
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MFCS '08 Proceedings of the 33rd international symposium on Mathematical Foundations of Computer Science
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Euro-Par'06 Proceedings of the 12th international conference on Parallel Processing
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This paper examines a number of variants of the sparse k- spanner problem, and presents hardness results concerning their approximability. Previously, it was known that most k-spanner problems are weakly inapproximable, namely, are NP-hard to approximate with ratio O(log n), for every k ≥ 2, and that the unit-length k-spanner problem for constant stretch requirement k ≥ 5 is strongly inapproximable, namely, is NP-hard to approximate with ratio O(2log∈n) [19]. The results of this paper significantly expand the ranges of hardness for k-spanner problems. In general, strong hardness is shown for a number of k-spanner problems, for certain ranges of the stretch requirement k depending on the particular variant at hand. The problems studied differ by the types of edge weights and lengths used, and include also directed, augmentation and client-server variants of the problem. The paper also considers k-spanner problems in which the stretch requirement k is relaxed (e.g., k = Ω (log n)). For these cases, no inapproximability results were known at all (even for a constant approximation ratio) for any spanner problem. Moreover, some versions of the k-spanner problem are known to enjoy the ratio degradation property, namely, their complexity decreases exponentially with the inverse of the stretch requirement. So far, no hardness result existed precluding any k-spanner problem from enjoying this property. This paper establishes strong inapproximability results for the case of relaxed stretch requirement (up to k = o(nδ), for any 0 k-spanner problems. It is also shown that these problems do not enjoy the ratio degradation property.