The Hardness of Approximating Spanner Problems
STACS '00 Proceedings of the 17th Annual Symposium on Theoretical Aspects of Computer Science
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This paper studies the approximability of the sparse k-spanner problem. An O(logn)-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 [24]. 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(logn), for every k ? 2 [19]. This lower bound is tight for k = 2 but leaves a considerable gap for small constants k e 1, unless NP I 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+e. This hardness result extends into a result of O(2loge n)-inapproximability for the k-spanner problem for k = logm n and 0 e 1-m, for any 0 m 1. This result is tight, in view of the O(2log1-m n)-approximation ratio for the problem, implied by the algorithm of [24] for the case k = logm 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 ``all-server'''' k-spanner problem for any constant k and the augmentation k-spanner problem for k = 3.