Strong inapproximability of the basic k-spanner problem

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(n^1/⌊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(2^{logϵ n}), for any fixed 0 < ϵ < 1, unless NP ⊆ DTIME (n^{polylog 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(2^{logϵ n})-inapproximability for the k-spanner problem for k = log^μ n and 0 < ϵ < 1 − μ, for any 0 < μ < 1. This result is tight, in view of the O(2^{log1−μ n})-approximation ratio for the problem, implied by the algorithm of [14] for the case k = log^μ 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.