The hardness of approximating spanner problems

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(2^logƐn) [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(n^δ), for any 0 < δ < 1), for a large variety of k-spanner problems. It is also shown that these problems do not enjoy the ratio degradation property.