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This paper examines a number of variants of the sparse k-spannerproblem and presents hardness results concerning theirapproximability. Previously, it was known that most k-spannerproblems are weakly inapproximable (namely, they are NP-hard toapproximate with ratio O(log n), for every k ≥ 2) and that theunit-length k-spanner problem for constant stretch requirement k≥ 5 is strongly inapproximable (namely, it is NP-hard toapproximate with ratio O(2log1-en))). The results of this paper significantly expand the ranges ofhardness for k-spanner problems. In general, strong hardness isshown for a number of k-spanner problems, for certain ranges of thestretch requirement k depending on the particular variant at hand.The problems studied differ by the types of edge weights andlengths used, and they include directed, augmentation andclient-server variants. The paper also considers k-spanner problemsin which the stretch requirement k is relaxed (e.g., k =Ω(logn)). For these cases, no inapproximabilityresults were known (even for a constant approximation ratio) forany spanner problem. Moreover, some versions of the k-spannerproblem are known to enjoy the ratio-degradation property; namely,their complexity decreases exponentially with the inverse of thestretch requirement. So far, no hardness result existed precludingany k-spanner problem from enjoying this property. This paperestablishes strong inapproximability results for the case ofrelaxed stretch requirement (up to k =O(n1 - δ), for any 0