Résumé  We study two variants of the problem of adding edges to a graph so as to reduce the resulting diameter. More precisely, given a graph G = (V, E), and two positive integers D and B, the MinimumCardinality BoundedDiameter Edge Addition (MCBD) problem is to find a minimum cardinality set F of edges to be added to G in such a way that the diameter of G+F is less than or equal to D, while the BoundedCardinality MinimumDiameter Edge Addition (BCMD) problem is to find a set F of B edges to be added to G in such a way that the diameter of G + F is minimized. Both problems are well known to be NPhard, as well as approximable within O(log n log D) and 4 (up to an additive term of 2), respectively. In this paper, we improve these longstanding approximation ratios to O(log n) and to 2 (up to an additive term of 2), respectively. As a consequence, we close, in an asymptotic sense, the gap on the approximability of the MCBD problem, which was known to be not approximable within c log n, for some constant c > 0, unless P = NP. Remarkably, as we further show in the paper, our approximation ratio remains asymptotically tight even if we allow for a solution 5 whose diameter is optimal up to a multiplicative factor approaching 3. On the other hand, on the positive side, we show that at most twice of the minimal number of additional edges suffices to get at most twice of the required diameter.
