STOC '97 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
Design networks with bounded pairwise distance
STOC '99 Proceedings of the thirty-first annual ACM symposium on Theory of computing
Graph classes: a survey
Bounded-Diameter Minimum-Cost Graph Problems
Theory of Computing Systems
Decreasing the diameter of bounded degree graphs
Journal of Graph Theory
Decreasing the diameter of cycles
Journal of Graph Theory
Augmenting outerplanar graphs to meet diameter requirements
CATS '12 Proceedings of the Eighteenth Computing: The Australasian Theory Symposium - Volume 128
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In this paper 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 Minimum-Cardinality Bounded-Diameter 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 Bounded-Cardinality Minimum-Diameter 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 NP-hard, as well as approximable within O(log n logD) and 4 (up to an additive term of 2), respectively. In this paper, we improve these long-standing 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 whose diameter is optimal up to a multiplicative factor approaching 5/3. 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.