Improved Approximability and Non-approximability Results for Graph Diameter Decreasing Problems

Abstract

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(logn logD) and 4 (up to an additive term of 2), respectively. In this paper, we improve these long-standing approximation ratios to O(logn) 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 logn, 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 $\frac{5}{3}$. 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.

This work was partially supported by the PRIN 2008 research project COGENT (COmputational and GamE-theoretic aspects of uncoordinated NeTworks), funded by the Italian Ministry of Education, University, and Research.