Algorithms, simulation, combinatorics and optimization for telecommunications

Abstract. Shortcutting is the operation of adding edges to a network with the intent to decrease its diameter. We are interested in shortcutting graphs while keeping degree increases bounded, a problem first posed by Chung and Garey. Improving on a result of Bokhari and Raza we show that, for any δ ≥ 1, every undirected graph G can be shortcut in linear time to a diameter of at most O(log1+δ n) by the addition of no more than O(n / log1+δ n) edges and degree increases bounded by δ. The result also improves on an estimate due to Alon et al. Degree increases can be limited to 1 at a small extra cost. For strongly connected, bounded-degree directed graphs Flaxman and Frieze proved that, if n random edges are added, then the resulting graph has diameter O(lnn) with high probability. We prove that O(n / lnn) edges suffice to shortcut any strongly connected directed graph to a graph with diameter less than O(lnn) while keeping degree increases bounded by O(1) per node. The result is proved in a stronger, parametrized form. For general directed graphs with stability number α, we show that all distances can be