Minimizing average shortest path distances via shortcut edge addition

Abstract

We consider adding k shortcut edges (i.e. edges of small fixed length δ ≥ 0) to a graph so as to minimize the weighted average shortest path distance over all pairs of vertices. We explore several variations of the problem and give O(1)-approximations for each. We also improve the best known approximation ratio for metric k-median with penalties, as many of our approximations depend upon this bound. We give a (1 + 2 (p+1)/β(p+1)-1, β)-approximation with runtime exponential in p. If we set β=1 (to be exact on the number of medians), this matches the best current k-median (without penalties) result. © 2009 Springer.