From balls and bins to points and vertices

Given a graph G = (V, E) with |V| = n, we consider the following problem. Place m = n points on the vertices of G independently and uniformly at random. Once the points are placed, relocate them using a bijection from the points to the vertices that minimizes the maximum distance between the random place of the points and their target vertices. We look for an upper bound on this maximum relocation distance that holds with high probability (over the initial placements of the points). For general graphs and in the case m = n, we prove the #P -hardness of the problem and that the maximum relocation distance is O(vn) with high probability. We present a Fully Polynomial Randomized Approximation Scheme when the input graph admits a polynomial-size family of witness cuts while for trees we provide a 2-approximation algorithm. Many applications concern the variation in which m = (1 - o)n for some 0 < o < 1. We provide several bounds for the maximum relocation distance according to different graph topologies.