Acquaintance Time of a Graph

We define the following parameter of connected graphs. For a given graph G = (V,E) we place one agent in each vertex v ∈ V . Every pair of agents sharing a common edge are declared to be acquainted. In each round we choose some matching of G (not necessarily a maximal matching), and for each edge in the matching the agents on this edge swap places. After the swap, again, every pair of agents sharing a common edge are acquainted, and the process continues. We define the acquaintance time of a graph G, denoted by AC(G), to be the minimal number of rounds required until every two agents are acquainted. We first study the acquaintance time for some natural families of graphs including the path, expanders, the binary tree, and the complete bipartite graph. We also show that for all functions f : N → N satisfying 1 ≤ f(n) ≤ n1.5 there is a family of graphs {Gn = (Vn, En)}n∈N with |Vn| = n such that AC(Gn) = Θ(f(n)). We also prove that for all n-vertex graphs G we have AC(G) = O ( n2 log(n)/ log log(n) ) , thus improving the trivial upper bound of O(n2) achieved by sequentially letting each agent perform depth-first search along some spanning tree of G. Studying the computational complexity of this problem, we prove that for any constant t ≥ 1 the problem of deciding that a given graph G has AC(G) ≤ t or AC(G) ≥ 2t is NP-complete. That is, AC(G) is NP-hard to approximate within multiplicative factor of 2, as well as within any additive constant factor. On the algorithmic side, we give a deterministic polynomial time algorithm that given an n-vertex graph G distinguishes between the cases AC(G) = 1 and AC(G) ≥ n − O(1). We also give a randomized polynomial time algorithm that distinguishes between the cases AC(G) = 1 and AC(G) = Ω(log(n)) with high probability. ∗{itai.benjamini,igor.shinkar,gilad.tsur}@weizmann.ac.il Faculty of Mathematics and Computer Science, Weizmann Institute of Science, Rehovot, Israel.