Dependent Random Graphs and Multi-Party Pointer Jumping

We initiate a study of a relaxed version of the standard Erdős-Rényi random graph model, where each edge may depend on a few other edges. We call such graphs dependent random graphs. Our main result in this direction is a thorough understanding of the clique number of dependent random graphs. We also obtain bounds for the chromatic number. Surprisingly, many of the standard properties of random graphs also hold in this relaxed setting. We show that with high probability, a dependent random graph will contain a clique of size p1´op1qq logpnq logp1{pq , and the chromatic number will be at most n logp1{p1´pqq log n . We expect these results to be of independent interest. As an application and second main result, we give a new communication protocol for the k-player Multi-Party Pointer Jumping (mpjk) problem in the number-on-the-forehead (NOF) model. Multi-Party Pointer Jumping is one of the canonical NOF communication problems, yet even for three players, its communication complexity...

• Type: Journal
• Year: 2015
• Where: APPROX
• Authors: Joshua Brody, Mario Sanchez
• Keywords: Algorithms, APPROX 2015