Plane Gossip: Approximating Rumor Spread in Planar Graphs

Abstract

We study the design of schedules for multi-commodity multicast. In this problem, we are given an undirected graph G and a collection of source-destination pairs, and the goal is to schedule a minimum-length sequence of matchings that connects every source with its respective destination. The primary communication constraint of the multi-commodity multicast model is the number of connections that a given node can make, not link bandwidth. Multi-commodity multicast and its special cases, (single-commodity) broadcast and multicast, are all NP-complete. Multi-commodity multicast is closely related to the problem of finding a subgraph of optimal poise, where the poise is defined as the sum of the maximum degree and the maximum distance between any source-destination pair. We show that for any instance of the multicast problem, the minimum poise subgraph can be approximated to within a factor of $O(\log k)$ with respect to the value of a natural LP relaxation in a graph with k source-destination pairs. This is the first upper bound on the integrality gap of the natural LP; all previous algorithms yielded approximations with respect to the integer optimum. Using this integrality gap upper bound and shortest-path separators in planar graphs, we obtain our main result: an $O(\log ^3 k \frac{\log n}{\log \log n})$-approximation for multi-commodity multicast for planar graphs which improves on the $2^{\widetilde{O}(\sqrt{\log n})}$-approximation for general graphs.

We also study the minimum-time radio gossip problem in planar graphs where a message from each node must be transmitted to all other nodes under a model where nodes can broadcast to all neighbors and only nodes with a single broadcasting neighbor get a non-interfered message. In earlier work Iglesias et al. (FSTTCS 2015), we showed a strong $\varOmega (n^{\frac{1}{2} - \epsilon })$-hardness of approximation for computing a minimum gossip schedule in general graphs. Using our techniques for the telephone model, we give an $O(\log ^2 n)$-approximation for radio gossip in planar graphs breaking this barrier. Moreover, this is the first bound for radio gossip given that doesn’t rely on the maximum degree of the graph.

This material is based upon research supported in part by the U. S. Office of Naval Research under award number N00014-12-1-1001 and National Science Foundation under award number CCF-1527032.

J. Iglesias—Now at Waymo. This material is based upon work supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. 2013170941.