Plane Gossip: Approximating rumor spread in planar graphs

Abstract: We study the design of schedules for multi-commodity multicast; 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. Multi-commodity multicast models a classic information dissemination problem in networks where the primary communication constraint is the number of connections that a node can make, not link bandwidth. Multi-commodity multicast is closely related to the problem of finding a subgraph, $H$, of optimal poise, where the poise is defined as the sum of the maximum degree of $H$ and the maximum distance between any source-destination pair in $H$. We first show that the minimum poise subgraph for single-commodity multicast can be approximated to within a factor of $O(\log k)$ with respect to the value of a natural LP relaxation in an instance with $k$ terminals. This is the first upper bound on the integrality gap of the natural LP. Using this poise result and shortest-path separators in planar graphs, we obtain a $O(\log^3 k\log n/(\log\log n))$-approximation for multi-commodity multicast for planar 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 in a single step but only nodes with a single broadcasting neighbor get a message. We give an $O(\log^2 n)$-approximation for radio gossip in planar graphs breaking previous barriers. This is the first bound for radio gossip that does not rely on the maximum degree of the graph. Finally, we show that our techniques for planar graphs extend to graphs with excluded minors. We establish polylogarithmic-approximation algorithms for both multi-commodity multicast and radio gossip problems in minor-free graphs.