Sending Secrets Swiftly: Approximation Algorithms for Generalized Multicast Problems.

We consider natural generalizations of the minimum broadcast time problem under the telephone model, where a rumor from a root node must be sent via phone calls to the whole graph in the minimum number of rounds; the telephone model implies that the set of edges involved in communicating in a round form a matching. The extensions we consider involve generalizing the number of calls that a vertex may participate in (the capacitated version), allowing conference calls (the hyperedge version) as well as a new multicommodity version we introduce where the rumors are no longer from a single node but from different sources and intended for specific destinations (the multicommodity version). Based on the ideas from[6,7], we present a very simple greedy algorithm for the basic multicast problem with logarithmic performance guarantee and adapt it to the extensions to design typically polylogarithmic approximation algorithms. For the multi-commodity version, we give the first approximation algorithm with performance ratio 2(O(log log k) (root log k)) for k source-sink pairs. We provide nearly matching lower bounds for the hypercasting problem. For the multicommodity multicasting problem, we present improved guarantees for other variants involving asymmetric capacities, small number of terminals and with larger additive guarantees.