A combinatorial logarithmic approximation algorithm for the directed telephone broadcast problem

Consider a synchronous network of processors, modeled by directed or undirected graph $G = (V,E)$, in which in each round every processor is allowed to choose one of its neighbors and to send a message to this neighbor. Given a processor $s \in V$ and a subset $T \subseteq V$ of processors, the telephone multicast problem requires computing the shortest schedule (in terms of the number of rounds) that delivers a message from $s$ to all the processors of $T$. The particular case $T = V$ is called the telephone broadcast problem. These problems have multiple applications in distributed computing. Several approximation algorithms with polylogarithmic ratio, including one with logarithmic ratio, for the undirected variants of these problems are known. However, all these algorithms involve solving large linear programs. Devising a polylogarithmic approximation algorithm for the directed variants of these problems is an open problem, posed by Ravi in [Proceedings of the 35th Annual IEEE Symposium on Foundations of Computer Science, (FOCS '94), 1994, pp. 202-213]. We devise a combinatorial logarithmic approximation algorithm for these problems that applies also for the directed broadcast problem. Our algorithm has significantly smaller running time and seems to reveal more information about the combinatorial structure of the solution than the previous algorithms that are based on linear programming. We also improve the lower bounds on the approximation threshold of these problems. Both problems are known to be 3/2-inapproximable. For the undirected (resp., directed) broadcast problem we show that it is NP-hard (resp., impossible unless $NP \subseteq DTIME(n^{O(\log n)})$) to approximate it within a ratio of $3 - \epsilon$ for any $\epsilon > 0$ (resp., $\Omega(\sqrt{\log n})$).