The Telephone $k$-Multicast Problem

TL;DR

This paper studies minimum time multicasting in graphs, providing new approximation algorithms for undirected and directed cases, and improving bounds for bounded degree Steiner Tree problems.

Contribution

It introduces improved approximation algorithms for the $k$-multicast problem in undirected and directed graphs, and extends results to bounded degree Steiner Tree with bounded treewidth.

Findings

Achieved a $ ilde{O}(t^{1/3})$ multiplicative approximation for undirected $k$-multicast.

Developed an additive $ ilde{O}(k^{1/2})$ approximation for directed $k$-multicast.

Provided improved polylogarithmic approximations for bounded degree Directed Steiner Tree in bounded treewidth graphs.

Abstract

We consider minimum time multicasting problems in directed and undirected graphs: given a root node and a subset of $t$ terminal nodes, multicasting seeks to find the minimum number of rounds within which all terminals can be informed with a message originating at the root. In each round, the telephone model we study allows the information to move via a matching from the informed nodes to the uninformed nodes. Since minimum time multicasting in digraphs is poorly understood compared to the undirected variant, we study an intermediate problem in undirected graphs that specifies a target $k < t$, and requires that only $k$ of the terminals be informed in the minimum number of rounds. For this problem, we improve the implications of the previous results and obtain a multiplicative approximation factor of $\tilde{O}(t^{1/3})$. For the directed version, we obtain an additive $\tilde{O}(k^{1/2})$ approximation algorithm (with a polylogarithmic multiplicative factor). Our algorithms are based on reductions to the related problems of finding $k$-trees of minimum poise (sum of maximum degree and diameter) and applying a combination of greedy network decomposition techniques and set covering under partition matroid constraints. We also study the problem of bounded degree Directed Steiner Tree, for which we obtain improved polylogarithmic approximations for the special case of bounded treewidth graphs. This extends prior work on the Group Steiner Tree problem.