The Telephone $k$-Multicast Problem

TL;DR

This work studies the Minimum Time Multicast problem (MTM) under the telephone model for directed and undirected graphs, introducing the intermediate k-MTM variant that asks to inform any $k$ terminals in minimum rounds. The authors connect MTM to the notion of poise in rooted Steiner trees, formalizing reductions to the Minimum Poise Steiner $k$-Tree and to partition-matroid constrained set coverage. They achieve a directed additive $\tilde{O}(k^{1/2})$-approximation and an undirected multiplicative $\tilde{O}(t^{1/3})$-approximation for k-MTM, both via novel algorithmic frameworks: greedy packing of low-poise trees combined with matroid-based submodular maximization and a partition-matroid set-coverage procedure. For undirected graphs, they further exploit contraction of low-poise substructures to obtain tighter guarantees, yielding a polynomial-time $\tilde{O}(t^{1/3})$-approximation. Overall, the paper advances the understanding of multicast timing by bridging directed and undirected cases and providing practical, theoretically grounded approximation schemes.

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 the only $k$ of the terminals be informed in the minimum number of rounds. For this problem, we improve implications of prior results and obtain an $\tilde{O}(t^{1/3})$ multiplicative approximation. For the directed version, we obtain an {\em additive} $\tilde{O}(k^{1/2})$ approximation algorithm (with a poly-logarithmic 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.

Figures (1)

• Figure 1: A depiction of the algorithm in the case that a $\rho$-additive partition is found. The set $A$ includes the root $r$ and all $\sqrt{k}$-good trees found, while $C$ contains the remaining vertices. Terminals are depicted in blue. Short paths from $r$ to the roots of the good trees are added (in red). Since $C$ is a $\sqrt{k}$-packing, each vertex $c \in C$ can reach less than $\sqrt{k}$ terminals within distance $D^*$. Hence, we can run the PMCover procedure, with each iteration enforcing a degree constraint of $B^*$ on each node in $A$, as shown.

Theorems & Definitions (34)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Remark 2.2
• Definition 3.1
• Definition 3.2
• Definition 3.3
• Claim 3.4
• proof
• Claim 3.5