# Multicast Communications in Tree Networks with Heterogeneous Capacity Constraints

## TL;DR

This paper investigates the complexity of scheduling maximum multicast requests in tree networks with heterogeneous capacities, providing approximation algorithms and polynomial solutions for special cases, thus advancing understanding of capacity-constrained multicast optimization.

## Contribution

It extends existing multicast scheduling algorithms to heterogeneous capacities and identifies polynomial solutions for specific tree path cases, highlighting the problem's increased complexity.

## Key findings

- The problem is harder to approximate with heterogeneous capacities.
- An M-approximation algorithm based on leaf count is proposed.
- Polynomial solutions exist for certain shared-vertex path sets.

## Abstract

A widely studied problem in communication networks is that of finding the maximum number of communication requests that can be scheduled concurrently, subject to node and/or link capacity constraints. In this paper, we consider the problem of finding the largest number of multicast communication requests that can be serviced simultaneously by a network of tree topology, subject to heterogeneous capacity constraints. This problem generalizes the following two problems studied in the literature: a) the problem of finding a largest induced $k$-colorable subgraph of a chordal graph, b) the maximum multi-commodity flow problem in tree networks. The problem is already known to be NP-hard and to admit a $c$-approximation ($c \approx 1.58$) in the case of homogeneous capacity constraints. We first show that the problem is much harder to approximate in the heterogeneous case. We then use a generalization of a classical algorithm to obtain an $M$-approximation where $M$ is the maximum number of leaves of the subtrees representing the multicast communications. Surprisingly, the same algorithm, though in various disguises, is used in the literature at least four times to solve related problems (though the analysis is different). The special case of the problem where instances are restricted to unicast communications in a star topology network is known to be polynomial-time solvable. We extend this result and show that the problem can be solved in polynomial time for a set of paths in a tree that share a common vertex.