Maximizing Communication Throughput in Tree Networks.

A widely studied problem in communication networks is that of finding the maximum number of communication requests that can be concurrently scheduled, provided that there are at most $k$ requests that pass through any given edge of the network. this work we consider the problem of finding the largest number of given subtrees of a tree that satisfy given load constraints. This is an extension of the problem of finding a largest induced $k$-colorable subgraph of a chordal graph (which is the intersection graph of subtrees of a tree). We extend a greedy algorithm that solves the latter problem for interval graphs, and obtain an $M$-approximation for chordal graphs where $M$ is the maximum number of leaves of the subtrees in the representation of the chordal graph. This implies a 2-approximation for textsc{Vpt}} graphs (vertex-intersection graphs of paths in a tree), and an optimal algorithm for the class of directed path graphs (vertex-intersection graphs of paths in a directed tree) which in turn extends the class of interval graphs. In fact, we consider a more general problem that is defined on the subtrees of the representation of chordal graphs, in which we allow any set of different bounds on the vertices and edges. Thus our algorithm generalizes the known one in two directions: first, it applies to more general graph classes, and second, it does not require the same bound for all the edges (of the representation). Last, we present a polynomial-time algorithm for the general problem where instances are restricted to paths in a star.