Computing Bounds on Network Capacity Regions as a Polytope Reconstruction Problem

TL;DR

This paper introduces a framework for defining and computing network capacity regions as polytopes, providing algorithms and heuristics, and connecting the problem to polytope reconstruction and combinatorial optimization.

Contribution

It formalizes network capacity regions as rational polytopes and offers algorithms for their exact and approximate computation, linking to polytope reconstruction problems.

Findings

Network capacity regions are closed, bounded, convex polytopes.

Algorithms for exact and approximate computation of these regions are provided.

Connections to polytope reconstruction and Steiner tree problems are established.

Abstract

We define a notion of network capacity region of networks that generalizes the notion of network capacity defined by Cannons et al. and prove its notable properties such as closedness, boundedness and convexity when the finite field is fixed. We show that the network routing capacity region is a computable rational polytope and provide exact algorithms and approximation heuristics for computing the region. We define the semi-network linear coding capacity region, with respect to a fixed finite field, that inner bounds the corresponding network linear coding capacity region, show that it is a computable rational polytope, and provide exact algorithms and approximation heuristics. We show connections between computing these regions and a polytope reconstruction problem and some combinatorial optimization problems, such as the minimum cost directed Steiner tree problem. We provide an example to illustrate our results. The algorithms are not necessarily polynomial-time.