Computing Bounds on Network Capacity Regions as a Polytope Reconstruction Problem

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.Comment: Appeared in the 2011 IEEE International Symposium on Information Theory, 5 pages, 1 figur

On the horseshoe conjecture for maximal distance minimizers

We study the properties of sets $\Sigma$ having the minimal length (one-dimensional Hausdorff measure) over the class of closed connected sets $\Sigma \subset \mathbb{R}^2$ satisfying the inequality \mbox{max}_{y \in M} \mbox{dist}(y,\Sigma) \leq r for a given compact set $M \subset \mathbb{R}^2$ and some given $r > 0$. Such sets can be considered shortest possible pipelines arriving at a distance at most $r$ to every point of $M$ which in this case is considered as the set of customers of the pipeline. We prove the conjecture of Miranda, Paolini and Stepanov about the set of minimizers for $M$ a circumference of radius $R>0$ for the case when $r < R/4.98$. Moreover we show that when $M$ is a boundary of a smooth convex set with minimal radius of curvature $R$, then every minimizer $\Sigma$ has similar structure for $r < R/5$. Additionaly we prove a similar statement for local minimizers.Comment: 25 pages, 21 figure