In this paper, we study the inapproximability of several well known optimization problems in network optimization. We show that the max directed vertex-disjoint paths problem cannot be approximated within ratio 2log1 n unless NP DTIME[2polylog n], the max directed edge-disjoint paths problem cannot be approximated within ratio 2log1 n unless NP DTIME[2polylog n], the integer multicommodity ow p...

Higher Dimensional Automata (HDA) are topological models for the study of concurrency phenomena. The state space for an HDA is given as a pre-cubical complex in which a set of directed paths (d-paths) is singled out. The aim of this paper is to describe a general method that determines the space of directed paths with given end points in a pre-cubical complex as the nerve of a particular catego...

Given a number of requests `, we propose a polynomial-time algorithm for finding ` disjoint paths in a symmetric directed graph. It is known that the problem of finding ` ≥ 2 disjoint paths in a directed graph is NP-hard [S. Fortune, J. Hopcroft, J. Wyllie, The directed subgraph homeomorphism problem, Journal of Theoretical Computer Science 10 (2) (1980) 111–121]. However, by studying minimal s...

Given an undirected graph G and four distinct special vertices s 1 ; s 2 ; t 1 ; t 2 , the Two Disjoint Paths Problem consists in determining whether there are two disjoint paths connecting s 1 to t 1 and s 2 to t 2 , respectively. There is an analogous version of the problem for acyclic directed graphs, in which it is required that the two paths be directed, as well. The known characterization...

Starting from an innocent Ramsey-theoretic question regarding directed paths in tournaments, we discover a series of rich and surprising connections that lead into the theory around a fundamental problem in Combinatorics: the Ruzsa-Szemerédi induced matching problem. Using these relationships, we prove that every coloring of the edges of the transitive n-vertex tournament using three colors con...

It is well-known that in a directed graph, if deleting any edge will not affect the shortest distance between two specific vertices s and t, then there are two edge-disjoint paths from s to t and both of them are shortest paths. In this paper, we generalize this to shortest k edge-disjoint s-t paths for any positive integer k.

We present a new methodology to create colorful textile antennas that can be embroidered within logos or other aesthetic shapes. Conductive threads (e-threads) have already been used in former embroidery unicolor approaches as attributed to the corresponding conductive material, viz. silver or copper. But so far, they have not been adapted to ‘print’ colorful textile antennas. For the first tim...

Counting lattice paths Maciej Dziemiańczuk A lattice path is a finite sequence of points p0, p1, . . . , pn in Z × Z, and a step of the path is the difference between two of its consecutive points, i.e., pi−pi−1. In this thesis, we consider lattice paths running between two fixed points and for which the set of allowable steps contains the vertical step (0,−1) and some number (possibly infinite...