Directed flow loops are highly undesirable because they are associated with wastage of energy for maintaining them and entail big losses to the world economy. It is shown that directed flow loops may appear in networks even if the dispatched commodity does not physically travel along a closed contour. Consequently, a theorem giving the necessary and sufficient condition of a directed flow loop ...

The correct wording for the text is: Trajectories of ants displaced laterally were analysed in four categories based on their search patterns: (a) zero turns: paths that did not have any turns 50°; (b) one turn: paths of ants characterised by a single turn 50°, these are usually 90° turns directed towards the route-mark corridor; (c) zero loops: paths of ants with one or more turn/s 50°, but th...

The colorful Carathéodory theorem asserts that if X1, X2, . . . , Xd+1 are sets in R, each containing the origin 0 in its convex hull, then exists a set S ⊆ X1 ∪ · · · ∪ Xd+1 with |S ∩ Xi| = 1 for all i = 1, 2, . . . , d + 1 and 0 ∈ conv(S) (we call conv(S) a colorful covering simplex). Deza, Huang, Stephen and Terlaky proved that if the Xi are in general position with respect to 0 (consequentl...

Finding disjoint paths in graphs is a problem that has attracted considerable attention from at least three perspectives: graph theory, VLSI design and network routing/flow. The corresponding literature is extensive. In this chapter we limit ourselves mostly to results on offline approximation algorithms for problems on general graphs as influenced from the network flow perspective. Surveys exa...

Computing shortest paths is one of the central problems in the theory of distributed computing. For the last few years, substantial progress has been made on the approximate single source shortest paths problem, culminating in an algorithm of Henzinger, Krinninger, and Nanongkai [STOC’16] which deterministically computes (1 + o(1))-approximate shortest paths in Õ(D + √ n) time, where D is the h...

Abstract In this article we consider the Directed Steiner Path Cover problem on directed co-graphs. Given a graph $$G=(V,E)$$ G = ( V , E ) and set $$T \subseteq V$$ T ...

Given a (directed or undirected) graph with costs on the edges, the power of a node is the maximum cost of an edge leaving it, and the power of the graph is the sum of the powers of its nodes. Motivated by applications for wireless networks, we consider some fundamental network design problems under the power minimization criteria. Let G = (V, E) be a graph with edge-costs {ce : e ∈ E} and let ...

We present near-optimal algorithms for two problems related to finding the replacement paths for edges with respect to shortest paths in sparse graphs. The problems essentially study how the shortest paths change as edges on the path fail, one at a time. Our technique improves the existing bounds for these problems on directed acyclic graphs, planar graphs, and non-planar integer-edge-weighted ...

A ~ P3-decomposition of a directed graph D is a partition of the arcs of D into directed paths of length 2. We characterize symmetric digraphs that do not admit a ~ P3-decomposition. We show that the only 2-regular, connected directed graphs that do not admit a ~ P3-decomposition are obtained from undirected odd cycles by replacing each edge by two oppositely directed arcs. In both cases, we gi...

We prove that for any finite point set P in the plane, a triangle T , and a positive integer k, there exists a coloring of P with k colors such that any homothetic copy of T containing at least 144k8 points of P contains at least one of each color. This is the first polynomial bound for range spaces induced by homothetic polygons. The only previously known bound for this problem applies to the ...