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A localized method to distribute paths on random graphs is devised, aimed at finding the shortest paths between given source/destination pairs while avoiding path overlaps at nodes. We propose a method based on message-passing techniques to process global information and distribute paths optimally. Statistical properties such as scaling with system size and number of paths, average path-length and the transition to the frustrated regime are analysed. The performance of the suggested algorithm is evaluated through a comparison against a greedy algorithm.
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In this paper, we continue our development of algorithms used for topological network discovery. We present native P syst
We study the classical Node-Disjoint Paths (NDP) problem: given an $n$-vertex graph $G$ and a collection $M={(s_1,t_1),ldots,(s_k,t_k)}$ of pairs of vertices of $G$ called demand pairs, find a maximum-cardinality set of node-disjoint paths connecting the demand pairs. NDP is one of the most basic routing problems, that has been studied extensively. Despite this, there are still wide gaps in our understanding of its approximability: the best currently known upper bound of $O(sqrt n)$ on its approximation ratio is achieved via a simple greedy algorithm, while the best current negative result shows that the problem does not have a better than $Omega(log^{1/2-delta}n)$-approximation for any constant $delta$, under standard complexity assumptions. Even for planar graphs no better approximation algorithms are known, and to the best of our knowledge, the best negative bound is APX-hardness. Perhaps the biggest obstacle to obtaining better approximation algorithms for NDP is that most currently known approximation algorithms for this type of problems rely on the standard multicommodity flow relaxation, whose integrality gap is $Omega(sqrt n)$ for NDP, even in planar graphs. In this paper, we break the barrier of $O(sqrt n)$ on the approximability of the NDP problem in planar graphs and obtain an $tilde O(n^{9/19})$-approximation. We introduce a new linear programming relaxation of the problem, and a number of new techniques, that we hope will be helpful in designing more powerful algorithms for this and related problems.
Consider a random regular graph with degree $d$ and of size $n$. Assign to each edge an i.i.d. exponential random variable with mean one. In this paper we establish a precise asymptotic expression for the maximum number of edges on the shortest-weight paths between a fixed vertex and all the other vertices, as well as between any pair of vertices. Namely, for any fixed $d geq 3$, we show that the longest of these shortest-weight paths has about $hat{alpha}log n$ edges where $hat{alpha}$ is the unique solution of the equation $alpha log(frac{d-2}{d-1}alpha) - alpha = frac{d-3}{d-2}$, for $alpha > frac{d-1}{d-2}$.
Traveling to different destinations is a big part of our lives. We visit a variety of locations both during our daily lives and when were on vacation. How can we find the best way to navigate from one place to another? Perhaps we can test all of the different ways of traveling between two places, but another method is to use mathematics and computation to find a shortest path. We discuss how to construct a shortest path and introduce Dijkstras algorithm to minimize the total cost of a path, where the cost may be the travel distance, travel time, or some other measurement. We also discuss how to use shortest paths in the real world to save time and increase traveling efficiency.
We solve the q-state Potts model with anti-ferromagnetic interactions on large random lattices of finite coordination. Due to the frustration induced by the large loops and to the local tree-like structure of the lattice this model behaves as a mean field spin glass. We use the cavity method to compute the temperature-coordination phase diagram and to determine the location of the dynamic and static glass transitions, and of the Gardner instability. We show that for q>=4 the model possesses a phenomenology similar to the one observed in structural glasses. We also illustrate the links between the positive and the zero-temperature cavity approaches, and discuss the consequences for the coloring of random graphs. In particular we argue that in the colorable region the one-step replica symmetry breaking solution is stable towards more steps of replica symmetry breaking.
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