The EdgeDisjoint Path Problem on Random Graphs by MessagePassing
Abstract
We present a messagepassing algorithm to solve the edge disjoint path problem (EDP) on graphs incorporating under a unique framework both traffic optimization and path length minimization. The minsum equations for this problem present an exponential computational cost in the number of paths. To overcome this obstacle we propose an efficient implementation by mapping the equations onto a weighted combinatorial matching problem over an auxiliary graph. We perform extensive numerical simulations on random graphs of various types to test the performance both in terms of path length minimization and maximization of the number of accommodated paths. In addition, we test the performance on benchmark instances on various graphs by comparison with stateoftheart algorithms and results found in the literature. Our messagepassing algorithm always outperforms the others in terms of the number of accommodated paths when considering non trivial instances (otherwise it gives the same trivial results). Remarkably, the largest improvement in performance with respect to the other methods employed is found in the case of benchmarks with meshes, where the validity hypothesis behind messagepassing is expected to worsen. In these cases, even though the exact messagepassing equations do not converge, by introducing a reinforcement parameter to force convergence towards a sub optimal solution, we were able to always outperform the other algorithms with a peak of 27% performance improvement in terms of accommodated paths. On random graphs, we numerically observe two separated regimes: one in which all paths can be accommodated and one in which this is not possible. We also investigate the behaviour of both the number of paths to be accommodated and their minimum total length.
 Publication:

PLoS ONE
 Pub Date:
 December 2015
 DOI:
 10.1371/journal.pone.0145222
 arXiv:
 arXiv:1503.00540
 Bibcode:
 2015PLoSO..1045222A
 Keywords:

 Condensed Matter  Disordered Systems and Neural Networks;
 Condensed Matter  Statistical Mechanics;
 Computer Science  Data Structures and Algorithms
 EPrint:
 14 pages, 8 figures