Abstract
We present a message-passing algorithm to solve a series of edge-disjoint path problems on graphs based on the zero-temperature cavity equations. Edge-disjoint paths problems are important in the general context of routing, that can be defined by incorporating under a unique framework both traffic optimization and total path length minimization. The computation of the cavity equations can be performed efficiently by exploiting a mapping of a generalized edge-disjoint path problem on a star graph onto a weighted maximum matching problem. 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 state-of-the-art algorithms and results found in the literature. Our message-passing 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 message-passing is expected to worsen. In these cases, even though the exact message-passing 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 behavior of both the number of paths to be accommodated and their minimum total length.
Highlights
The optimization of routing and connection requests is one of the main problems faced in traffic engineering and communication networks [1]
We compared the performance with a multi-start greedy algorithm (MSG) [44]
A bounded-length version [64] of MSG has been used to develop an iterative algorithm to solve the routing and wavelength assignments (RWA) using Edge-Disjoint Paths (EDP) in [18]: its performance was comparable to the one obtained using a linear programming solver on graphs of small sizes (V 40) but with faster execution times
Summary
The optimization of routing and connection requests is one of the main problems faced in traffic engineering and communication networks [1]. We propose here an efficient method to perform this calculation, by mapping it into a minimum-weight matching problem on a complete auxiliary graph with vertices in the set @i of neighbors of i, that can be solved by classical algorithms [54] With this construction, each iteration of the MP equations can be computed in a time which is polynomial in the number of graph edges (and linear in average for sparse random graphs). In order to allow communications to be absent, as mentioned in the mapping from EDP into MWEDP in the introduction, one can add an extra edge or path between sender and receiver Adding this extra path can in general worsen the approximation (e.g. even if the original graph was a tree, the modified one will normally be not). By paying cost 2w~ (w~ for the sender and w~ for the receiver), the communication can be always “accommodated” through these extra vertices without using the original graph edges
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