The Edge-Disjoint Path Problem on Random Graphs by Message-Passing

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.

Solving the edge‐disjoint paths problem using a two‐stage method

“…In the second experiment, we compare the three methods proposed in this paper with the extended artificial ant colony (E‐ACO; Blesa and Blum, ); the local search method combined with the simple greedy algorithm (LS‐SGA) and the recursive local search (LS‐R) method, both introduced (Dung et al., ); and two versions of the MP algorithm (MP and MP with reinforcement) described in Altarelli et al. (). Table summarizes average results for Set 2 (54 instances) .…”

“…Considering that the computing time is not included in Altarelli et al. (), we run our method without a time limit, denoted as ILP + EA*, to determine whether our method is able to outperform MP with reinforcement. In particular, ILP + EA* reaches the objective function values of its competitor when it is executed for 217.15 seconds (mesh25x25) and 186.76 seconds (planar200).…”

“…Recently, Altarelli et al. () propose a distributed method to solve the EDP based on message‐passing (MP) techniques. This algorithm has been previously used in other scientific fields (Mezard et al., ; Mezard and Montanari, ; de Bacco et al., ).…”

“…In virtual circuit routing, a path for each communication request is needed to be reserved in advance, in the sense that once a communication is established no interruptions will occur. Virtual circuit routing presents a practical application (Altarelli et al., ) for database servers and large‐scale video servers.…”

Coordinating dynamical routes with statistical physics on space-time networks

“…For instance, models are built to understand the growth of road networks [7,8]; traffic dynamics are modeled by cellular automata [9], diffusion [10], random walks [11] and user equilibrium [12]. Principled statistical physics tools are applied to optimize transportation networks [13][14][15][16]. These fundamental understandings always lead to applications to improve transportation networks.…”