Parallel primal-dual methods for the minimum cost flow problem

Abstract

In this paper we discuss the parallel asynchronous implementation of the classical primal-dual method for solving the linear minimum cost network flow problem. Multiple augmentations and price rises are simultaneously attempted starting from several nodes with possibly outdated price and flow information.The results are then merged asynchronously subject to rather weak compatibility conditions. We show that this algorithm is valid, terminating finitely to an optimal solution. We also present computational results using an Encore Multimax that illustrate the speedup that can be obtained by parallel implementation.

Convergence Properties for the Physarum Solver

“…The convergence of the physarum solver to an optimal flow can be discussed in the context of primal-dual methods (see [12], [3]) to solve mathematical programming problems. By the Theorem of Complementarity of Slackness ([12, Theorem 13.4]), a primal-dual pair of feasible solutions (p, φ), corresponds to optimal solutions of the dual and primal problems, respectively, precisely when…”

Linear Assignment Problems and Extensions

“…They are parallelized versions of primal-dual algorithms based on shortest path computations, due to Kennington and Wang [113], Zaki [170], Balas, Miller, Pekny, and Toth [14], Bertsekas and Castañon [32,33], parallelized versions of the auction algorithm, due to Zaki [170], Wein and Zenios [167,168], Bertsekas and Castañon [31], and parallelizations of primal simplex-based methods, due to Miller, Pekny, and Thompson [128], Peters [138], Barr and Hickman [27]. For a good review on parallel algorithms for the LSAP and network flow problems in general the reader is referred to Bertsekas, Castañon, Eckstein, and Zenios [34].…”