Efficient Parallel Algorithms for the Minimum Cost Flow Problem

In this paper, we propose efficient parallel implementations of the auction/sequential shortest path and the ∈-relaxation algorithms for solving the linear minimum cost flow problem. In the parallel auction algorithm, several augmenting paths can be found simultaneously, each of them starting from a different node with positive surplus. Convergence results of an asynchronous version of the algorithm are also given. For the ∈-relaxation method, there exist already parallel versions implemented on CM-5 and CM-2; our implementation is the first on a shared memory multiprocessor. We have obtained significant speedup values for the algorithms considered; it turns out that our implementations are effective and efficient.

[1]  D. R. Fulkerson,et al.  A primal‐dual algorithm for the capacitated Hitchcock problem , 1957 .

[2]  D. R. Fulkerson,et al.  Flows in Networks. , 1964 .

[3]  Ellis L. Johnson,et al.  Networks and Basic Solutions , 1966, Oper. Res..

[4]  Gerald L. Thompson,et al.  Benefit-Cost Analysis of Coding Techniques for the Primal Transportation Algorithm , 1973, JACM.

[5]  Darwin Klingman,et al.  NETGEN: A Program for Generating Large Scale Capacitated Assignment, Transportation, and Minimum Cost Flow Network Problems , 1974 .

[6]  Fred Glover,et al.  A Computation Study on Start Procedures, Basis Change Criteria, and Solution Algorithms for Transportation Problems , 1974 .

[7]  Hanif D. Sherali,et al.  Linear Programming and Network Flows , 1977 .

[8]  D. Bertsekas.  A distributed asynchronous relaxation algorithm for the assignment problem , 1985, 1985 24th IEEE Conference on Decision and Control.

[9]  D. Bertsekas.  Distributed relaxation methods for linear network flow problems , 1986, 1986 25th IEEE Conference on Decision and Control.

[10]  D. Bertsekas,et al.  Distributed asynchronous relaxation methods for convex network flow problems , 1987 .

[11]  Andrew V. Goldberg,et al.  Solving minimum-cost flow problems by successive approximation , 1987, STOC.

[12]  Paul Tseng,et al.  Relaxation Methods for Minimum Cost Ordinary and Generalized Network Flow Problems , 1988, Oper. Res..

[13]  D. Bertsekas.  The auction algorithm: A distributed relaxation method for the assignment problem , 1988 .

[14]  John N. Tsitsiklis,et al.  Parallel and distributed computation , 1989 .

[15]  Jörg Peters,et al.  The network simplex method on a multiprocessor , 1990, Networks.

[16]  Andrew V. Goldberg,et al.  Finding Minimum-Cost Circulations by Successive Approximation , 1990, Math. Oper. Res..

[17]  Donald L. Miller,et al.  Solution of large dense transportation problems using a parallel primal algorithm , 1990 .

[18]  Dimitri P. Bertsekas,et al.  Linear network optimization - algorithms and codes , 1991 .

[19]  Dimitri P. Bertsekas,et al.  An Auction Algorithm for Shortest Paths , 1991, SIAM J. Optim..

[20]  Prasoon Tiwari,et al.  An implementation of the ε-relaxation algorithm on the CM-5 , 1993, SPAA '93.

[21]  Dimitri P. Bertsekas,et al.  Parallel primal-dual methods for the minimum cost flow problem , 1993, Comput. Optim. Appl..

[22]  Dimitri P. Bertsekas,et al.  Parallel Shortest Path Auction Algorithms , 1994, Parallel Comput..

[23]  Richard S. Barr,et al.  Parallel Simplex for Large Pure Network Problems: Computational Testing and Sources of Speedup , 1994, Oper. Res..

[24]  Xiaoye S. Li,et al.  Data-level parallel solution of min-cost network flow problems using ϵ-relaxations , 1994 .

[25]  D. Bertsekas.  An auction algorithm for the max-flow problem , 1995 .