A polynomial time primal network simplex algorithm for minimum cost flows
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Developing a polynomial time primal network simplex algorithm for the minimum cost flow problem has been a long standing open problem. In this paper, we develop one such algorithm that runs in O(min(n 2m lognC, n 2m2 logn)) time, wheren is the number of nodes in the network,m is the number of arcs, andC denotes the maximum absolute arc costs if arc costs are integer and ∞ otherwise. We first introduce a pseudopolynomial variant of the network simplex algorithm called the “premultiplier algorithm”. We then develop a cost-scaling version of the premultiplier algorithm that solves the minimum cost flow problem in O(min(nm lognC, nm 2 logn)) pivots. With certain simple data structures, the average time per pivot can be shown to be O(n). We also show that the diameter of the network polytope is O(nm logn).
KeywordsMinimum cost flows Network simplex Polynomial time Simplex algorithm Premultipliers
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- R.K. Ahuja, T.L. Magnanti and J.B. Orlin,Network Flows: Theory, Algorithms and Applications (Prentice Hall, Englewood Cliffs, NJ, 1993).Google Scholar
- P.T. Sokkalingam, P. Sharma and R.K. Ahuja, A new primal simplex algorithm for network flow problem, Unpublished manuscript, 1993; Presented at NETFLOW’93 at San Miniato, Italy.Google Scholar
- M. Akgul, Shortest path and simplex method, Research Report, Department of Computer Science and Operations Research, North Carolina State University, Raleigh, NC, 1985.Google Scholar
- J.B. Orlin, Genuinely polynomial simplex and nonsimplex algorithms for the minimum cost flow problem, Technical Report No. 1615-84, Sloan School of Management, MIT, Cambridge, MA, 1984.Google Scholar
- S.A. Plotkin and E. Tardos, Improved dual network simplex, in:Proceedings of the 1st ACM - SIAM Symposium on Discrete Algorithms (1990) 367–376.Google Scholar
- R.K. Ahuja, T.L. Magnanti and J.B. Orlin, Network flows, in: G.L. Nemhauser, A.H.G. Rinnooy Kan and M.J. Todd, eds.,Handbooks in Operations Research and Management Science, Vol. 1: Optimization (North-Holland, Amsterdam, 1989) 211–369.Google Scholar