A new dual simplex type algorithm for the Minimum Cost Network Flow Problem (MCNFP) is presented. The proposed algorithm belongs to a special 'exterior- point simplex type' category. Similarly to the classical network dual simplex algorithm (NDSA), this algorithm starts with a dual feasible tree-solution and reduces the primal infeasibility, iteration by iteration. However, contrary to the NDSA, the new algorithm does not always maintain a dual feasible solution. Instead, the new algorithm might reach a basic point (tree-solution) outside the dual feasible area (exterior point - dual infeasible tree)

Abstract. The minimum cost network flow problem, (MCNFP) constitutes a wide category of network flow problems. Recently a new dual network exterior point simplex algorithm (DNEPSA) for the MCNFP has been developed. This algorithm belongs to a special "exterior point simplex type" category. Similar to the classical dual network simplex algorithm (DNSA), this algorithm starts with a dual feasible tree-solution and after a number of iterations, it produces a solution that is both primal and dual feasible, i.e. it is optimal. However, contrary to the DNSA, the new algorithm does not always maintain a dual feasible solution. Instead, it produces tree-solutions that can be infeasible for the dual problem and at the same time infeasible for the primal problem. In this paper, we present for the first time, the mathematical proof of correctness of DNEPSA, a detailed comparative computational study of DNEPSA and DNSA on sparse and dense random problem instances, a statistical analysis of the experimental results, and finally some new results on the empirical complexity of DNEPSA. The analysis proves the superiority of DNEPSA compared to DNSA in terms of cpu time and iterations.Keywords. Network flows, minimum cost network flow problem, dual network exterior point simplex algorithm.Mathematics Subject Classification. 90C27, 65K05, 90B10, 91A90.

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