A network simplex algorithm with O(n) consecutive degenerate pivots

  • Authors:

  • Ravindra K. Ahuja;James B. Orlin;Prabha Sharma;P. T. Sokkalingam

  • Affiliations:

  • Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL 32611, USA;Sloan School of Management, Massachusetts Institute of Technology, Cambridge, MA 02139, USA;Department of Mathematics, Indian Institute of Technology, Kanpur 208016, India;HCL-CISCO ODC Centre, Nelson Mannicken Road, Chennai 600 029, India

  • Venue:
  • Operations Research Letters
  • Year:
  • 2002

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Abstract

In this paper, we suggest a new pivot rule for the primal simplex algorithm for the minimum cost flow problem, known as the network simplex algorithm. Due to degeneracy, cycling may occur in the network simplex algorithm. The cycling can be prevented by maintaining strongly feasible bases proposed by Cunningham (Math. Programming 11 (1976) 105; Math. Oper. Res. 4 (1979) 196); however, if we do not impose any restrictions on the entering variables, the algorithm can still perform an exponentially long sequence of degenerate pivots. This phenomenon is known as stalling. Researchers have suggested several pivot rules with the following bounds on the number of consecutive degenerate pivots: m,n^2,k(k+1)/2, where n is the number of nodes in the network, m is the number of arcs in the network, and k is the number of degenerate arcs in the basis. (Observe that k=