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

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. By maintaining strongly feasible bases due to Cunningham [1976, 1979], cycling can be prevented but without restrictions on the entering variable, the algorithm can still perform an exponentially long sequence of non-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, 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 ≤ n.) In this paper, we describe an anti-stalling pivot rule that ensures that the network simplex algorithm performs at most k consecutive degenerate pivots. This rule uses a negative cost augmenting cycle to identify a sequence of entering variables. 1 Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL 32611, USA. 2 Sloan School of Management, MIT, Cambridge, MA 02139, USA. 3 Department of Mathematics, Indian Institute of Technology, Kanpur 208016, INDIA. 4 P. T. Sokkalingam, HCL-CISCO ODC Centre, Nelson Mannicken Road, Chennai 600 029, INDIA.