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

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⩽ 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.