Escaping Degeneracy by Following Shadow Edges

Abstract: The Simplex method is among the most widely used approaches for solving linear programs. Starting at a vertex solution of the feasible region, the algorithm proceeds through a sequence of basis exchanges (known as pivots), each corresponding to a move along an improving edge of the polyhedron toward a better vertex. A central challenge affecting the efficiency of the Simplex method is degeneracy, which can cause long sequences of pivot operations that fail to change the current vertex solution. In this paper, we prove the existence of a pivot rule which is able to escape degeneracy and follow any given shadow edge-direction when initialized at some compatible basis, with a linear number of degenerate Simplex pivots. As a byproduct of our result, we obtain an improved bound on the number of degenerate Simplex pivots needed to solve linear programs defined on 0/1 polytopes.