Escaping Degeneracy by Following Shadow Edges

2026-10-08

AI Summary

This study addresses the efficiency bottleneck in the simplex method for linear programming, where degeneracy frequently induces long sequences of non-improving pivot operations. Drawing on polyhedral combinatorial optimization theory, this work proposes a novel pivot rule capable of escaping degenerate vertices along specified shadow edge directions. It is rigorously proven that this escape process requires only a linear number of pivots. Consequently, this approach yields an improved upper bound on the number of degenerate pivots for linear programs over 0/1 polytopes. Overall, this research provides both a theoretical foundation and a promising new pathway for overcoming the degeneracy bottleneck inherent in the simplex method.

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.

Problem

• Simplex method
• degeneracy
• linear programming
• pivot rule
• shadow edges

Innovation

• Simplex method
• Degeneracy
• Pivot rule
• Shadow edges
• 0/1 polytopes