On the Simplex Method for 0/1-Polytopes

Alexander E. Black (), Jesús A. De Loera (), Sean Kafer () and Laura Sanità ()
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Alexander E. Black: Department of Mathematics, University of California, Davis, Davis, California 95616
Jesús A. De Loera: Department of Mathematics, University of California, Davis, Davis, California 95616
Sean Kafer: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332
Laura Sanità: Department of Computing Sciences, Bocconi University of Milan, 20136 Milan, Italy

Mathematics of Operations Research, 2025, vol. 50, issue 2, 1398-1420

Abstract: We present three new pivot rules for the Simplex method for Linear Programs over 0/1-polytopes. We show that the number of nondegenerate steps taken using these three rules is strongly polynomial, linear in the number of variables, and linear in the dimension. Our bounds on the number of steps are asymptotically optimal on several well-known combinatorial polytopes. Our analysis is based on the geometry of 0/1-polytopes and novel modifications to the classical steepest-edge and shadow-vertex pivot rules. We draw interesting connections between our pivot rules and other well-known algorithms in combinatorial optimization.

Keywords: Primary: 90C05; 90C08; Secondary: 90C27; 52B12; Simplex method; 0/1-polytopes; steepest edge; shadow vertex pivot rule (search for similar items in EconPapers)
Date: 2025
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Persistent link: https://EconPapers.repec.org/RePEc:inm:ormoor:v:50:y:2025:i:2:p:1398-1420

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