2018/02/12 by Patricia Hersh, Hersh, Patricia · 1 citation
Computer Science · Mathematics · #05C12 #05E45 #06A07 #52B05 #52B12 #90C05 #Advanced Combinatorial Mathematics #Algebraic Topology (math.AT) #Combinatorics (math.CO) #Commutative Algebra and Its Applications #FOS: Mathematics #Optimization and Control (math.OC) #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.1802.04342
openalex publication_date 2018/02/12 · openalex created_date 2023/08/11 · openalex updated_date 2026/07/28
Given any polytope P and any generic linear functional \bf c , one obtains a directed graph G(P,\bf c) from the 1-skeleton of P by orienting each edge e(u,v) from u to v for \bf c (u) < \bf c ( v). For P a simple polytope and G(P,\bf c) the Hasse diagram of a lattice L, the join of any collection S of elements which all cover a common element u in L is proven to equal the sink of the smallest face of P containing u and all of the elements of S. The author conjectures for such G(P,\bf c) that no directed path in G(P,\bf c) ever revisits any facet of P. This would imply for such P and \bf c that the simplex method for linear programming is efficient under all possible pivot rules. This conjecture is proven for 3-polytopes and for spindles. For simple polytopes in which G(P,\bf c) is the Hasse diagram of a lattice L, the order complex of each open interval in L is proven homotopy equivalent to a ball or a sphere. Applications are given to the weak Bruhat order, the Tamari lattice, and the Cambrian lattices. This paper concludes with an appendix by Dominik Preußproving the monotone Hirsch conjecture for P a simple polytope and G(P,\bf c) the Hasse diagram of a lattice. This confirms one of the main consequences that the author's conjecture would have.