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- W4287901035 abstract "Motivated by the problem of bounding the number of iterations of the Simplex algorithm we investigate the possible lengths of monotone paths followed by the Simplex method inside the oriented graphs of polyhedra (oriented by the objective function). We consider both the shortest and the longest monotone paths and estimate the monotone diameter and height of polyhedra. Our analysis applies to transportation polytopes, matroid polytopes, matching polytopes, shortest-path polytopes, and the TSP, among others. We begin by showing that combinatorial cubes have monotone and Bland pivot height bounded by their dimension and that in fact all monotone paths of zonotopes are no larger than the number of edge directions of the zonotope. We later use this to show that several polytopes have polynomial-size pivot height, for all pivot rules. In contrast, we show that many well-known combinatorial polytopes have exponentially-long monotone paths. Surprisingly, for some famous pivot rules, e.g., greatest improvement and steepest edge, these same polytopes have polynomial-size simplex paths." @default.
- W4287901035 created "2022-07-26" @default.
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- W4287901035 date "2020-01-26" @default.
- W4287901035 modified "2023-10-03" @default.
- W4287901035 title "On the Length of Monotone Paths in Polyhedra" @default.
- W4287901035 hasPublicationYear "2020" @default.
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