We present a construction of deformed products of polytopes that has as special cases all the known constructions of linear programs with \many pivots," starting with the famous Klee-Minty cubes from 1972. Thus we obtain sharp estimates for the following geometric quantities for d-dimensional simple polytopes with at most n facets: the maximal number of vertices on an increasing path, the maxim...

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The purpose of this paper is to introduce a new pivot rule of the simplex algorithm. The simplex algorithm first presented by George B. Dantzig, is a widely used method for solving a linear programming problem (LP). One of the important steps of the simplex algorithm is applying an appropriate pivot rule to select the basis-entering variable corresponding to the maximum reduced cost. Unfortunat...

We prove that the Random-Edge simplex algorithm requires an expected number of at most 13n/ √ d pivot steps on any simple d-polytope with n vertices. This is the first nontrivial upper bound for general polytopes. We also describe a refined analysis that potentially yields much better bounds for specific classes of polytopes. As one application, we show that for combinatorial d-cubes, the trivi...

We prove that the Random-Edge simplex algorithm requires an expected number of at most 13n/ √ d pivot steps on any simple d-polytope with n vertices. This is the first nontrivial upper bound for general polytopes. We also describe a refined analysis that potentially yields much better bounds for specific classes of polytopes. As one application, we show that for combinatorial d-cubes, the trivi...

Scoring rules are compared by the equilibria that they generate for simple elections with three candidates and voters drawn from large Poisson distributions. A calculus for comparing pivot probabilities in Poisson voting games is applied. For a symmetric Condorcet cycle, nonsymmetric discriminatory equilibria exist under best-rewarding scoring rules like plurality voting. A candidate who is uni...

We introduce the simple extension complexity of a polytope P as the smallest number of facets of any simple (i.e., non-degenerate in the sense of linear programming) polytope which can be projected onto P . We devise a combinatorial method to establish lower bounds on the simple extension complexity and show for several polytopes that they have large simple extension complexities. These example...

In this paper we give an exponential lower bound for Cunningham’s least recently considered (round-robin) rule as applied to parity games, Markhov decision processes and linear programs. This improves a recent subexponential bound of Friedmann for this rule on these problems. The round-robin rule fixes a cyclical order of the variables and chooses the next pivot variable starting from the previ...