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... probabilities a neighboring vertex with a higher value of the objective function. Unfortunately, it seems very dicult to analyze this rule for general problems. Recently, Gartner, Henk and Ziegler =-=[9]-=- managed to analyze the behavior of random pivoting on the "Klee-Minty cube". The Hirsch conjecture Let (d; n) denotes the maximal diameter of the graphs of d-polyhedra P with n facets and b (d; n)...

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...y choose a pivot rule uniformly at random from all incident edges which improve the objective function. Gartner and Ziegler have recently analyzed this algorithm's performance of the Klee-Minty cubes =-=[7]-=-, and established quadratic upper bounds on the expected numberofsteps for this instance. The expected time for this algorithm can be modeled as the time ittakes for a random walk on a directed acycli...

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...d that there exists a descent path whose length O(n). Variants of the original argument by Klee and Minty [KM72] are cited in [GHZ98, page 2]. Randomized pivot rules appear to do better. According to =-=[GHZ98]-=-, several of the most popular randomized pivot rules appear to have polynomial – even quadratic – running time. Rigorous and general results on these, however, have been hard to come by. When one rest...

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... Then we add a perfect matching between the vertices of K and K ′ having identical coordinates and orient these edges from K ′ towards K . Note that by Lemma 1 the resulting orientation is an AUSO: We interconnect 2n−1 copies of a 1-dimensional USO using the two frames K and K ′. Originally the Klee-Minty cube was defined as a geometric object. It is a polytope combinatorially equivalent to the cube, 0 1 2 3 4 5 6 7 Fig. 2. An AUSO A with a graph Tba(A) of height 4. which produces the orientation defined above if we evaluate an appropriate linear objective function on its vertices. (See, e.g, [4] for a more detailed study on KleeMinty cubes.) As we mentioned in the introduction the Klee-Minty cube is a worst-case example for many of the natural pivot rules of the simplex algorithm. As it turns out, BottomAntipodal and BottomTop are very efficient on the Klee-Minty cube, even on a much wider class of AUSOs called decomposable orientations (See [15] for an introduction to decomposable AUSOs). Lemma 3. Starting at an arbitrary vertex, BottomTop needs at most n + 1 steps to find the sink of the n-dimensional Klee-Minty cube. Due to the construction of the Klee-Minty cubes for the coordina...

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... Despite the fact that Random-Edge is the most natural randomized variant of the simplex method, and that it has received considerable attention ([11], [16, Exercise 8.10∗ ], [15, Section 9.10], [1], =-=[5]-=-, [13]), nontrivial bounds for its expected performance could only be derived for specific instances [5]. In particular, the case of general n-facet polytopes in dimension d could only be handled for ...