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...r was supported by the DFG-Forschergruppe Algorithmen, Struktur, Zufall (FOR 413/1-1, Zi 475/3-1) and by the DFG Leibniz grant of G. M. Ziegler. 1s2 RAFAEL GILLMANN that Dantzig’s original pivot-rule =-=[6]-=- could visit all vertices in a cube and thus requires an exponential number of steps. In fact for most deterministic pivot-rules such examples are known, c.f. the overview by Goldfarb 1994 [12]. Many ...

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...thms and we will refer to the simplex algorithm with a certain pivot-rule also by the name of the pivot-rule. In 1980 Khachiyan proved that LP can be solved in polynomial time by the ellipsoid method =-=[21]-=- depending on d, n and the bit-size of the input (Turing-Machine model). Until now there is no (combinatorial) strongly polynomial algorithm known to solve LP in running time bounded by a polynomial d...

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...ct cubes are not realizable with high probability. A trivial general upper bound is the maximal number of vertices of any dpolytope with n facets. This number is given by the Upper Bound Theorem (cf. =-=[29]-=-) as the number of vertices of the dual cyclic d-polytopes. Gärtner and Kaibel gave the first non-trivial general upper bound of O(N/ √ d) in [10], where N denotes the number of vertices of the given ...

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...fficult inputs have been constructed on which an exponential number of pivot-steps are required (in the dimension d). The first examples were the famous Klee-Minty Cubes due to Klee and Minty in 1972 =-=[22]-=- which showed Date: May 4, 2006. 2000 Mathematics Subject Classification. Primary 90C05; Secondary 52B12, 68W20. Key words and phrases. random edge, simplex algorithm, dual cyclic polytopes, 4-polytop...

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...e rather then the worst case. Borgwardt showed in 1987 that over random LPs w.r.t. a certain probability distribution the shadow-vertex simplexalgorithm needs only polynomial many pivot-steps [3]. In =-=[27]-=- Spielman and Teng introduced the smoothed analysis which combines advantages of worst-case and average-case analysis. Smoothed analysis measures the maximum of the expected running time over inputs u...

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...cted running-time of randomized pivot-rules. The first substantial progress on upperbounds on randomized pivot rules was obtained by Kalai in [17] and independently by Matouˇsek, Sharir, and Welzl in =-=[23]-=-. Kalai proved that random facet needs at most (in expectation) exp(O( √ d log d)) steps. This was the first subexponential running time for any pivot rule. Matouˇsek, Sharir, and Welzl had a similar ...

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... find some extremal point in P w.r.t. a given linear function c t x. The simplex algorithm is the oldest linear programming algorithm. It was devised by Dantzig in 1947 and first published in 1951 in =-=[5]-=-. In terms of geometry it finds the minimal vertex of the given simple d-Polytope P by starting at a given starting vertex and iteratively moving to an improving neighbor until the minimal vertex is r...

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... are essentially the same. The same concept was also introduced independently by Williamson Hoke in [28]. Besides AUSOs there are also other abstract settings like Sharir and Welzl’s LP-type problems =-=[26]-=- and Gärtner’s abstract optimization problems [7]. The upper bounds on random facet established in [17] and [23] are nearly tight in the setting of AUSOs. In [24] Matouˇsek constructs a family of abst...

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...so introduced independently by Williamson Hoke in [28]. Besides AUSOs there are also other abstract settings like Sharir and Welzl’s LP-type problems [26] and Gärtner’s abstract optimization problems =-=[7]-=-. The upper bounds on random facet established in [17] and [23] are nearly tight in the setting of AUSOs. In [24] Matouˇsek constructs a family of abstract cubes (AUSOs on cubes) such that random face...

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...atorial framework in which the proof works. In this paper we will call those orientations acyclic unique sink orientations (AUSO) though Kalai introduced them as abstract objective functions (AOF) in =-=[16]-=-. AOFs and AUSOs are essentially the same. The same concept was also introduced independently by Williamson Hoke in [28]. Besides AUSOs there are also other abstract settings like Sharir and Welzl’s L...

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...s. In fact for most deterministic pivot-rules such examples are known, c.f. the overview by Goldfarb 1994 [12]. Many of these constructions have been unified by Amenta and Ziegler’s deformed products =-=[1]-=-. Two strategies have mainly been followed to try to overcome the exponential worst-case behavior of the simplex algorithm. The first idea is to investigate the average case rather then the worst case...

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...shadow-vertex pivot rule has a polynomial smoothed complexity, i.e. the running time is polynomial in the input size and the standard deviation of Gaussian perturbations. Recently Kelner and Spielman =-=[20]-=- introduced a randomized “simplex like” algorithm which runs in polynomial (but not strongly polynomial) time. Their algorithm solves a randomized sequence of LPs using the shadow-vertex simplex algor...

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...ns (AUSO) though Kalai introduced them as abstract objective functions (AOF) in [16]. AOFs and AUSOs are essentially the same. The same concept was also introduced independently by Williamson Hoke in =-=[28]-=-. Besides AUSOs there are also other abstract settings like Sharir and Welzl’s LP-type problems [26] and Gärtner’s abstract optimization problems [7]. The upper bounds on random facet established in [...

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...t. the given linear objective function. On Klee-Minty cubes random edge needs Θ(d 2 ) steps only. This is a result of Balogh and Pemantle [2] improving an earlier result of Gärtner, Henk, and Ziegler =-=[9]-=-. Gärtner et al. in [11] analyzed random edge on d-polytopes with d + 2 facets–that is one facet more than the simplex. It can be shown that on the abstract cubes from [24] random edge only needs O(d ...

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...st among six pivot rules suggested for deeper study. Up to quite recently the hope was that random edge could be quadratic, e.g. in O(dn). But this hope was partially destroyed by Matouˇsek and Szabó =-=[25]-=- who constructed a family of abstract cubes on which random edge would need at least exp(Ω(d 1/3 )) steps 1 with high probability. Thus random edge is exponential. 2 It seems reasonable to believe tha...

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...inimal to the unique maximal vertex of the given polytope w.r.t. the given linear objective function. On Klee-Minty cubes random edge needs Θ(d 2 ) steps only. This is a result of Balogh and Pemantle =-=[2]-=- improving an earlier result of Gärtner, Henk, and Ziegler [9]. Gärtner et al. in [11] analyzed random edge on d-polytopes with d + 2 facets–that is one facet more than the simplex. It can be shown th...

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...rientation is an AUSO, but not vice versa. Thus AUSOs are a more general model than linear orientations. If P is simple, then it suffices to require that only all 2-faces of P have a unique sink (see =-=[14]-=-). There are two important properties which follow from this fact for AUSOs on simple polytopes. First there are also unique sources in every non-empty face of P, and secondly the reverse orientation ...

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...to define a partition ΠQ of Q such that D[Q]/ΠQ is acyclic. To get this partition and a useful characterization, we will first define the following partition of the vertex-set V \ F0 (c.f. Figure 7). =-=(18)-=- (19) (20) (21) W1 := (Fm+1 ∪ Fm+2 ∪ . . . ∪ Ft−1) ∩ (Fs+1 ∪ Fs+2 ∪ . . . ∪ Fm−1) W2 := � F V s+1 ∪ F V s+2 ∪ . . . ∪ F V � � H m ∩ Fs+1 ∪ F H s+2 ∪ . . . ∪ F H� m W3 := � F V m ∪ F V m+1 ∪ . . . ∪ F ...

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...steps. In [15] Kaibel, Mechtel, Sharir, and Ziegler compute the coefficients of linearity of various pivot rules and random edge turns out to be the most difficult to analyze. Broder et al. showed in =-=[4]-=- that random edge can be exponential in the height. The height is the shortest directed path from the unique minimal to the unique maximal vertex of the given polytope w.r.t. the given linear objectiv...

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... 0,1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19,20 Figure 2. The graph of C △ (21). The 2-faces F7, F8, and F13 are indicated. The 2-faces F7 and F8 span a facet, which is also indicated. See also =-=[13]-=- for a description of the graphs of dual cyclic 4-polytopes. The terms vertical and horizontal refer to the vertically and horizontally drawn parts of the 2-faces in the figures (e.g. Figure 2). 2.4. ...

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...it is alreadysTHE RANDOM EDGE SIMPLEX ALGORITHM ON DUAL CYCLIC 4-POLYTOPES 3 quite difficult to analyze random edge on 3-polytopes. On 3-polytopes all pivot rules need at most linearly many steps. In =-=[15]-=- Kaibel, Mechtel, Sharir, and Ziegler compute the coefficients of linearity of various pivot rules and random edge turns out to be the most difficult to analyze. Broder et al. showed in [4] that rando...

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... This number is given by the Upper Bound Theorem (cf. [29]) as the number of vertices of the dual cyclic d-polytopes. Gärtner and Kaibel gave the first non-trivial general upper bound of O(N/ √ d) in =-=[10]-=-, where N denotes the number of vertices of the given polytope. Thus in contrast to most exponential examples for deterministic pivot rules, random edge skips a substantial amount of vertices. A subst...

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...rage case rather then the worst case. Borgwardt showed in 1987 that over random LPs w.r.t. a certain probability distribution the shadow-vertex simplexalgorithm needs only polynomial many pivot-steps =-=[3]-=-. In [27] Spielman and Teng introduced the smoothed analysis which combines advantages of worst-case and average-case analysis. Smoothed analysis measures the maximum of the expected running time over...

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...ot-rule [6] could visit all vertices in a cube and thus requires an exponential number of steps. In fact for most deterministic pivot-rules such examples are known, c.f. the overview by Goldfarb 1994 =-=[12]-=-. Many of these constructions have been unified by Amenta and Ziegler’s deformed products [1]. Two strategies have mainly been followed to try to overcome the exponential worst-case behavior of the si...

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...4] Matouˇsek constructs a family of abstract cubes (AUSOs on cubes) such that random facet requires exp(Ω( √ d)) steps. So geometry must help to get under the sub-exponential bound. Gärtner showed in =-=[8]-=- that on the realizable examples of [24] random facet needs O(d 2 ) steps only. An AUSO is called realizable if there exist an embedded polytope and a linear function such that the orientation induced...

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...ective function. On Klee-Minty cubes random edge needs Θ(d 2 ) steps only. This is a result of Balogh and Pemantle [2] improving an earlier result of Gärtner, Henk, and Ziegler [9]. Gärtner et al. in =-=[11]-=- analyzed random edge on d-polytopes with d + 2 facets–that is one facet more than the simplex. It can be shown that on the abstract cubes from [24] random edge only needs O(d 2 ) steps. In a survey a...

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...-polytopes with d + 2 facets–that is one facet more than the simplex. It can be shown that on the abstract cubes from [24] random edge only needs O(d 2 ) steps. In a survey article by Kalai from 2001 =-=[19]-=- random edge is the first among six pivot rules suggested for deeper study. Up to quite recently the hope was that random edge could be quadratic, e.g. in O(dn). But this hope was partially destroyed ...

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...e Sharir and Welzl’s LP-type problems [26] and Gärtner’s abstract optimization problems [7]. The upper bounds on random facet established in [17] and [23] are nearly tight in the setting of AUSOs. In =-=[24]-=- Matouˇsek constructs a family of abstract cubes (AUSOs on cubes) such that random facet requires exp(Ω( √ d)) steps. So geometry must help to get under the sub-exponential bound. Gärtner showed in [8...