## Randomized Simplex Algorithms on Klee-Minty Cubes (1994)

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Venue: | COMBINATORICA |

Citations: | 19 - 6 self |

### BibTeX

@ARTICLE{Gärtner94randomizedsimplex,

author = {Bernd Gärtner and Martin Henk and Günter M. Ziegler},

title = {Randomized Simplex Algorithms on Klee-Minty Cubes},

journal = {COMBINATORICA},

year = {1994},

volume = {18},

pages = {502--510}

}

### Years of Citing Articles

### OpenURL

### Abstract

We investigate the behavior of randomized simplex algorithms on special linear programs. For this, we use combinatorial models for the Klee-Minty cubes [22] and similar linear programs with exponential decreasing paths. The analysis of two most natural randomized pivot rules on the Klee-Minty cubes leads to (nearly) quadratic lower bounds for the complexity of linear programming with random pivots. Thus we disprove two bounds (for the expected running time of the random-edge simplex algorithm on Klee-Minty cubes) conjectured in the literature. At the same time, we establish quadratic upper bounds for the expected length of a path for a simplex algorithm with random pivots on the classes of linear programs under investigation. In contrast to this, we find that the average length of an increasing path in a Klee-Minty cube is exponential when all paths are taken with equal probability.

### Citations

1276 |
Combinatorial Optimization: Algorithms and Complexity
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(Show Context)
Citation Context ... cyclic polytopes, for which the actual program has not been constructed, yet. This relates to the unsolved "upper bound problem for linear programs." 2 Combinatorial Models The Klee-Minty c=-=ubes [22] [29] are the polyto-=-pes of the linear programs in IR n with m = 2n facets given by min x n : 0 x 1s1; "x i\Gamma1 x is1 \Gamma "x i\Gamma1 for 2sisn and 0 ! " ! 1=2. Our illustration shows the 3-dimensiona... |

816 |
Linear Programming and Extensions
- Dantzig
- 1963
(Show Context)
Citation Context ...). 1 algorithm problem "Is there an algorithm which quickly finds a (short) path to the lowest vertex?" The diameter problem is closely related to the "Hirsch conjecture" (from 195=-=7) and its variants [8] [20] [34]-=-. Currently there is no counterexample to the "Strong monotone Hirsch conjecture" [34] that there always has to be a decreasing path, from the vertex which maximizes x n to the lowest vertex... |

259 | Convex Polytopes
- Grünbaum
- 1967
(Show Context)
Citation Context ... parameter settings, like m = 2n, there might be substantially longer paths — see the following construction. Cyclic programs. Here the construction starts with the polars Cn(m) ∆ of cyclic polytopes =-=[15]-=- [34]. These simple polytopes have the maximal number of vertices for given m and n, namely V (n, m) = � n m − ⌈ 2 ⌉ ⌊ n 2 ⌋ � + � n−1 m − 1 − ⌈ 2 ⌉ ⌊ n−1 2 ⌋ � , according to McMullen’s upper bound t... |

194 | Linear programming in linear time when the dimension is fixed
- Megiddo
- 1984
(Show Context)
Citation Context ...n;m (x)s` `(F ) + 1 2 ' \Gamma ` n + 1 2 ' : (The proof for this result is similar to that of Proposition 12, and thus omitted.) 20 Since both the diameter problem [23] [20] and the algorithm problem =-=[27]-=- [25] have upper bounds that are linear in m, it would be interesting to know that E n;m (x) indeed grows at most linearly in m for such problems. On the other hand, it is certainly challenging to str... |

181 |
Lectures on polytopes. Graduate texts in mathematics
- Ziegler
- 2006
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Citation Context ...rithm problem "Is there an algorithm which quickly finds a (short) path to the lowest vertex?" The diameter problem is closely related to the "Hirsch conjecture" (from 1957) and it=-=s variants [8] [20] [34]. Currentl-=-y there is no counterexample to the "Strong monotone Hirsch conjecture" [34] that there always has to be a decreasing path, from the vertex which maximizes x n to the lowest vertex, of lengt... |

171 |
How good is the simplex algorithm, in
- Klee, Minty
- 1972
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Citation Context ... ∗ Martin Henk ∗∗ Günter M. Ziegler ∗∗ Abstract We investigate the behavior of randomized simplex algorithms on special linear programs. For this, we use combinatorial models for the Klee-Minty cubes =-=[22]-=- and similar linear programs with exponential decreasing paths. The analysis of two most natural randomized pivot rules on the Klee-Minty cubes leads to (nearly) quadratic lower bounds for the complex... |

166 | A subexponential bound for linear programming, Algorithmica 16 (4–5
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- 1996
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Citation Context ...aximum expected number of steps performed by random-facet is bounded by O(ne O( p k log n) ), which leads to a remarkable subexponential bound if k is small, see Kalai [16]. (Matousek, Sharir & Welzl =-=[25]-=- prove a good bound if k is large, in a very similar dual setting [13].) The random-shadow rule is a randomized version of Borgwardt's shadow vertex algorithm [4] (also known as the Gass-Saaty rule [2... |

146 |
The maximum numbers of faces of a convex polytope, Mathematika 17
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- 1970
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Citation Context ...ve the maximal number of vertices for given m and n, namely V (n; m) = ` m \Gamma d n 2 e b n 2 c ' + ` m \Gamma 1 \Gamma d n\Gamma1 2 e b n\Gamma1 2 c ' ; according to McMullen's upper bound theorem =-=[26] [34]. The-=- facets of C n (m) \Delta are identified with [m] := f1; 2; : : : ; mg; the vertices correspond to those n-subsets F ` [m] which 19 satisfy "Gale's evenness condition": if i; k 2 [m]nF , the... |

92 |
A subexponential randomized simplex algorithm
- Kalai
- 1992
(Show Context)
Citation Context ...ose a random unit vector c orthogonal to e n . Now take the path from y to the lowest vertex given by fx 2 P : cxscz for all z 2 P with z n = x n g. random-facet is a randomized version, due to Kalai =-=[16]-=-, of Bland's procedure A [3], which assumes that the facets are numbered, and always restricts to the facet with the smallest index. Interestingly enough, on an n-dimensional linear program with m = n... |

68 |
Shellable decompositions of cells and spheres
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- 1971
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Citation Context ...ytope C n (m) \Delta such that the x n -coordinate orders the vertices according to twisted lexicographic order. (Equivalently, in general this order does not correspond to a Bruggesser-Mani shelling =-=[7] [34]-=- of some realization of the cyclic polytope. In fact, Carl Lee has observed that for n = 7 and m = 10 the twisted lexicographic order is not a shelling order.) Thus the following "upper bound pro... |

51 |
The Simplex Method, A Probabilistic Analysis
- Borgwardt
- 1987
(Show Context)
Citation Context ...alai [16]. (Matousek, Sharir & Welzl [25] prove a good bound if k is large, in a very similar dual setting [13].) The random-shadow rule is a randomized version of Borgwardt's shadow vertex algorithm =-=[4]-=- (also known as the Gass-Saaty rule [21]), for which the auxiliary function c is deterministically obtained, depending on the starting vertex. Borgwardt [4] has successfully analyzed this algorithm un... |

50 | A quasi-polynomial bound for the diameter of graphs of polyhedra
- Kalai, Kleitman
- 1992
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Citation Context ...rtex which maximizes x n to the lowest vertex, of length at most m \Gamma n. On the other hand, the best arguments known for upper bounds establish paths whose length is roughly bounded by m log 2 2n =-=[17]-=-, see also [34]. The algorithm problem includes the quest for a strongly polynomial algorithm for linear programming. Klee & Minty [22] showed in 1972 that linear programs with exponentially long decr... |

48 |
The d-step conjecture and its relatives
- Klee, Kleinschmidt
- 1987
(Show Context)
Citation Context ... algorithm problem "Is there an algorithm which quickly finds a (short) path to the lowest vertex?" The diameter problem is closely related to the "Hirsch conjecture" (from 1957) a=-=nd its variants [8] [20] [34]. Cur-=-rently there is no counterexample to the "Strong monotone Hirsch conjecture" [34] that there always has to be a decreasing path, from the vertex which maximizes x n to the lowest vertex, of ... |

43 | New finite pivoting rules for the Simplex method
- Bland
- 1977
(Show Context)
Citation Context ...thogonal to e n . Now take the path from y to the lowest vertex given by fx 2 P : cxscz for all z 2 P with z n = x n g. random-facet is a randomized version, due to Kalai [16], of Bland's procedure A =-=[3]-=-, which assumes that the facets are numbered, and always restricts to the facet with the smallest index. Interestingly enough, on an n-dimensional linear program with m = n+k inequalities, the maximum... |

37 |
A Las Vegas algorithm for linear programming when the dimension is small
- Clarkson
(Show Context)
Citation Context ... this, we set `(F ) := m+ 1 \Gamma min(F ), with ns`(F )sm, and obtain `(F ) \Gamma nsE n;m (x)s` `(F ) + 1 2 ' \Gamma ` n + 1 2 ' : Since both the diameter problem [17, 14] and the algorithm problem =-=[4, 11]-=- have upper bounds that are linear in m, it would be interesting to know that En;m (x) indeed grows at most linearly in m for such problems. On the other hand, it is certainly challenging to strive fo... |

26 | A survey of linear programming in randomized subexponential time
- Goldwasser
- 1995
(Show Context)
Citation Context ...y O(ne O( p k log n) ), which leads to a remarkable subexponential bound if k is small, see Kalai [16]. (Matousek, Sharir & Welzl [25] prove a good bound if k is large, in a very similar dual setting =-=[13]-=-.) The random-shadow rule is a randomized version of Borgwardt's shadow vertex algorithm [4] (also known as the Gass-Saaty rule [21]), for which the auxiliary function c is deterministically obtained,... |

20 | Notes on Bland’s pivoting rule
- Avis, Chvatal
- 1978
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Citation Context ...rial model for the Klee-Minty cubes, which describes the random-edge algorithm as a random walk on an acyclic directed graph, see Section 2. (This model was first, it seems, derived by Avis & Chvátal =-=[2]-=-.) The combinatorial model also makes it possible to do simulation experiments. Our tests in the range n ≤ 1, 000 suggested that the O(n 2 ) upper bound is close to the truth, although the constant 1/... |

15 |
Paths on polytopes
- Larman
- 1970
(Show Context)
Citation Context ...(F )sm, and obtain `(F ) \Gamma nsE n;m (x)s` `(F ) + 1 2 ' \Gamma ` n + 1 2 ' : (The proof for this result is similar to that of Proposition 12, and thus omitted.) 20 Since both the diameter problem =-=[23]-=- [20] and the algorithm problem [27] [25] have upper bounds that are linear in m, it would be interesting to know that E n;m (x) indeed grows at most linearly in m for such problems. On the other hand... |

14 |
Deformed products and maximal shadows
- Amenta, Ziegler
- 1996
(Show Context)
Citation Context ...g such a path. Using variations of the Klee-Minty constructions, it has been shown that the simplex algorithm may take an exponential number of steps for virtually every deterministic pivot rule [20] =-=[1]-=-. (A notable exception is Zadeh’s rule [33] [20], locally minimizing revisits, for which Zadeh’s $1,000.– prize [20, p. 730] has not been collected, yet.) No such evidence exists for some natural rand... |

12 |
Worst case complexity of the shadow vertex simplex algorithm
- Goldfarb
- 1983
(Show Context)
Citation Context ...effect in even dimensions ns18, while x seems to be the worst starting vertex in all odd dimensions. The random-shadow algorithm has not yet been analyzed on special programs. Murty [28] and Goldfarb =-=[12]-=- have constructed variants of the Klee-Minty cubes for which the deterministic shadow vertex algorithm takes an exponential number of steps. 4 There is hope for a successful analysis since Borgwardt's... |

10 |
unbaum, Convex Polytopes
- Gr
- 1967
(Show Context)
Citation Context ...r settings, like m = 2n, there might be substantially longer paths --- see the following construction. Cyclic programs. Here the construction starts with the polars C n (m) \Delta of cyclic polytopes =-=[15]-=- [34]. These simple polytopes have the maximal number of vertices for given m and n, namely V (n; m) = ` m \Gamma d n 2 e b n 2 c ' + ` m \Gamma 1 \Gamma d n\Gamma1 2 e b n\Gamma1 2 c ' ; according to... |

10 |
What is the worst case behavior of the simplex algorithm
- Zadeh
- 1980
(Show Context)
Citation Context ...e-Minty constructions, it has been shown that the simplex algorithm may take an exponential number of steps for virtually every deterministic pivot rule [20] [1]. (A notable exception is Zadeh's rule =-=[33]-=- [20], locally minimizing revisits, for which Zadeh's $1,000.-- prize [20, p. 730] has not been collected, yet.) No such evidence exists for some natural randomized pivot rules, among them the followi... |

9 |
Paths on polyhedra I
- Klee
- 1965
(Show Context)
Citation Context ...s that every vertex is adjacent to the previous one. Thus the digraph is acyclic with unique source and sink, and with a directed path through all the vertices. (The construction is derived from Klee =-=[19]-=-, where the order is constructed and described recursively.) In general one cannot realize the polytope C n (m) \Delta such that the x n -coordinate orders the vertices according to twisted lexicograp... |

9 | Combinatorial linear programming: geometry can help
- Gärtner
- 1998
(Show Context)
Citation Context ... — more general than linear programs — for which the subexponential analysis is tight. For all actual linear programs in Matouˇsek’s class, however, a polynomial (in fact quadratic) upper bound holds =-=[9]-=-. In this paper we concentrate on the analysis of the “Klee-Minty cubes,” see Section 2. These are very interesting linear programs whose polytope is a deformed n-cube, but for which some pivot rules ... |

8 |
Lower bound for a subexponential optimization algorithm
- Matouˇsek
- 1994
(Show Context)
Citation Context ... Kalai [16] are essentially best possible. An interesting open problem in this context is to find linear programs on which the algorithms in [16] and [25] actually behave superpolynomially; Matouˇsek =-=[24]-=- has constructed a class of abstract optimization problems — more general than linear programs — for which the subexponential analysis is tight. For all actual linear programs in Matouˇsek’s class, ho... |

7 | The worst-case running time of the random simplex algorithm is exponential in the height
- Broder, Dyer, et al.
- 1995
(Show Context)
Citation Context ...idence contradicts the possibility that the expected running time of all three randomized algorithms we consider is bounded from above by a polynomial, even a quadratic function, in n and m. (But see =-=[6].) In this-=- connection, we 2 report investigations on the performance of such algorithms on infinite families of "test problems": specific linear programs which have decreasing paths of exponential len... |

7 |
Some results on random linear programs
- Kelly
- 1981
(Show Context)
Citation Context ...dge one gets an upper bound E n (x)s\Gamma n+1 2 \Delta for the expected number of steps starting at any vertex x of the n-dimensional Klee-Minty cube, see Section 2. This was first observed by Kelly =-=[18]-=-, see also [32]. 3 Theorem 2. The expected number E n of steps that the random-edge rule will take, starting at a random vertex on the n-dimensional Klee-Minty cube, is bounded by n 2 4(H n+1 \Gamma 1... |

6 |
Schrijver: Theory of Linear and
- unknown authors
- 1986
(Show Context)
Citation Context ...obability. 1 Introduction Linear programming is the problem of minimizing a linear objective function over a polyhedronsP ` IR n given by a system of m linear inequalities. Without loss of generality =-=[30]-=- we may assume that the problem is primally and dually nondegenerate, that the feasible region is full-dimensional and bounded, and that the objective function is given by the last coordinate. In othe... |

4 |
A primal simplex algorithm
- Goldfarb, Hao
- 1990
(Show Context)
Citation Context ...expectation values (in rational arithmetic, using REDUCE) that the smallest dimension in which this fails is n = 18. The random-shadow algorithm has not yet been studied on special programs. Goldfarb =-=[7, 8]-=- has constructed a variant of the Klee-Minty cubes for which the deterministic shadow vertex algorithm takes an exponential number of steps. There is hope for a successful analysis since Borgwardt's w... |

3 |
Geometry of the Gass-Saaty parametric cost LP algorithm
- Klee, Kleinschmidt
- 1990
(Show Context)
Citation Context ...5] prove a good bound if k is large, in a very similar dual setting [13].) The random-shadow rule is a randomized version of Borgwardt's shadow vertex algorithm [4] (also known as the Gass-Saaty rule =-=[21]-=-), for which the auxiliary function c is deterministically obtained, depending on the starting vertex. Borgwardt [4] has successfully analyzed this algorithm under the assumption that P is random in a... |

3 |
Computational complexity of parametric linear programming
- G
- 1980
(Show Context)
Citation Context ...that one has this effect in even dimensions ns18, while x seems to be the worst starting vertex in all odd dimensions. The random-shadow algorithm has not yet been analyzed on special programs. Murty =-=[28]-=- and Goldfarb [12] have constructed variants of the Klee-Minty cubes for which the deterministic shadow vertex algorithm takes an exponential number of steps. 4 There is hope for a successful analysis... |

3 |
The efficiency of the simplex method: a survey, Management Sci
- Shamir
- 1987
(Show Context)
Citation Context ...starting vertex of P along edges in such a way that the objective function decreases, until the unique lowest vertex of P is found. The (theoretical and practical) efficiency of the simplex algorithm =-=[31] depends on a s-=-uitable choice of decreasing edges that "quickly leads to the lowest vertex." Connected to this are two major problems of linear programming: the diameter problem "Is there a short path... |

3 |
Deformed products and maximal shadows, Preprint 502/1996
- Amenta, Ziegler
- 1996
(Show Context)
Citation Context ...g such a path. Using variations of the Klee-Minty constructions, it has been shown that the simplex algorithm may take an exponential number of steps for virtually every deterministic pivot rule [18] =-=[1]-=-. (A notable exception is Zadeh's rule [31] [18], locally minimizing revisits, for which Zadeh's $1,000.-- prize [18, p. 730] has not been collected, yet.) No such evidence exists for some natural ran... |

2 |
Hoke: Completely unimodal numberings of a simple polytope
- Williamson
- 1988
(Show Context)
Citation Context ... upper bound E n (x)s\Gamma n+1 2 \Delta for the expected number of steps starting at any vertex x of the n-dimensional Klee-Minty cube, see Section 2. This was first observed by Kelly [18], see also =-=[32]-=-. 3 Theorem 2. The expected number E n of steps that the random-edge rule will take, starting at a random vertex on the n-dimensional Klee-Minty cube, is bounded by n 2 4(H n+1 \Gamma 1)sE ns` n + 1 2... |

2 | Randomized Optimization by Simplex Type Methods - Gartner - 1995 |

1 |
Deformed products and maximal shadows, in: "Advances in Discrete and Computational Geometry
- Ziegler
- 1998
(Show Context)
Citation Context ...g such a path. Using variations of the Klee-Minty constructions, it has been shown that the simplex algorithm may take an exponential number of steps for virtually every deterministic pivot rule [20] =-=[1]-=-. (A notable exception is Zadeh's rule [33] [20], locally minimizing revisits, for which Zadeh's $1,000.-- prize [20, p. 730] has not been collected, yet.) No such evidence exists for some natural ran... |

1 |
Chv' atal: Notes on Bland's pivoting rule, in: "Polyhedral Combinatorics
- Avis
- 1978
(Show Context)
Citation Context ...ial model for the Klee-Minty cubes, which describes the random-edge algorithm as a random walk on an acyclic directed graph, see Section 2. (This model was first, it seems, derived by Avis & Chv'atal =-=[2]-=-.) The combinatorial model also makes it possible to do simulation experiments. Our tests in the range ns1; 000 suggested that the O(n 2 ) upper bound is close to the truth, although the constant 1=2 ... |

1 |
elou: The number and average length of decreasing paths in the KleeMinty cube
- Bousquet-M'
- 1996
(Show Context)
Citation Context ...the following table it seems likely that \Phi n ? 2 n\Gamma1 . Using generating function techniques, Bousquet-M'elou has now been able to prove the existence of a constant C ? 0 such that \Phi nsC2 n =-=[5]-=-. n 2 3 4 5 6 7 8 9 10 11 12 13 \Phi n 2 n\Gamma1 1 1.036 1.075 1.085 1.0875 1.0887 1.0893 1.0896 1.08981 1.08988 1.08992 1.08994 5 Related Models In this final section, we provide two more combinator... |

1 |
artner: Combinatorial Linear Programming: Geometry Can Help, in
- G
- 1998
(Show Context)
Citation Context ...- more general than linear programs --- for which the subexponential analysis is tight. For all actual linear programs in Matousek's class, however, a polynomial (in fact quadratic) upper bound holds =-=[9]. In this -=-paper we concentrate on the analysis of the "Klee-Minty cubes," see Section 2. These are very interesting linear programs whose polytope is a deformed n-cube, but for which some pivot rules ... |

1 | artner: Randomized Optimization by Simplex Type Methods - G - 1995 |

1 |
Randomized simplex algorithms on Klee-Minty cubes
- artner
- 1994
(Show Context)
Citation Context ...to this are two major problems of linear programming: the diameter problem "Is there a short path to the lowest vertex?", and the z A shorter "extended abstract" version of this pa=-=per has appeared in [11]. Supported by -=-the ESPRIT Basic Research Action No. 7141 of the EU (project ALCOM II). Supported by a grant in the "Gerhard-Hess-Program" of the German Science Foundation (DFG). 1 algorithm problem "I... |

1 |
How good is the simplex algorithm?, in: "Inequalities III
- Klee
- 1972
(Show Context)
Citation Context ... Gartner Martin Henk Gunter M. Ziegler Abstract We investigate the behavior of randomized simplex algorithms on special linear programs. For this, we use combinatorial models for the Klee-Minty cubes =-=[22]-=- and similar linear programs with exponential decreasing paths. The analysis of two most natural randomized pivot rules on the Klee-Minty cubes leads to (nearly) quadratic lower bounds for the complex... |

1 |
sek: Lower bounds for a subexponential optimization algorithm
- Matou
- 1994
(Show Context)
Citation Context ...y Kalai [16] are essentially best possible. An interesting open problem in this context is to find linear programs on which the algorithms in [16] and [25] actually behave superpolynomially; Matousek =-=[24]-=- has constructed a class of abstract optimization problems --- more general than linear programs --- for which the subexponential analysis is tight. For all actual linear programs in Matousek's class,... |

1 |
et al.: Randomized simplex algorithms on special linear programs, in preparation
- artner, Ziegler
- 1994
(Show Context)
Citation Context ... m for such problems. On the other hand, it is certainly challenging to strive for a nonlinear lower bound for these models. More details for the analysis of the models in this section will appear in =-=[6]-=-. Thanks We wish to thank M. Reiss, E. Welzl and M. Henk for helpful discussions, comments and contributions, and D. Kuhl, W. Schlickenrieder, C. Betz-Haubold, T. Takkula and W. Neun for extensive com... |

1 |
Note on the Klee-Minty cube, manuscript
- Henk
- 1993
(Show Context)
Citation Context ...stance, one can derive that the average length \Phi n of a decreasing path from the highest to the lowest vertex --- taking all paths with equal probability --- satisfies \Phi ns(1 + 1= p 5) n\Gamma1 =-=[10]: it is ex-=-ponential. Thus, the "average" path is exponentially long, but 3 the random-edge and random-facet pivot rules take the long paths with low probability. The random-edge algorithm moves on the... |

1 |
Lectures on Polytopes, SpringerVerlag New York 1994, in preparation. Bernd G artner Institut fur Informatik Freie Universitat Berlin Takustr
- Ziegler
(Show Context)
Citation Context ...the algorithm problem "Is there an algorithm which quickly finds a (short) path to the lowest vertex?". The diameter problem is closely related to the "Hirsch conjecture" (from 195=-=7) and its variants [5, 14, 26]. Currentl-=-y there is no counterexample to the "Strong monotone Hirsch conjecture" [26] that there always has to be a decreasing path, from the vertex which maximizes xn to the lowest vertex, of length... |

1 |
The number and average length of decreasing paths
- Bousquet-Mélou
- 1996
(Show Context)
Citation Context ... In view of the following table it seems likely that Φn > 2 n−1 . Using generating function techniques, Bousquet-Mélou has now been able to prove the existence of a constant C > 0 such that Φn ≥ C2 n =-=[5]-=-. n 2 3 4 5 6 7 8 9 10 11 12 13 Φn 2 n−1 1 1.036 1.075 1.085 1.0875 1.0887 1.0893 1.0896 1.08981 1.08988 1.08992 1.08994 5 Related Models In this final section, we provide two more combinatorial model... |