On the Graph-Density of Random 0/1-Polytopes

Kaibel, Volker; Remshagen, Anja

doi:10.1007/978-3-540-45198-3_27

Cited by 9 publications

(15 citation statements)

References 13 publications

(21 reference statements)

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“…In addition to Theorem 2, we give the assertion which immediately follows from Theorem 1 in [21]: if k.m/ .3=2/ m for sufficiently large m 2 N, then lim m!1 P k.m/;m D 0.…”

Section: Remarkmentioning

confidence: 95%

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On random 2-adjacent 0/1-polyhedra

Бондаренко¹,

Бродский²

2008

Discrete Mathematics and Applications

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We estimate the probability P k;m that, as k vertices of the unit cube I m D f0; 1g m are randomly chosen, their convex hull is a polyhedron whose graph is complete. In particular, we establish that, as n ! 1, the probability P k.m/;m tends to one if k.m/ D O.2 .m=6/ and P k.m/;m tends to zero if k.m/ .3=2/ m .The results given in this paper, first, to a great extent explain why the intractable discrete problems are so widely spread, and second, support the well-known Gale's hypothesis published in 1956.This research was supported by the Russian Foundation for Basic Research, grant 03-01-00822a. 1.Let M be a convex polyhedron, X D ext M be the set of its vertices. Two vertices x; y 2 X are said to be adjacent [1] if the segment OEx; y is a one-dimensional face of the polyhedron (or an edge). What this means is that OEx; y does not intersect the convex hull of the remaining vertices in X . The graph of the polyhedron M (the polyhedron graph) is the graph with the sets of its geometrical vertices and edges. By the density of a graph [2] is meant the maximum number of its pairwise adjacent vertices. The density of a graph is of especial importance among the numerical characteristics of polyhedra graphs which reflect various aspects of computational complexity of the corresponding problems. In particular, bounds for density of polyhedra graphs for a great body of combinatorial problems found in [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] show that this characteristic grows exponentially as the dimensionality of polyhedra increases for intractable problems and grows polynomially for polynomially solvable ones. Furthermore, polyhedra graphs for the problems on maximal cut (see [4,18]), on clique (vertex covering, independent set, see [8,13]), on covering a matrix (see [11]) are complete. Recall that the polyhedra whose any two vertices are adjacent are referred to as 2-adjacent.In [19], existence of 2-adjacent polyhedra in R 4 was established with the number of vertices as large as desired. Later, this fact was re-established in [20], where a hypothesis was posed that, as k points on a sphere in R m were randomly chosen, their convex hull must, with high probability, be a 2-adjacent polyhedron even for k m.Recently, a great body of studies have come to the light dealing with polyhedra with vertices from the set I m D f0; 1g m , in particular, in connection with various aspects of combinatorial optimisation. A series of works (see, e.g., [21,22]) are devoted to the graphs G D .V; E/ of so-called random 0=1-polyhedra whose set V of vertices is randomly chosen from I m . In [21], the following theorem was proved.Theorem 1. Let .k; m/ be the mathematical expectation of the variable jEj= k 2 as k

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“…In addition to Theorem 2, we give the assertion which immediately follows from Theorem 1 in [21]: if k.m/ .3=2/ m for sufficiently large m 2 N, then lim m!1 P k.m/;m D 0.…”

Section: Remarkmentioning

confidence: 95%

“…A series of works (see, e.g., [21,22]) are devoted to the graphs G D .V; E/ of so-called random 0=1-polyhedra whose set V of vertices is randomly chosen from I m . In [21], the following theorem was proved.…”

mentioning

confidence: 99%

On random 2-adjacent 0/1-polyhedra

Бондаренко¹,

Бродский²

2008

Discrete Mathematics and Applications

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“…[6,13,15,23,24] and the references therein). The presence of long cycles has been studied ( [21], see also [14,12]), as has the graph density / vertex degrees (e.g., [19], see also [9,7]). …”

Section: Introductionmentioning

confidence: 99%

The Graph of the Pedigree Polytope is Asymptotically Almost Complete (Extended Abstract)

Makkeh

Pourmoradnasseri

Theis

2017

Algorithms and Discrete Applied Mathematics

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Abstract. Graphs (1-skeletons) of Traveling-Salesman-related polytopes have attracted a lot of attention. Pedigree polytopes are extensions of the classical Symmetric Traveling Salesman Problem polytopes (Arthanari 2000) whose graphs contain the TSP polytope graphs as spanning subgraphs (Arthanari 2013). Unlike TSP polytopes, Pedigree polytopes are not "symmetric", e.g., their graphs are not vertex transitive, not even regular. We show that in the graph of the pedigree polytope, the quotient minimum degree over number of vertices tends to 1 as the number of cities tends to infinity.

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“…этот факт был переоткрыт Д. Гейлом [20], где им сформулирована также гипотеза о том, что при случайном выборе k точек на сфере в R m их выпуклой оболочкой с высокой вероятностью будет служить 2-смежностный многогранник даже при k m. В последние годы появилось значительное число работ, посвященных изучению мно-гогранников с вершинами из множества I m D f0; 1g m , в частности, в связи с различными аспектами комбинаторной оптимизации. Ряд работ (см., например, [21,22]) направлен на исследование графов G D .V; E/ так называемых случайных 0=1-многогранников, множество V вершин которых случайно выбирается из I m . Так, в статье [21] доказана следующая теорема.…”

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“…Ряд работ (см., например, [21,22]) направлен на исследование графов G D .V; E/ так называемых случайных 0=1-многогранников, множество V вершин которых случайно выбирается из I m . Так, в статье [21] доказана следующая теорема.…”

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О Случайных 2-Смежностных 0/1-Многогранниках

Бондаренко¹,

Бродский²,

Brodskiy³

2008

Дискрет. матем.

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Оценивается вероятность P k;m того, что при случайном выборе k вершин единич-ного куба I m D f0; 1g m их выпуклой оболочкой служит многогранник, граф которого является полным. Устанавливается, в частности, что вероятностьРезультаты работы, во-первых, в значительной мере объясняют распространен-ность труднорешаемых дискретных задач и, во-вторых, подтверждают известную ги-потезу Д. Гейла, опубликованную им в 1956 г.Работа выполнена при поддержке Российского фонда фундаментальных исследо-ваний, проект 03-01-00822а.

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On the Graph-Density of Random 0/1-Polytopes

Cited by 9 publications

References 13 publications

On random 2-adjacent 0/1-polyhedra

On random 2-adjacent 0/1-polyhedra

The Graph of the Pedigree Polytope is Asymptotically Almost Complete (Extended Abstract)

О Случайных 2-Смежностных 0/1-Многогранниках

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