Random half-integral polytopes☆
Introduction
Given a polytope and a cutting-plane procedure , the rank of with respect to is the minimum number of rounds (equivalently, applications of ) needed in order to compute the integral hull of . Whereas the rank problem has been studied for various polytopes in a deterministic setting, in this note we will show that the rank of random polytopes can be high (e.g., where is the dimension of ) although the polytopes have only a relatively small number of vertices. In order to disentangle this effect from a possibly high facet complexity of the integral hull of , we confine ourselves to polytopes with empty integral hull.
To simultaneously study the problem for large classes of cutting-plane procedures, we use the abstract model of cutting-plane procedures introduced in [16]. This model contains well-known procedures such as the Gomory–Chvátal procedure (cf., [4], [11], [12], [13]), the lift-and-project operators (cf., [2]), in particular the Lovász–Schrijver operators (cf., [14]), and the split cut operator (cf., [7]) and characterizes when a cutting-plane procedure has high rank. A sufficient condition is the existence of obstructing sub-polytopes that have high rank with respect to any cutting-plane procedure. We then use the probabilistic method (see e.g., [9], [10], or [1]) to infer the existence of an obstructing sub-polytope.
More precisely, we consider polytopes given as the convex hull of a random subset of the half-integral points (i.e., with coordinates ) of the -dimensional 0/1 cube except the vertices. Roughly speaking, our main result (Theorem 3.1) is that for any , the rank of the convex hull of is with probability at least 1/2 whenever as will contain an obstruction set ensuring the high rank. Here the on the right-hand side is the exact threshold number: for larger numbers will almost never contain an . This implies, e.g., that whenever we have ; see Corollary 3.2. In particular, although the relative number of points compared to the total number of points tends to 0, the rank of the resulting polytopes can be quite high.
Section snippets
Preliminaries
Throughout this note, we consider polytopes with . Let denote the logarithm to the base of 2 and denote the natural logarithm. For convenience we define for .
A lower bound on the rank
Our random construction of polytopes is the following. We uniformly choose a fixed-size random set of half-integral points of the -dimensional 0/1 cube excluding the vertices. We consider the limit of the probability of the event ‘’ that contains the of a -dimensional face. The dimension will tend to infinity, and and the size of may also vary. We will prove that the limit depends on the behavior of an expression of these parameters.
Theorem 3.1 Let be a natural number and
References (16)
- et al.
On the chvátal rank of polytopes in the 0/1 cube
Discrete Applied Mathematics
(1999) - et al.
On cutting-plane proofs in combinatorial optimization
Linear Algebra and its Applications
(1989) - et al.
The Probabilistic Method
(2000) - et al.
A lift-and-project cutting plane algorithm for mixed 0–1 programs
Mathematical Programming
(1993) Edmonds polytopes and a hierarchy of combinatorial problems
Discrete Mathematics
(1973)- et al.
On the matrix-cut rank of polyhedra
Mathematics of Operations Research
(2001) - et al.
Chvátal closures for mixed integer programming problems
Mathematical Programming
(1990) Valid inequalities for mixed integer linear programs
Mathematical Programming
(2008)
Cited by (2)
Expansion of random 0/1 polytopes
2024, Random Structures and AlgorithmsTHE EXPANSION OF HALF-INTEGRAL POLYTOPES
2024, arXiv
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An extended abstract of this note appeared in the proceedings of the International Symposium on Combinatorial Optimization 2010.