arXiv subjects
Publications and source records attributed to Aleksandr Maksimenko.
We give a complete enumeration of all 2-neighborly 0/1-polytopes of dimension 7. There are 13 959 358 918 different 0/1-equivalence classes of such polytopes. They form 5 850 402 014 combinatorial classes and 1 274 089 different f-vectors. It enables us to list some of their combinatorial properties. In particular, we have found a 2-neighborly polytope with 14 vertices and 16 facets.
Let $μ_{\text{2n}}(d,v)$ (respectively, $μ^{\text{s}}_{\text{2n}}(d,v)$) be the minimal number of facets of a (simplicial) 2-neighborly $d$-polytope with $v$ vertices, $v > d \ge 4$. It is known that $μ_{\text{2n}}(4,v) = v (v-3)/2$, $μ_{\text{2n}}(d, d+2) = d+5$, $μ_{\text{2n}}(d,d+3) = d+7$ for $d \ge 5$, and $μ_{\text{2n}}(d,d+4) \in [d+5, d+8]$ for $d \ge 6$. We show that $μ_{\text{2n}}(5, v) = Ω(v^{4/3})$, $μ_{\text{2n}}(6, v) \ge v$, and the equality $μ_{\text{2n}}(6, v) = v$ holds only for a simplex and for a dual 2-neighborly 6-polytope (if it exists) with $v \ge 27$. By using $g$-theorem, we get $μ^{\text{s}}_{\text{2n}}(d, v) = Δ(Δ(d-3) + 3d - 5)/2 + d + 1$, where $Δ= v - d - 1$. Also we show that $μ_{\text{2n}}(d, v) \ge d+7$ for $v \ge d+4$.
In this paper, we consider the notion of a direct type algorithm introduced by V.A. Bondarenko in 1983. A direct type algorithm is a linear decision tree with some special properties. Until recently, it was thought that the class of direct type algorithms is wide and includes many classical combinatorial algorithms, including the branch and bound algorithm for the traveling salesman problem, proposed by J.D.C. Little, K.G. Murty, D.W. Sweeney, C. Karel in 1963. We show that this algorithm is not a direct type algorithm.
We have found the minimal difference $Δ(k) = \min\limits_P (f_{d-1}(P) - f_{0}(P))$ between the number of facets and the number of vertices of a $k$-neighborly $d$-polytope $P$ for the case $f_{0}(P) = d+3$: $Δ(2) = 4$, $Δ(3) = 15$, and $Δ(k) = 2 (k^2 - 1)$ for $k \ge 4$.
We consider two polytopes. The quadratic assignment polytope $QAP(n)$ is the convex hull of the set of tensors $x\otimes x$, $x \in P_n$, where $P_n$ is the set of $n\times n$ permutation matrices. The second polytope is defined as follows. For every permutation of vertices of the complete graph $K_n$ we consider appropriate $\binom{n}{2} \times \binom{n}{2}$ permutation matrix of the edges of $K_n$. The Young polytope $P((n-2,2))$ is the convex hull of all such matrices. In 2009, S. Onn showed that the subgraph isomorphism problem can be reduced to optimization both over $QAP(n)$ and over $P((n-2,2))$. He also posed the question whether $QAP(n)$ and $P((n-2,2))$, having $n!$ vertices each, are isomorphic. We show that $QAP(n)$ and $P((n-2,2))$ are not isomorphic. Also, we show that $QAP(n)$ is a face of $P((2n-2,2))$, but $P((n-2,2))$ is a projection of $QAP(n)$.
Let $BQP(n)$ be a boolean quadric polytope, $LOP(m)$ be a linear ordering polytope. It is shown that $BQP(n)$ is linearly isomorphic to a face of $LOP(2n)$.
We consider several families of combinatorial polytopes associated with the following NP-complete problems: maximum cut, Boolean quadratic programming, quadratic linear ordering, quadratic assignment, set partition, set packing, stable set, 3-assignment. For comparing two families of polytopes we use the following method. We say that a family $P$ is affinely reduced to a family $Q$ if for every polytope $p\in P$ there exists $q\in Q$ such that $p$ is affinely equivalent to $q$ or to a face of $q$, where $\dim q = O((\dim p)^k)$ for some constant $k$. Under this comparison the above-mentioned families are splitted into two equivalence classes. We show also that these two classes are simpler (in the above sence) than the families of poytopes of the following problems: set covering, traveling salesman, 0-1 knapsack problem, 3-satisfiability, cubic subgraph, partial ordering. In particular, Boolean quadratic polytopes appear as faces of polytopes in every of the mentioned families.
We provide an extended formulation of size O(log n)^{\lfloor d/2 \rfloor} for the cyclic polytope with dimension d and n vertices (i,i^2,\ldots,i^d), i in [n]. First, we find an extended formulation of size log(n) for d= 2. Then, we use this as base case to construct small-rank nonnegative factorizations of the slack matrices of higher-dimensional cyclic polytopes, by iterated tensor products. Through Yannakakis's factorization theorem, these factorizations yield small-size extended formulations for cyclic polytopes of dimension d>2.
Let $X$ be a finite set in $Z^d$. We consider the problem of optimizing linear function $f(x) = c^T x$ on $X$, where $c\in Z^d$ is an input vector. We call it a problem $X$. A problem $X$ is related with linear program $\max\limits_{x \in P} f(x)$, where polytope $P$ is a convex hull of $X$. The key parameters for evaluating the complexity of a problem $X$ are the dimension $d$, the cardinality $|X|$, and the encoding size $S(X) = \log_2 \left(\max\limits_{x\in X} \|x\|_{\infty}\right)$. We show that if the (time and space) complexity of some algorithm $A$ for solving a problem $X$ is defined only in terms of combinatorial structure of $P$ and the size $S(X)$, then for every $d$ and $n$ there exists polynomially (in $d$, $\log n$, and $S$) solvable problem $Y$ with $\dim Y = d$, $|Y| = n$, such that the algorithm $A$ requires exponential time or space for solving $Y$.