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Gennadiy Averkov

Publications and source records attributed to Gennadiy Averkov.

At least 37 records · Page 2Linked to original sources
arXiv2021

Computational Aspects of Relaxation Complexity: Possibilities and Limitations

The relaxation complexity $\mathrm{rc}(X)$ of the set of integer points $X$ contained in a polyhedron is the smallest number of facets of any polyhedron $P$ such that the integer points in $P$ coincide with $X$. It is a useful tool to investigate the existence of compact linear descriptions of $X$. In this article, we derive tight and computable upper bounds on $\mathrm{rc}_{\mathbb{Q}}(X)$, a variant of $\mathrm{rc}(X)$ in which the polyhedra $P$ are required to be rational, and we show that $\mathrm{rc}(X)$ can be computed in polynomial time if $X$ is 2-dimensional. Further, we investigate computable lower bounds on $\mathrm{rc}(X)$ with the particular focus on the existence of a finite set $Y \subseteq \mathbb{Z}^d$ such that separating $X$ and $Y \setminus X$ allows us to deduce $\mathrm{rc}(X) \geq k$. In particular, we show for some choices of $X$ that no such finite set $Y$ exists to certify the value of $\mathrm{rc}(X)$, providing a negative answer to a question by Weltge (2015). We also obtain an explicit formula for $\mathrm{rc}(X)$ for specific classes of sets $X$ and present the first practically applicable approach to compute $\mathrm{rc}(X)$ for sets $X$ that admit a finite certificate.

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arXiv2021

Sharp bounds on fake weighted projective spaces with canonical singularities

We give a sharp upper bound on the multiplicity of a fake weighted projective space with at worst canonical singularities. This is equivalent to giving a sharp upper bound on the index of the sublattice generated by the vertices of a lattice simplex containing only the origin as an interior lattice point. We also completely characterise when equality occurs and discuss related questions and conjectures.

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arXiv2020

The lattice of cycles of an undirected graph

We study bases of the lattice generated by the cycles of an undirected graph, defined as the integer linear combinations of the 0/1-incidence vectors of cycles. We prove structural results for this lattice, including explicit formulas for its dimension and determinant, and we present efficient algorithms to construct lattice bases, using only cycles as generators, in quadratic time. By algebraic considerations, we relate these results to the more general setting with coefficients from an arbitrary Abelian group. Our results generalize classical results for the vector space of cycles of a graph over the binary field to the case of an arbitrary field.

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arXiv2020

Classification of triples of lattice polytopes with a given mixed volume

We present an algorithm for the classification of triples of lattice polytopes with a given mixed volume $m$ in dimension 3. It is known that the classification can be reduced to the enumeration of so-called irreducible triples, the number of which is finite for fixed $m$. Following this algorithm, we enumerate all irreducible triples of normalized mixed volume up to 4 that are inclusion-maximal. This produces a classification of generic trivariate sparse polynomial systems with up to 4 solutions in the complex torus, up to monomial changes of variables. By a recent result of Esterov, this leads to a description of all generic trivariate sparse polynomial systems that are solvable by radicals.

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arXiv2020

Optimizing Sparsity over Lattices and Semigroups

Motivated by problems in optimization we study the sparsity of the solutions to systems of linear Diophantine equations and linear integer programs, i.e., the number of non-zero entries of a solution, which is often referred to as the $\ell_0$-norm. Our main results are improved bounds on the $\ell_0$-norm of sparse solutions to systems $A x = b$, where $A \in \mathbb{Z}^{m \times n}$, $b \in \mathbb{Z}^m$ and $x$ is either a general integer vector (lattice case) or a non-negative integer vector (semigroup case). In the lattice case and certain scenarios of the semigroup case, we give polynomial time algorithms for computing solutions with $\ell_0$-norm satisfying the obtained bounds.

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arXiv2020

A local maximizer for lattice width of $3$-dimensional hollow bodies

The second and fourth authors have conjectured that a certain hollow tetrahedron $Δ$ of width $2+\sqrt2$ attains the maximum lattice width among all three-dimensional convex bodies. We here prove a local version of this conjecture: there is a neighborhood $U$ of $Δ$ in the Hausdorff distance such that every convex body in $U \setminus \{Δ\}$ has width strictly smaller than $Δ$. When the search space is restricted to tetrahedra, we compute an explicit such neighborhood. We also limit the space of possible counterexamples to the conjecture. We show, for example, that their width must be smaller than $3.972$ and their volume must lie in $[2.653, 19.919]$.

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arXiv2020

Complexity of linear relaxations in integer programming

For a set $X$ of integer points in a polyhedron, the smallest number of facets of any polyhedron whose set of integer points coincides with $X$ is called the relaxation complexity $\mathrm{rc}(X)$. This parameter was introduced by Kaibel & Weltge (2015) and captures the complexity of linear descriptions of $X$ without using auxiliary variables. Using tools from combinatorics, geometry of numbers, and quantifier elimination, we make progress on several open questions regarding $\mathrm{rc}(X)$ and its variant $\mathrm{rc}_{\mathbb{Q}}(X)$, restricting the descriptions of $X$ to rational polyhedra. As our main results we show that $\mathrm{rc}(X) = \mathrm{rc}_{\mathbb{Q}}(X)$ when: (a) $X$ is at most four-dimensional, (b) $X$ represents every residue class in $(\mathbb{Z}/2\mathbb{Z})^d$, (c) the convex hull of $X$ contains an interior integer point, or (d) the lattice-width of $X$ is above a certain threshold. Additionally, $\mathrm{rc}(X)$ can be algorithmically computed when $X$ is at most three-dimensional, or $X$ satisfies one of the conditions (b), (c), or (d) above. Moreover, we obtain an improved lower bound on $\mathrm{rc}(X)$ in terms of the dimension of $X$.

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arXiv2020

Inequalities between mixed volumes of convex bodies: volume bounds for the Minkowski sum

In the course of classifying generic sparse polynomial systems which are solvable in radicals, Esterov recently showed that the volume of the Minkowski sum $P_1+\dots+P_d$ of $d$-dimensional lattice polytopes is bounded from above by a function of order $O(m^{2^d})$, where $m$ is the mixed volume of the tuple $(P_1,\dots,P_d)$. This is a consequence of the well-known Aleksandrov-Fenchel inequality. Esterov also posed the problem of determining a sharper bound. We show how additional relations between mixed volumes can be employed to improve the bound to $O(m^d)$, which is asymptotically sharp. We furthermore prove a sharp exact upper bound in dimensions 2 and 3. Our results generalize to tuples of arbitrary convex bodies with volume at least one.

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arXiv2019

Generalized flatness constants, spanning lattice polytopes, and the Gromov width

In this paper we motivate some new directions of research regarding the lattice width of convex bodies. We show that convex bodies of sufficiently large width contain a unimodular copy of a standard simplex. This implies that every lattice polytope contains a minimal generating set of the affine lattice spanned by its lattice points such that the number of generators is bounded by a constant which only depends on the dimension. We also discuss relations to recent results on spanning lattice polytopes and how our results could be viewed as the beginning of the study of generalized flatness constants. Regarding symplectic geometry, we point out how the lattice width of a Delzant polytope is related to upper and lower bounds on the Gromov width of its associated symplectic toric manifold. Throughout, we include several open questions.

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arXiv2019

Computing the volume of the convex hull of the graph of a trilinear monomial using mixed volumes

Speakman and Lee (2017) gave a formula for the volume of the convex hull of the graph of a trilinear monomial, $y=x_1x_2x_3$, over a box in the nonnegative orthant, in terms of the upper and lower bounds on the variables. This was done in the context of using volume as a measure for comparing alternative convexifications to guide the implementation of spatial branch-and-bound for mixed integer nonlinear optimization problems. Here, we introduce an alternative method for computing this volume, making use of the rich theory of mixed volumes. This new method may lead to a natural approach for considering extensions of the problem.

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arXiv2019

Second-order cone representable slices of the positive semidefinite cone of size three

To demonstrate the discrepancy between second-order cone and semidefinite programming, Hamza Fawzi showed that the cone S_+^3 of symmetric positive semidefinite matrices of size 3 is not second-order cone representable (socr). A slice of S_+^3 is intersection of S_+^3 and a linear sub-space of the space S^3 of 3 \times 3 symmetric matrices. It is known that some slices of S_+^3 are socr, while some others are not. We classify socr slices of S_+^3 by showing that a slice of S_+^3 is socr if and if it has dimension at most 4 or is orthogonal to a non-zero singular matrix (where the orthogonality is considered with respect to the standard trace scalar product).

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arXiv2019

Optimal size of linear matrix inequalities in semidefinite approaches to polynomial optimization

The abbreviations LMI and SOS stand for `linear matrix inequality' and `sum of squares', respectively. The cone $Σ_{n,2d}$ of SOS polynomials in $n$ variables of degree at most $2d$ is known to have a semidefinite extended formulation with one LMI of size $\binom{n+d}{n}$. In other words, $Σ_{n,2d}$ is a linear image of a set described by one LMI of size $\binom{n+d}{n}$. We show that $Σ_{n,2d}$ has no semidefinite extended formulation with finitely many LMIs of size less than $\binom{n+d}{n}$. Thus, the standard extended formulation of $Σ_{n,2d}$ is optimal in terms of the size of the LMIs. As a direct consequence, it follows that the cone of $k \times k$ symmetric positive semidefinite matrices has no extended formulation with finitely many LMIs of size less than $k$. We also derive analogous results for further cones considered in polynomial optimization such as truncated quadratic modules, the cones of copositive and completely positive matrices and the cone of sums of non-negative circuit polynomials.

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arXiv2018

Difference between families of weakly and strongly maximal integral lattice-free polytopes

A $d$-dimensional closed convex set $K$ in $\mathbb{R}^d$ is said to be lattice-free if the interior of $K$ is disjoint with $\mathbb{Z}^d$. We consider the following two families of lattice-free polytopes: the family $\mathcal{L}^d$ of integral lattice-free polytopes in $\mathbb{R}^d$ that are not properly contained in another integral lattice-free polytope and its subfamily $\mathcal{M}^d$ consisting of integral lattice-free polytopes in $\mathbb{R}^d$ which are not properly contained in another lattice-free set. It is known that $\mathcal{M}^d = \mathcal{L}^d$ holds for $d \le 3$ and, for each $d \ge 4$, $\mathcal{M}^d$ is a proper subfamily of $\mathcal{L}^d$. We derive a super-exponential lower bound on the number of polytopes in $\mathcal{L}^d \setminus \mathcal{M}^d$ (with standard identification of integral polytopes up to affine unimodular transformations).

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arXiv2018

Local optimality of Zaks-Perles-Wills simplices

In 1982, Zaks, Perles and Wills discovered a d-dimensional lattice simplex S_{d,k} with k interior lattice points, whose volume is linear in k and doubly exponential in the dimension d. It is conjectured that, for all d \ge 3 and k \ge 1, the simplex S_{d,k} is a volume maximizer in the family P^d(k) of all d-dimensional lattice polytopes with k interior lattice points. To obtain a partial confirmation of this conjecture, one can try to verify it for a subfamily of P^d(k) that naturally contains S_{d,k} as one of the members. Currently, one does not even know whether S_{d,k} is optimal within the family S^d(k) of all d-dimensional lattice simplices with k interior lattice points. In view of this, it makes sense to look at even narrower families, for example, some subfamilies of S^d(k). The simplex S_{d,k} of Zaks, Perles and Wills has a facet with only one lattice point in the relative interior. We show that S_{d,k} is a volume maximizer in the family of simplices S \in S^d(k) that have a facet with one lattice point in its relative interior. We also show that, in the above family, the volume maximizer is unique up to unimodular transformations.

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arXiv2018

Equality case in van der Corput's inequality and collisions in multiple lattice tilings

Van der Corput's provides the sharp bound vol(C) \le m 2^d on the volume of a d-dimensional origin-symmetric convex body C that has 2m-1 points of the integer lattice in its interior. For m=1, a characterization of the equality case vol(C)= m 2^d is equivalent to the well-known problem of characterizing tilings by translations of a convex body. It is rather surprising that so far, for m \ge 2, no characterization of the equality case has been available, though a hint to the respective characterization problem can be found in the 1987 monograph of Gruber and Lekkerkerker. We give an explicit characterization of the equality case for all m \ge 2. Our result reveals that, the equality case for m \ge 2 is more restrictive than for $m=1$. We also present consequences of our characterization in the context of multiple lattice tilings.

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arXiv2017

Approximation of corner polyhedra with families of intersection cuts

We study the problem of approximating the corner polyhedron using intersection cuts derived from families of lattice-free sets in $\mathbb{R}^n$. In particular, we look at the problem of characterizing families that approximate the corner polyhedron up to a constant factor, which depends only on $n$ and not the data or dimension of the corner polyhedron. The literature already contains several results in this direction. In this paper, we use the maximum number of facets of lattice-free sets in a family as a measure of its complexity and precisely characterize the level of complexity of a family required for constant factor approximations. As one of the main results, we show that, for each natural number $n$, a corner polyhedron with $n$ basic integer variables and an arbitrary number of continuous non-basic variables is approximated up to a constant factor by intersection cuts from lattice-free sets with at most $i$ facets if $i> 2^{n-1}$ and that no such approximation is possible if $i \leq 2^{n-1}$. When the approximation factor is allowed to depend on the denominator of the fractional vertex of the linear relaxation of the corner polyhedron, we show that the threshold is $i > n$ versus $i \leq n$. The tools introduced for proving such results are of independent interest for studying intersection cuts.

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arXiv2016

Maximum Semidefinite and Linear Extension Complexity of Families of Polytopes

We relate the maximum semidefinite and linear extension complexity of a family of polytopes to the cardinality of this family and the minimum pairwise Hausdorff distance of its members. This result directly implies a known lower bound on the maximum semidefinite extension complexity of 0/1-polytopes. We further show how our result can be used to improve on the corresponding bounds known for polygons with integer vertices. Our geometric proof builds upon nothing else than a simple well-known property of maximum volume inscribed ellipsoids of convex bodies. In particular, it does not rely on factorizations over the semidefinite cone and thus avoids involved procedures of balancing them as required, e.g., in [Briet, Dadush & Pokutta 2015]. We hope that revealing the geometry behind the phenomenon opens doors for further results. Moreover, we show that the linear extension complexity of every d-dimensional 0/1-polytope is bounded from above by O(2^d / d).

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