This research focuses on problems of combinatorial optimization necessary for mapping combinatorial sets into arithmetic Euclidean space. The analysis shows that there is a class of vertex-located sets that coincide with the set of vertices within their convex hull. The author has proved the theorems on the existence of convex, strongly convex, and differentiable extensions for functions defined on vertex-located sets. An equivalent problem of mathematical programming with convex objective function and functional constraints has been formulated. The author has studied the properties of convex function extremes on vertex-located sets. The research contains the examples of vertex-located combinatorial sets and algorithms for constructing convex, strongly convex, and differentiable extensions for functions defined on these sets. The conditions have been formulated that are sufficient for a minimum value of functions, as well as lower bounds of functions have been defined on the permutation set. The results obtained can be well used for developing new methods of combinatorial optimization.

Convex Extensions in Combinatorial Optimization and Their Applications

The class of combinatorial optimization problems over polyhedral-spherical sets is considered. The results of theory of convex extensions are generalized to certain classes of functions defined on sphere-located and vertex-located sets. The original problem is equivalently formulated as a mathematical programming problem with convex both objective function and functional constraints. A numerical illustration and possible applications of the results to solving combinatorial optimization problems are given.

Properties of Combinatorial Optimization Problems Over Polyhedral-Spherical Sets

$2$-level polytopes naturally appear in several areas of pure and applied mathematics, including combinatorial optimization, polyhedral combinatorics, communication complexity, and statistics. In t...

On 2-Level Polytopes Arising in Combinatorial Settings

/pdf/four-dimensional-polytopes-of-minimum-positive-semidefinite-5ft7cy40lk.pdf

Four-dimensional polytopes of minimum positive semidefinite rank

The positive semidefinite (psd) rank of a polytope is the size of the smallest psd cone that admits an affine slice that projects linearly onto the polytope. The psd rank of a d-polytope is at least d+1, and when equality holds we say that the polytope is psd-minimal. In this paper we develop new tools for the study of psd-minimality and use them to give a complete classification of psd-minimal 4-polytopes. The main tools introduced are trinomial obstructions, a new algebraic obstruction for psd-minimality, and the slack ideal of a polytope, which encodes the space of realizations of a polytope up to projective equivalence. 
Our central result is that there are 31 combinatorial classes of psd-minimal 4-polytopes. We provide combinatorial information and an explicit psd-minimal realization in each class. For 11 of these classes, every polytope in them is psd-minimal, and these are precisely the combinatorial classes of the known projectively unique 4-polytopes. We give a complete characterization of psd-minimality in the remaining classes, encountering in the process counterexamples to some open conjectures.

Four Dimensional Polytopes of Minimum Positive Semidefinite Rank

We propose the first algorithm for enumerating all combinatorial types of 2-level polytopes of a given dimension d, and provide complete experimental results for $d \leqslant 6$. Our approach is based on the notion of a simplicial core, that allows us to reduce the problem to the enumeration of the closed sets of a discrete closure operator, along with some convex hull computations and isomorphism tests.

Enumeration of 2-Level Polytopes

A (convex) polytope $P$ is said to be $2$-level if for every direction of hyperplanes which is facet-defining for $P$, the vertices of $P$ can be covered with two hyperplanes of that direction. The study of these polytopes is motivated by questions in combinatorial optimization and communication complexity, among others. In this paper, we present the first algorithm for enumerating all combinatorial types of $2$-level polytopes of a given dimension $d$, and provide complete experimental results for $d \leqslant 7$. Our approach is inductive: for each fixed $(d-1)$-dimensional $2$-level polytope $P_0$, we enumerate all $d$-dimensional $2$-level polytopes $P$ that have $P_0$ as a facet. This relies on the enumeration of the closed sets of a closure operator over a finite ground set. By varying the prescribed facet $P_0$, we obtain all $2$-level polytopes in dimension $d$.

Enumeration of $2$-level polytopes

We define a biclique to be the complement of a bipartite graph, consisting of two cliques joined by a number of edges. In this paper we study algebraic aspects of the chromatic polynomials of these graphs. We derive a formula for the chromatic polynomial of an arbitrary biclique, and use this to give certain conditions under which two of the graphs have chromatic polynomials with the same splitting field. Finally, we use a subfamily of bicliques to prove the cubic case of the α + n conjecture, by showing that for any cubic integer α, there is a natural number n such that α + n is a chromatic root.

Chromatic Polynomials of Complements of Bipartite Graphs

A dense set of chromatic roots which is closed under multiplication by positive integers

A chromatic root is a zero of the chromatic polynomial of a graph. At a Newton Institute workshop on Combinatorics and Statistical Mechanics in 2008, two conjectures were proposed on the subject of which algebraic integers can be chromatic roots, known as the ``$α +n$ conjecture'' and the ``$nα$ conjecture''. These say, respectively, that given any algebraic integer α there is a natural number $n$ such that $α +n$ is a chromatic root, and that any positive integer multiple of a chromatic root is also a chromatic root. By computing the chromatic polynomials of two large families of graphs, we prove the $α +n$ conjecture for quadratic and cubic integers, and show that the set of chromatic roots satisfying the nα conjecture is dense in the complex plane.

https://hal.archives-ouvertes.fr/hal-01283106/document

Chromatic roots as algebraic integers

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