### Article

## The width and integer optimization on simplices with bounded minors of the constraint matrices

In this paper, we will show that the width of simplices defined by systems of linear inequalities can be computed in polynomial time if some minors of their constraint matrices are bounded. Additionally, we present some quasi-polynomial-time and polynomial-time algorithms to solve the integer linear optimization problem defined on simplices minus all their integer vertices assuming that some minors of the constraint matrices of the simplices are bounded.

We consider the problem of planning the ISS cosmonaut training with different objectives. A pre-defined set of minimum qualification levels should be distributed between the crew members with minimum training time differences, training expenses or a maximum of the training level with a limitation of the budget. First, a description of the cosmonaut training process is given. The model are considered for the volume planning problem. The objective of the model is to minimize the differences between the total time of the preparation of all crew members. Then two models are considered for the timetabling planning problem. For the volume planning problem, two algorithms are presented. The first one is aheuristic with a complexity of O(n) operations. The second one consists of a heuristic and exact parts, and it is based on the npartition problem approach.

We consider the problem of trainings planning on ISS. Shown that the problem is a combination of a k Partition Problem and an Assignment Problem. NP-compleeteness is proofed. A heuristic and an exact algorithms are proposed.

The central question that motives this paper is the problem of making up a freight train and the routes on the railway. It is necessary from the set of orders available at the stations to determine time-scheduling and destination routing by railways in order to minimize the total completion time. In this paper it was suggested formulation of this problem by applying integer programming.

The notion of a tolerance of an element of a combinatorial optimization problem is often used for stability analysis of an optimal solution and it is a base for design branch-and-bound algorithms solving such problems. In this paper we show that for the weighted independent set problem on trees with *n* vertices all upper and lower tolerances related to this solution can be computed with *O*(*n*) time.

Using a certain Pick-type formula for the mixed volume of Cayley polyhedra, we compute the multiplicity of the isolated common zero of the maximal minors for a matrix of generic homogeneous polynomials of given degrees.

We consider certain spaces of functions on the circle, which naturally appear in harmonic analysis, and superposition operators on these spaces. We study the following question: which functions have the property that each their superposition with a homeomorphism of the circle belongs to a given space? We also study the multidimensional case.

We consider the spaces of function on the circle whose Fourier transform is p-summable. We obtain estimates for the norms of exponential functions deformed by a C1 -smooth phase.

We consider the spaces of functions on the m-dimensional torus, whose Fourier transform is p -summable. We obtain estimates for the norms of the exponential functions deformed by a C1 -smooth phase. The results generalize to the multidimensional case the one-dimensional results obtained by the author earlier in “Quantitative estimates in the Beurling—Helson theorem”, Sbornik: Mathematics, 201:12 (2010), 1811 – 1836.