Some Zero-One Linear Programming Reformulations for the Maximum Clique Problem

Many combinatorial optimization problems can be expressed in terms of zero-one linear programs. For the maximum clique problem the so-called edge reformulation is applied most commonly. Two less frequently used LP equivalents are the independent set and edge covering set reformulations. The number of the constraints (as a function of the number of vertices of the ground graph) is asymptotically quadratic in the edge and the edge covering set LP reformulations and it is exponential in the independent set reformulation, respectively. F. D. Croce and R. Tadei proposed an approach in which the number of the constraints is equal to the number of the vertices. In this paper we are looking for possible tighter variants of these linear programs.

One method for investigating the solvability of boundary value problems for an implicit differential equation

The article concernes a boundary value problem with linear boundary conditions of general form for the scalar differential equation f(t,x(t),x ̇(t))=y ̂(t), not resolved with respect to the derivative x ̇ of the required function. It is assumed that the function f satisfies the Caratheodory conditions, and the function y ̂ is measurable. The method proposed for studying such a boundary value problem is based on the results about operator equation with a mapping acting from a metric space to a set with distance (this distance satisfies only one axiom of a metric: it is equal to zero if and only if the elements coincide). In terms of the covering set of the function f(t,x_1,•): R→R and the Lipschitz set of the function f(t,•,x_2): R →R, conditions for the existence of solutions and their stability to perturbations of the function f generating the differential equation, as well as to perturbations of the right-hand sides of the boundary value problem: the function y ̂ and the value of the boundary condition, are obtained.

On Some Features of the Problem of Solvability of Systems of Boolean Equations

The purpose of the article is to present new results on combinatorial characteristics of systems of Boolean equations, on which such properties of systems as compatibility, solvability, number of solutions and a number of others depend. The research method is the reduction of applied problems to combinatorial models with the subsequent application of classical methods of combinatorics: the method of generating functions, the method of coefficients, methods for obtaining asymptotics, etc. Obtained result. In this paper, we obtain results concerning the solvability of systems of Boolean equations. The complexity of the problem of “ transformation” of an incompatible system into a joint one is analyzed. An approach to solving the problem of separating the minimum number of joint subsystems from an incompatible system is described and justified. The problem is reduced to the problem of finding the minimum covering set. The system compatibility criterion is obtained. Using the method of coefficients, formulas for finding and estimating the number of solutions for parameterizing the problem on the right-hand sides of equations are derived. The maximum of this number is also investigated depending on the parameter. Formulas for the number of solutions for two special cases are obtained: with a restriction on the number of equations and on the size of the problem parameters

Sensor Coverage Strategy in Underwater Wireless Sensor Networks

This paper mainly describes studies hydrophone placement strategy in a complex underwater environment model to compute a set of "good" locations where data sampling will be most effective. Throughout this paper it is assumed that a 3-D underwater topographic map of a workspace is given as input.Since the negative gradient direction is the fastest descent direction, we fit a complex underwater terrain to a differentiable function and find the minimum value of the function to determine the low-lying area of the underwater terrain.The hydrophone placement strategy relies on gradient direction algorithm that solves a problem of maximize underwater coverage: Find the maximize coverage set of hydrophone inside a 3-D workspace. After finding the maximize underwater coverage set, to better take into account the optimal solution to the problem of data sampling, the finite VC-dimension algorithm computes a set of hydrophone that satisfies hydroacoustic signal energy loss constraints. We use the principle of the maximize splitting subset of the coverage set and the ”dual” set of the coverage covering set, so as to find the hitting set, and finally find the suboptimal set (i.e., the sensor suboptimal coverage set).Compared with the random deployment algorithm, although the computed set of hydrophone is not guaranteed to have minimum size, the algorithm does compute with high network coverage quality.

A Novel Composite Method using the Simplified Cuckoo Optimization Algorithm and Harmony Search for Cancer Classification

For each malignant development type, simply slight characteristics are using. The quality perseverance work remains a difficult 1. To conquer this issue, all of us propose the double degree quality dedication Technique known as MRMR-SCOAHS. Within the principal stage, the base repeating and max-imam pertinence (MRMR) highlight willpower is employed to pick a subsection, subdivision, subgroup, subcategory, subclass of substantial qualities. The actual favored features are after that nourished right into a covering set up that combine another computation, SCOA-HS, making use of the help vector machine like a classifier. The particular strategy had been implemented in order to four microarray datasets, and also the exhibition has been broke down through forget about one particular crossacknowledgment method. Temporary performance investigation from the expert introduced strategy to developmental computations suggested that this proposed calculations amazing is better than other program in selecting a less amount of qualities whilst safeguarding the greatest order accuracy. The methods in the pre-owned attributes were furthermore explored, also it was accepted that they select qualities tend to be organically vital that you every malignancy type.

On coincidence points of mappings in generalized metric spaces

Let 𝑋 be a space with ∞-metric 𝜌 (a metric with possibly infinite value) and 𝑌 a space with ∞-distance 𝑑 satisfying the identity axiom. We consider the problem of coincidence point for mappings 𝐹,𝐺:𝑋→𝑌, i.e. the problem of existence of a solution for the equation 𝐹(𝑥)=𝐺(𝑥). We provide conditions of the existence of coincidence points in terms of a covering set for the mapping 𝐹 and a Lipschitz set for the mapping 𝐺 in the space 𝑋×𝑌. An 𝛼-covering set (𝛼>0) of the mapping 𝐹 is a set of (𝑥,𝑦) such that ∃𝑢∈𝑋 𝐹(𝑢)=𝑦, 𝜌(𝑥,𝑢)≤𝛼−1𝑑(𝐹(𝑥),𝑦), 𝜌(𝑥,𝑢)<∞, and a 𝛽 - Lipschitz set (𝛽≥0) for the mapping 𝐺 is a set of (𝑥,𝑦) such that ∀𝑢∈𝑋 𝐺(𝑢)=𝑦⇒𝑑(𝑦,𝐺(𝑥))≤𝛽𝜌(𝑢,𝑥). The new results are compared with the known theorems about coincidence points.