arXiv subjects

Alexander Prolubnikov

Publications and source records attributed to Alexander Prolubnikov.

5 recordsLinked to original sources

arXiv2023

Finding the connected components of the graph using perturbations of the adjacency matrix

The problem of finding the connected components of a graph is considered. The algorithms addressed to solve the problem are used to solve such problems on graphs as problems of finding points of articulation, bridges, maximin bridge, etc. A natural approach to solving this problem is a breadth-first search, the implementations of which are presented in software libraries designed to maximize the use of the capabi\-lities of modern computer architectures. We present an approach using perturbations of adjacency matrix of a graph. We check wether the graph is connected or not by comparing the solutions of the two systems of linear algebraic equations (SLAE): the first SLAE with a perturbed adjacency matrix of the graph and the second SLAE with~unperturbed matrix. This approach makes it possible to use effective numerical implementations of SLAE solution methods to solve connectivity problems on graphs. Iterations of iterative numerical methods for solving such SLAE can be considered as carrying out a graph traversal. Generally speaking, the traversal is not equivalent to the traversal that is carried out with breadth-first search. An algorithm for finding the connected components of a graph using such a traversal is presented. For any instance of the problem, this algorithm has no greater computational complexity than breadth-first search, and for~most individual problems it has less complexity.

cs.DM↗

arXiv2022

On the representativeness of approximate solutions of discrete optimization problems with interval objective function

We consider discrete optimization problems with interval uncertatinty of objective function coefficients. The interval uncertainty models measurements errors. A pos\-sible optimal solution is a solution that is optimal for some possible values of the coefficients. Pro\-ba\-bi\-li\-ty of a possible solution is the probability to obtain such coefficients that the solution is optimal. Similarly we define the notion of a possible approximate solution with given accuracy and probability of the solution. A possible approximate solution is an approximate solution that is obtained for some possible values of the coefficients by some fixed approximate algorithm, e.g. by the greedy algorithm. Pro\-ba\-bi\-li\-ty of a such solution is the probability to obtain such coefficients that the algorithm produces the solution as its output. We consider optimal or approximate possible solution un\-re\-pre\-sen\-ta\-ti\-ve if its probability less than some boundary value. The mean approximate solution is a possible approximate solution for midpoints of the coefficients intervals. The solution may be treated as approximate solution for exact values of the coefficients. We show that the share of individual discrete optimization problems instances with unrepresentative mean approximate solution may be wide enough for rather small values of error and the boundary value. The same is true for any other possible approximate solution: all of them may be unrepresentative.

math.OC↗

arXiv2020

The interval greedy algorithm for discrete optimization problems with interval objective function

We consider a wide class of the discrete optimization problems with interval objective function. We give a generalization of the greedy algorithm for the problems. Using the algorithm, we obtain the set of all possible greedy solutions and the set of all possible values of the objective function for the solutions. For a given probability distribution on intervals of objective function' coefficients, we compute probabilities of the solutions, compute expected values of the objective function for them and other probabilistic characteristics of the problem.

cs.DS↗

arXiv2019

An estimation of the greedy algorithm's accuracy for a set cover problem instance

Considering the set cover problem, by modifying the approach that gives a logarithmic approximation guarantee for the greedy algorithm, we obtain an estimation of the greedy algorithm's accuracy for a particular input. We compare the presented estimation to another estimations of this type. We give such examples of the set cover problem instances that the presented estimation sagnificantly improves over linear programming relaxation based estimation.

cs.DS↗

arXiv2015

Reduction of the graph isomorphism problem to equality checking of $n$-variables polynomials and the algorithms that use the reduction

The graph isomorphism problem is considered. We assign modified $n$-variable characteristic polynomials for graphs and reduce the graph isomorphism problem to the problem of the polynomials isomorphism. It is required to find out, is there such a numbering of the second graph's vertices that the polynomials of the graphs are equal. We present algorithms for the graph isomorphism problem that use the reduction. We prove the propositions that justify the possibility of numerical realization of the algorithms for the general case of the graph isomorphism problem. The algorithms perform equality checking of graphs polynomials. We show that probability of obtaining a wrong solution of the graph isomorphism problem by comparing values of graph polynomials is negligible if the mantissa length is sufficiently large. Since, for a graph on $n$ vertices, the graph polynomial has $2^n$ coefficients, its value at some point cannot be evaluated directly for large enough $n$. We show that we can check the equality of the polynomials at some points without direct evaluation of the polynomials values at these points. We prove that it is required $O(n^4)$ elementary machine operations and machine numbers with mantissas length $O(n^2)$ to check equality of the values for the graphs on $n$ vertices. For the worst, it needs an exponential from $n$ time to solve the graph isomorphism problem instance using the presented approach, but in practice, it is efficient even for well known computationally hard instances of the graph isomorphism problem.

cs.DM↗