Heuristic Algorithms for The Problem of Integer Balancing of a Three-dimensional Matrix with Constraints of Second Type

A. V. Smirnov

Modelirovanie i analiz informacionnyh sistem, Tome 21 (2014) no. 4, pp. 132-147.

Résumé

The problem of integer balancing of a three-dimensional matrix with constraints of second type is studied. The elements of the inner part (all three indices are greater than zero) of the three-dimensional matrix are summed in each direction and each section of the matrix; the total sum is also found. These sums are placed into the elements where one or more indices are equal to zero (according to the summing directions). The problem is to find an integer matrix of the same structure, which can be produced from the initial one by replacing the elements of the inner part with the largest previous or the smallest following integer. At the same time, variations of the sums of elements from those in the initial matrix should be less than 2 and the element with three zero indices should be produced with standard rules of rounding-off. Heuristic algorithms for this problem are suggested in the article: layering algorithm being got as generalization of a similar algorithm for the problem with constraints of first type and a new matrix algorithm. The last one consists of three parts: search for the base matrix, search for the maximal matrix and matrix correcting. Each of them is a cyclic change of the integer matrix using from 1 to 3 elements from the inner part. A modification of the matrix algorithm is suggested. The algorithm is directed to more uniform filling of the inner part of the integer matrix. Also, the complexity of all three algorithms is estimated in the article. The comparative analysis of matrix algorithms based on the results of computing experiments is adduced.

Détail

Keywords: integer balancing, three-dimensional matrices, constraints of second type, multiple networks, multiple flows, generalized labeling algorithm, layering algorithm
Mots-clés : matrix algorithm.

Bibliographie

[1] V. S. Roublev, A. V. Smirnov, “$NP$-Completeness of the Integer Balancing Problem for a Three-Dimensional Matrix”, Doklady Mathematics, 82:3 (2010), 912–914 | DOI | MR | Zbl

[2] Roublev V. S., Smirnov A. V., “The Problem of Integer-Valued Balancing of a Three-Dimensional Matrix and Algorithms of Its Solution”, MAIS, 17:2 (2010), 72–98 (in Russian)

[3] Smirnov A. V., “The Problem of Integer-valued Balancing of a Three-dimensional Matrix and Network Model”, MAIS, 16:3 (2010), 70–76 (in Russian)

[4] Smirnov A. V., “Some Solvability Classes for The Problem of Integer Balancing of a Three-dimensional Matrix with Constraints of Second Type”, MAIS, 20:2 (2013), 54–69 (in Russian)

[5] Smirnov A. V., “Zadacha tselochislennogo sbalansirovaniya tryokhmernoy matritsy s ogranicheniyami vtorogo roda”, Modelirovanie i analiz informatsionnykh sistem, Trudy mezhdunarodnoy nauchnoy konferentsii, posvyashchyonnoy 35-letiyu matematicheskogo fakulteta i 25-letiyu fakulteta informatiki i vychislitelnoy tekhniki Yaroslavskogo gosudarstvennogo universiteta im. P. G. Demidova (Yaroslavl, 2012), 164–167 (in Russian)

[6] Rublev V. S., Smirnov A. V., “Flows in Multiple Networks”, Yaroslavsky Pedagogichesky Vestnik, 3:2 (2011), 60–68 (in Russian)

[7] Kondakov A. S., Roublev V. S., “Zadacha sbalansirovaniya matritsy plana”, Doklady Odesskogo seminara po diskretnoy matematike, 2, Astroprint, Odessa, 2005, 24–26 (in Russian)

[8] Korshunova N. M., Roublev V. S., “Zadacha tselochislennogo sbalansirovaniya matritsy”, Sovremennye problemy matematiki i informatiki, 3, Yaroslavl, 2000, 145–150 (in Russian)

[9] L. R. Ford, D. R. Fulkerson, Flows in Networks, Princeton University Press, 1962, 194 pp. | MR | Zbl | Zbl

[10] Roublev V. S., Smirnov A. V., “Zadacha optimalnogo okrugleniya plana valyutnykh schetov”, Kibernetika i vysokie tekhnologii XXI veka, v. 1, NPF “SAKVOEE”, Voronezh, 2008, 112–123 (in Russian)

[11] Mikhaylov G. A., Voytishek A. V., Chislennoe statisticheskoe modelirovanie. Metody Monte-Karlo, Akademiya, M., 2006, 368 pp. (in Russian)