
This article is cited in 1 scientific paper (total in 1 paper)
Network model for the problem of integer balancing of a fourdimensional matrix
A. V. Smirnov^{} ^{} P.G. Demidov Yaroslavl State University, Sovetskaya str., 14, Yaroslavl, 150000, Russia
Abstract:
The problem of integer balancing of a fourdimensional matrix is studied. The elements of the inner part (all four indices are greater than zero) of the given real matrix are summed in each direction and each two and threedimensional 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 with the largest previous or the smallest following integer. At the same time, the element with four zero indices should be produced with standard rules of roundingoff. In the article the problem of finding the maximum multiple flow in the network of any natural multiplicity $k$ is also studied. There are arcs of three types: ordinary arcs, multiple arcs and multiarcs. Each multiple and multiarc is a union of $k$ linked arcs, which are adjusted with each other. The network constructing rules are described. The definitions of a divisible network and some associated subjects are stated. There are defined the basic principles for reducing the integer balancing problem of an $l$dimensional matrix ($l\geq3$) to the problem of finding the maximum flow in a divisible multiple network of multiplicity $k$. There are stated the rules for reducing the fourdimensional balancing problem to the maximum flow problem in the network of multiplicity 5. The algorithm of finding the maximum flow, which meets the solvability conditions for the integer balancing problem, is formulated for such a network.
Keywords:
integer balancing, multiple networks, multiple flows, divisible networks, $NP$completeness, generalized labeling algorithm, fourdimensional matrices.
DOI:
https://doi.org/10.18255/1818101520164466478
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UDC:
519.179.2, 519.854.3 Received: 18.04.2016
Citation:
A. V. Smirnov, “Network model for the problem of integer balancing of a fourdimensional matrix”, Model. Anal. Inform. Sist., 23:4 (2016), 466–478
Citation in format AMSBIB
\Bibitem{Smi16}
\by A.~V.~Smirnov
\paper Network model for the problem of integer balancing of a fourdimensional matrix
\jour Model. Anal. Inform. Sist.
\yr 2016
\vol 23
\issue 4
\pages 466478
\mathnet{http://mi.mathnet.ru/mais515}
\crossref{https://doi.org/10.18255/1818101520164466478}
\mathscinet{http://www.ams.org/mathscinetgetitem?mr=3549347}
\elib{http://elibrary.ru/item.asp?id=26561564}
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