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Book Sections Year : 2007

Reconstruction of binary matrices under adjacency constraints

Abstract

Given a binary matrix, its horizontal and vertical projections are defined as the sum of its elements for each row and each column, respectively. It is well-known that the basic problem, where the only constraints to verify are both projections, can be solved in polynomial time. Numerous studies deal with this problem when additional constraints have to be taken into account. In this chapter we study two additional constraints for this problem. In a first part we take into account the “non-adjacency” constraint that depends on the definition of neighborhood: if a cell value is 1, then the values of each one of its neighbors must be 0. This problem arises especially in statistical physics to determine the microscopic properties (energy, density, entropy). In the model of Hard Square Gas, two adjacent cells cannot be occupied simultaneously by particles because the energy of repulsion between them is very high. In the second part, we are concerned with a different constraint that imposes that in every row of the binary matrix no isolated cell with value 1 is permitted, and the maximum number of consecutive cells of value 0 is not greater than a prescribed integer k. Taking into account this constraint, the reconstruction of binary matrices has natural applications in timetabling and workforce scheduling.
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hal-01125187 , version 1 (06-03-2015)

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  • HAL Id : hal-01125187 , version 1

Cite

S. Brunetti, Marie-Christine Costa, Andrea Frosini, Fethi Jarray, Christophe Picouleau. Reconstruction of binary matrices under adjacency constraints. Birkhauser/Gabor and Attila. Advances in Discrete Tomography and Its Applications, pp.125-150, 2007, 978-0-8176-3614-2. ⟨hal-01125187⟩

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