Reconstructing 3-Colored Grids from Horizontal and Vertical Projections is NP-Hard: A Solution to the 2-Atom Problem in Discrete Tomography

Abstract

International audienceWe consider the problem of coloring a grid using k colors with the restriction that each row and each column has a specific number of cells of each color. This problem has been known as the $(k-1)$-atom problem in the discrete tomography community. In an already classical result, Ryser obtained a necessary and sufficient condition for the existence of such a coloring when two colors are considered. This characterization yields a linear time algorithm for constructing such a coloring when it exists. Gardner et al. showed that for $k\geqslant 7$ the problem is NP-hard. Afterward Chrobak and Dürr improved this result by proving that it remains NP-hard for $k\geqslant 4$. We close the gap by showing that for $k=3$ colors the problem is already NP-hard. In addition, we give some results on tiling tomography problems

On the Degree Sequences of Uniform Hypergraphs

“…, v n ). Numerous papers give some generalizations of this problem for the graphs with colored edges (see [3,6,9,10,13] ) with distinct rows, i.e., A is the incidence matrix of H where rows and columns correspond to hyperedges and vertices, respectively. To our knowledge the problem of the reconstruction of a binary matrix with distinct rows has not be studied in discrete tomography.…”

Solving Multicolor Discrete Tomography Problems by Using Prior Knowledge

“…In the general case, even the 2-color problem results to be NP-complete ( [16]). In this section we describe a particular case where the problem can be solved efficiently.…”

“…Another case could be the one where through a scan of a material we desire to determine impurities in the object represented by atoms of a different type. Unfortunately even considering only two atoms the problem results to be NP-complete, as proven in [16] 1 . Several solvable cases have been studied in [9], [10] and [14], and in [5] it has been shown that for two colors heuristic algorithms can often furnish exact solutions or very good approximations, but anyway it would be desirable to obtain exact algorithms for sets for which we dispose some prior knowledge.…”

Discrete Tomography Reconstruction Algorithms for Images with a Blocking Component

“…To tackle these problems and to include some other information that may model effectively properties of the image to be rebuilt, often discrete tomography problems include some prior knowledge that gives birth to many variations of the reconstruction problem. Some examples that have been studied in literature include connectivity and convexity [4,2], cell coloring [10,3] or skeletal properties [14,15]. Often, with appropriate assumptions, the arising problems result to be connected to other fields of study as timetabling [18], image compression [1], network flow [5], graph theory [7] and combinatorics [13].…”