Reconstructing 3-Colored Grids from Horizontal and Vertical Projections Is NP-hard

Dürr, Christoph; Guiñez, Flavio; Matamala, Martı́n

doi:10.1007/978-3-642-04128-0_69

Cited by 23 publications

(28 citation statements)

References 8 publications

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“…Later and with different techniques, in [2], this result was improved, by showing that the presence of three different types of atoms is sufficient to maintain the c-color problem NP-hard. Recently, this result has been definitely extended to c = 2 in [5].…”

Section: Introductionmentioning

confidence: 79%

Solving Some Instances of the 2-Color Problem

Brocchi

Frosini

Rinaldi

2009

Discrete Geometry for Computer Imagery

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Abstract. In the field of Discrete Tomography, the 2-color problem consists in determining a matrix whose elements are of two different types, starting from its horizontal and vertical projections. It is known that the one color problem has a polynomial time reconstruction algorithm, while, with k ≥ 2, the k-color problem is NP-complete. Thus, the 2-color problem constitutes an interesting example of a problem just in the frontier between hard and easy problems.In this paper we define a linear time algorithm to solve a set of its instances, where some values of the horizontal and vertical projections are constant, while the others are upper bounded by a positive number proportional to the dimension of the problem. Our algorithm relies on classical studies for the solution of the one color problem.

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Section: Introductionmentioning

confidence: 79%

Solving Some Instances of the 2-Color Problem

Brocchi

Frosini

Rinaldi

2009

Discrete Geometry for Computer Imagery

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“…Then by keeping the colors of all edges [x, y] of K X∪Y , we get a solution for P in K X,Y . From Lemma 7.1 and from the NP-completeness of the image reconstruction problem for any fixed k ≥ 3 (see Dürr et al 2009;Chrobak and Dürr 2001), we obtain the following.…”

Section: Constraints On Unions Of Colorsmentioning

confidence: 93%

“…, n). For the general case with k ≥ 3 colors, the following has been obtained recently in Dürr et al (2009). (m, n, k, H, V) is NP-complete for any fixed k ≥ 3.…”

Section: The Basic Image Reconstruction Problem From Projectionsmentioning

confidence: 99%

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On the use of graphs in discrete tomography

et al. 2009

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In this tutorial paper, we consider the basic image reconstruction problem which stems from discrete tomography. We derive a graph theoretical model and we explore some variations and extensions of this model. This allows us to establish connections with scheduling and timetabling applications. The complexity status of these problems is studied and we exhibit some polynomially solvable cases. We show how various classical techniques of operations research like matching, 2-SAT, network flows are applied to derive some of these results.

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“…Let ≤ |T | be an integer. Suppose that G(m S , m T ) is non-empty, that is, Condition Observe that the conditions (58) and (9) in Theorem 68 can be united as follows.…”

Section: Packing Branchings With Bounds On Sizes On Total Indegreesmentioning

confidence: 99%

Supermodularity in Unweighted Graph Optimization I: Branchings and Matchings

Bérczi

Frank

2018

Mathematics of OR

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The main result of the paper is motivated by the following two, apparently unrelated graph optimization problems: (A) as an extension of Edmonds' disjoint branchings theorem, characterize digraphs comprising k disjoint branchings B i each having a specified number µ i of arcs, (B) as an extension of Ryser's maximum term rank formula, determine the largest possible matching number of simple bipartite graphs complying with degree-constraints. The solutions to these problems and to their generalizations will be obtained from a new min-max theorem on covering a supermodular function by a simple degree-constrained bipartite graph. A specific feature of the result is that its minimum cost extension is already NP-hard. Therefore classic polyhedral tools themselves definitely cannot be sufficient for solving the problem, even though they make some good service in our approach.

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Reconstructing 3-Colored Grids from Horizontal and Vertical Projections Is NP-hard

Cited by 23 publications

References 8 publications

Solving Some Instances of the 2-Color Problem

Solving Some Instances of the 2-Color Problem

On the use of graphs in discrete tomography

Supermodularity in Unweighted Graph Optimization I: Branchings and Matchings

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