Optimal partial discretization orders for discretizable distance geometry

Gonçalves, Douglas; Mucherino, Antonio

doi:10.1111/itor.12249

Cited by 12 publications

(13 citation statements)

References 34 publications

(46 reference statements)

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“…Figure 17 shows an example of this scenario. Given these considerations, we can see that the vertex orderings are important in the determination of solutions, which also happens in other DGPs (Gonçalves and Mucherino, 2016).…”

Section: Exactness Of Bpb Methodsmentioning

confidence: 85%

On distance graph coloring problems

Freitas

Dias

Maculan

et al. 2019

Int Trans Operational Res

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One of the most important classes of combinatorial optimization problems is graph coloring, and there are several variations of this general problem involving additional constraints either on vertices or edges. They constitute models for real applications, such as channel assignment in mobile wireless networks. In this work, we consider some coloring problems involving distance constraints as weighted edges, modeling them as distance geometry problems (DGPs). Thus, the vertices of the graph are considered as embedded on the real line and the coloring is treated as an assignment of positive integers to the vertices, while the distances correspond to line segments, where the goal is to find their feasible intersection. We formulate these coloring problem variants and show feasibility conditions for some problems. We also propose implicit enumeration methods for some of the optimization problems based on branch-and-prune algorithms proposed for DGPs in the literature. An empirical analysis was undertaken, considering equality and inequality constraints, and uniform and arbitrary set of distances. As the main contributions, we propose new variations of vertex coloring problems in graphs, involving a new theoretical model in distance geometry (DG) for vertex coloring problems with generalized adjacency constraints, promoting the correlation between graph theory and DG fields. We also give a characterization and formal proof of polynomial cases for special graph classes, since the general main problem is NP-complete.

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Section: Exactness Of Bpb Methodsmentioning

confidence: 85%

On distance graph coloring problems

Freitas

Dias

Maculan

et al. 2019

Int Trans Operational Res

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“…In the following, we will refer to assumptions (a) and (b) as the discretization assumptions. Such assumptions can be verified only if a vertex ordering is associated to V [10], which is generally referred to as a discretization order when the two assumptions above are satisfied.…”

Section: Definition 1 Given a Simple Weighted Undirected Graph G = (Vmentioning

confidence: 99%

An Efficient Exhaustive Search for the Discretizable Distance Geometry Problem with Interval Data

Mucherino

Lin

2019

Proceedings of the 2019 Federated Conference on Computer Science and Information Systems

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The Distance Geometry Problem (DGP) asks whether a simple weighted undirected graph can be realized in a given space (generally Euclidean) so that a given set of distance constraints (associated to the edges of the graph) is satisfied. The Discretizable DGP (DDGP) represents a subclass of instances where the search space can be reduced to a discrete domain having the structure of a tree. In the ideal case where all distances are precise, the tree is binary and one singleton, representing one possible position for a vertex of the graph, is associated to every tree node. When the distance information is however not precise, the uncertainty on the distance values implies that a three-dimensional region of the search space needs to be assigned to some nodes of the tree. By using a recently proposed coarse-grained representation for DDGP solutions, we extend in this work the branch-and-prune (BP) algorithm so that it can efficiently perform an exhaustive search of the search domain, even when the uncertainty on the distances is important. Instead of associating singletons to nodes, we consider a pair consisting of a box and of a most-likely position for the vertex in this box. Initial estimations of the vertex positions in every box can be subsequently refined by using local optimization. The aim of this paper is twofold: (i) we propose a new simple method for the computation of the three-dimensional boxes to be associated to the nodes of the search tree; (ii) we introduce the resolution parameter ρ, with the aim of controling the similarity between pairs of solutions in the solution set. Some initial computational experiments show that our algorithm extension, differently from previously proposed variants of the BP algorithm, is actually able to terminate the enumeration of the solution set by providing solutions that differ from one another accordingly to the given resolution parameter.

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“…However, when this cardinality is strictly larger than K + 1, then there are many choices for the setû i . This is likely to have an impact on the performance of the BP algorithm [2].…”

Section: The K-discretization Graphmentioning

confidence: 99%

The K-discretization and K-incident graphs for discretizable Distance Geometry

et al. 2018

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Optimal partial discretization orders for discretizable distance geometry

Cited by 12 publications

References 34 publications

On distance graph coloring problems

On distance graph coloring problems

An Efficient Exhaustive Search for the Discretizable Distance Geometry Problem with Interval Data

The K-discretization and K-incident graphs for discretizable Distance Geometry

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