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Article Dans Une Revue Discrete Applied Mathematics Année : 2024

An impossible combinatorial counting method in distance geometry

Résumé

The Distance Geometry Problem asks for a geometric representation of a given weighted graph in R K so that vertices are points and edges are segments with lengths equal to the corresponding weights. Two problem variants are defined by a vertex order given as part of the input, which allows the use of a branching algorithm based on K-lateration: find two possible positions for the next vertex j using the positions of K predecessors and their distances to j, then explore each position recursively, pruning positions at need. Whereas the first variant only requires the K predecessors to exist, the second variant also requires them to be contiguous and immediately preceding j. For this variant, fixed-parameter tractability of the algorithm can be established by means of a solution counting method that only depends on the graph edges (rather than their weights). Only in the first variant, however, it is possible to efficiently construct a suitable vertex order directly from the graph. Since both fixed-parameter tractability and efficient vertex order construction are desirable properties, one would need an analogous solution counting method for the first variant. In this paper we prove that such a counting method cannot be devised for the first variant.

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Dates et versions

hal-04680675 , version 1 (28-08-2024)

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Germano Abud, Jorge Alencar, Carlile Lavor, Leo Liberti, Antonio Mucherino. An impossible combinatorial counting method in distance geometry. Discrete Applied Mathematics, 2024, 354, pp.83-93. ⟨10.1016/j.dam.2024.02.018⟩. ⟨hal-04680675⟩
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