Algorithms for the decomposition of a polygon into convex polygons

Fernández, José María Pérez; Cánovas, Lázaro; Pelegrı́n, Blas

doi:10.1016/s0377-2217(99)00033-8

Cited by 27 publications

(27 citation statements)

References 12 publications

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“…The decomposition algorithm does not introduce additional vertices, and is based on Fernandéz et al (2000). Considering the non-convex polygon P, described by a sequence of vertices < v 1 , .…”

Section: Geometric Constraints Based On the Nofit Polygonmentioning

confidence: 99%

A global constraint for nesting problems

Ribeiro

Carravilla

2008

Artif Intell Rev

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Nesting problems are particularly hard combinatorial problems. They involve the positioning of a set of small arbitrarily-shaped pieces on a large stretch of material, without overlapping them. The problem constraints are bidimensional in nature and have to be imposed on each pair of pieces. This all-to-all pattern results in a quadratic number of constraints. Constraint programming has been proven applicable to this category of problems, particularly in what concerns exploring them to optimality. But it is not easy to get effective propagation of the bidimensional constraints represented via finite-domain variables. It is also not easy to achieve incrementality in the search for an improved solution: an available bound on the solution is not effective until very late in the positioning process. In the sequel of work on positioning non-convex polygonal pieces using a CLP model, this work is aimed at improving the expressiveness of constraints for this kind of problems and the effectiveness of their resolution using global constraints. A global constraint "outside" for the non-overlapping constraints at the core of nesting problems has been developed using the constraint programming interface provided by Sicstus Prolog. The global constraint has been applied together with a specialized backtracking mechanism to the resolution of instances of the problem where optimization by Integer Programming techniques is not considered viable. The use of a global constraint for nesting problems is also regarded as a first step in the direction of integrating Integer Programming techniques within a Constraint Programming model.

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Section: Geometric Constraints Based On the Nofit Polygonmentioning

confidence: 99%

A global constraint for nesting problems

Ribeiro

Carravilla

2008

Artif Intell Rev

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“…References and other applications can be found in Fernández et al (2000), Lien and Amato (2004). The problem that led us to consider such a decomposition was the definition of the feasible set in constrained planar location problems.…”

mentioning

confidence: 99%

“…Their technique obtains convex decompositions by diagonals in O(n + r log r) time, and its inefficiency in terms of the number of pieces is bounded with respect to the optimum: no more than four times the optimal number of convex pieces, as any other algorithm producing partitions which do not contain inessential diagonals (see Corollary 2). Fernández et al (2000) presented new algorithms with the same aim and bound. Although with higher theoretical complexity (see Sect.…”

mentioning

confidence: 99%

“…5). The reason for this is that the algorithms in Fernández et al (2000) try to generate convex polygons with as many vertices as possible, and so the partitions they produce have low cardinality. On the other hand, Hertel and Mehlhorn's algorithm does not follow any strategy with that aim apart from removing inessential diagonals of the triangulation.…”

mentioning

confidence: 99%

“…On the other hand, Hertel and Mehlhorn's algorithm does not follow any strategy with that aim apart from removing inessential diagonals of the triangulation. An implementation of the algorithms in Fernández et al (2000) can be found in Fernández et al (1997).…”

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confidence: 99%

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