Non-overlapping Constraints between Convex Polytopes

Beldiceanu, Nicolas; Guo, Qi; Thiel, Sven

doi:10.1007/3-540-45578-7_27

Cited by 11 publications

(17 citation statements)

References 6 publications

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“…Finally, we shall mention that an exact formula for the overlapping set S has been given in [BGT01] in the case where objects are polytopes. The set, in this case, is the convex hull of the points obtained by summing one vertex of the first polytope to one vertex of the second one.…”

Section: Contribution and Related Workmentioning

confidence: 99%

The Non-overlapping Constraint between Objects Described by Non-linear Inequalities

Salas¹,

Chabert²,

Goldsztejn

2014

Lecture Notes in Computer Science

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International audiencePacking 2D objects in a limited space is an ubiquitous problem with many academic and industrial variants. In any case, solving this problem requires the ability to determine where a first object can be placed so that it does not intersect a second, previously placed, object. This subproblem is called the non-overlapping constraint. The complexity of this non-overlapping constraint depends on the type of objects considered. It is simple in the case of rectangles. It has also been studied in the case of polygons. This paper proposes a numerical approach for the wide class of objects described by non-linear inequalities. Our goal here is to calculate the non-overlapping constraint, that is, to describe the set of all positions and orientations that can be assigned to the first object so that intersection with the second one is empty. This is done using a dedicated branch & prune approach. We first show that the non-overlapping constraint can be cast into a Minkowski sum, even if we take into account orientation. We derive from this an inner contractor, that is, an operator that removes from the current domain a subset of positions and orientations that necessarily violate the non-overlapping constraint. This inner contractor is then embedded in a sweeping loop, a pruning technique that was only used with discrete domains so far. We finally come up with a branch & prune algorithm that outperforms Rsolver, a generic state-of-the-art solver for continuous quantified constraints

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Section: Contribution and Related Workmentioning

confidence: 99%

The Non-overlapping Constraint between Objects Described by Non-linear Inequalities

Salas¹,

Chabert²,

Goldsztejn

2014

Lecture Notes in Computer Science

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“…Y Constraints between n-dimensional boxes like two_quad_are_in_contact [32] or two_quad_do_not_overlap [36].…”

Section: Designing Automaton Reformulations For Global Constraintsmentioning

confidence: 99%

Reformulation of Global Constraints Based on Constraints Checkers

et al. 2005

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This article deals with global constraints for which the set of solutions can be recognized by an extended finite automaton whose size is bounded by a polynomial in n, where n is the number of variables of the corresponding global constraint. By reducing the automaton to a conjunction of signature and transition constraints we show how to systematically obtain an automaton reformulation. Under some restrictions on the signature and transition constraints, this reformulation maintains arc-consistency. An implementation based on some constraints as well as on the metaprogramming facilities of SICStus Prolog is available. For a restricted class of automata we provide an automaton reformulation for the relaxed case, where the violation cost is the minimum number of variables to unassign in order to get back to a solution.

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“…-Constraints between S -dimensional boxes like two quad are in contact [17] or two quad do not overlap [7].…”

Section: Applications Of This Techniquementioning

confidence: 99%

Deriving Filtering Algorithms from Constraint Checkers

Beldiceanu

Carlsson

Petit

2004

Principles and Practice of Constraint Programming – CP 2004

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Abstract. This reportdeals with global constraints for which the set of solutions can be recognized by an extended finite automaton whose size is bounded by a polynomial in ¦ , where ¦ is the number of variables of the corresponding global constraint. By reformulating the automaton as a conjunction of signature and transition constraints we show how to systematically obtain a filtering algorithm. Under some restrictions on the signature and transition constraints this filtering algorithm achieves arc-consistency. An implementation based on some constraints as well as on the metaprogramming facilities of SICStus Prolog is available. For a restricted class of automata we provide a filtering algorithm for the relaxed case, where the violation cost is the minimum number of variables to unassign in order to get back to a solution.

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Non-overlapping Constraints between Convex Polytopes

Cited by 11 publications

References 6 publications

The Non-overlapping Constraint between Objects Described by Non-linear Inequalities

The Non-overlapping Constraint between Objects Described by Non-linear Inequalities

Reformulation of Global Constraints Based on Constraints Checkers

Deriving Filtering Algorithms from Constraint Checkers

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