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26
Function Variables for Constraint Programming
, 2003
"... We introduce function variables to constraint programs (CP), variables whose values are one of (exponentially many) possible functions between two sets. Such variables are useful for modelling problems from domains such as configuration, planning, scheduling, etc. We show that a function variable ca ..."
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Cited by 42 (5 self)
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We introduce function variables to constraint programs (CP), variables whose values are one of (exponentially many) possible functions between two sets. Such variables are useful for modelling problems from domains such as configuration, planning, scheduling, etc. We show that a function variable can be mapped into different representations in terms of integer and set variables, and illustrate how to map constraints stated on a function variable into constraints on integer and set variables. As a result, a constraint model expressed using function variables allows for the generation of alternate CP models. Furthermore, we present an extensive theoretical comparison of models of problems involving injective functions supported by asymptotic and empirical studies. Finally, we present and evaluate a practical modelling tool that is based on a highlevel language that supports function variables. The tool helps users explore different alternate CP models starting from a function model that is easy to develop, understand, and maintain.
Consistency and the Quantified Constraint Satisfaction Problem
, 2007
"... Constraint satisfaction is a very well studied and fundamental artificial intelligence technique. Various forms of knowledge can be represented with constraints, and reasoning techniques from disparate fields can be encapsulated within constraint reasoning algorithms. However, problems involving unc ..."
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Cited by 16 (1 self)
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Constraint satisfaction is a very well studied and fundamental artificial intelligence technique. Various forms of knowledge can be represented with constraints, and reasoning techniques from disparate fields can be encapsulated within constraint reasoning algorithms. However, problems involving uncertainty, or which have an adversarial nature (for example, games), are difficult to express and solve in the classical constraint satisfaction problem. This thesis is concerned with an extension to the classical problem: the Quantified Constraint Satisfaction Problem (QCSP). QCSP has recently attracted interest. In QCSP, quantifiers are allowed, facilitating the expression of uncertainty. I examine whether QCSP is a useful formalism. This divides into two questions: whether QCSP can be solved efficiently; and whether realistic problems can be represented in QCSP. In attempting to answer these questions, the main contributions of this thesis are the following: • the definition of two new notions of consistency; • four new constraint propagation algorithms (with eight variants in total), along with empirical evaluations;
Search strategies for rectangle packing
 of Lecture Notes in Computer Science
, 2008
"... Abstract. Rectangle (square) packing problems involve packing all squares with sizes 1 × 1 to n × n into the minimum area enclosing rectangle (respectively, square). Rectangle packing is a variant of an important problem in a variety of realworld settings. For example, in electronic design automati ..."
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Cited by 15 (2 self)
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Abstract. Rectangle (square) packing problems involve packing all squares with sizes 1 × 1 to n × n into the minimum area enclosing rectangle (respectively, square). Rectangle packing is a variant of an important problem in a variety of realworld settings. For example, in electronic design automation, the packing of blocks into a circuit layout is essentially a rectangle packing problem. Rectangle packing problems are also motivated by applications in scheduling. In this paper we demonstrate that an “offtheshelf ” constraint programming system, SICStus Prolog, outperforms recently developed adhoc approaches by over three orders of magnitude. We adopt the standard CP model for these problems, and study a variety of search strategies and improvements to solve large rectangle packing problems. As well as being over three orders of magnitude faster than the current stateoftheart, we close eight open problems: two rectangle packing problems and six square packing problems. Our approach has other advantages over the stateoftheart, such as being trivially modifiable to exploit multicore computing platforms to parallelise search, although we use only a singlecore in our experiments. We argue that rectangle packing is a domain where constraint programming significantly outperforms handcrafted adhoc systems developed for this problem. This provides the CP community with a convincing success story. 1
Exploiting symmetry in SMT problems
 In Proceedings of CADE23, volume 6803 of LNCS
, 2011
"... Abstract. Methods exploiting problem symmetries have been very successful in several areas including constraint programming and SAT solving. We here recast a technique to enhance the performance of SMTsolvers by detecting symmetries in the input formulas and use them to prune the search space of ..."
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Cited by 7 (2 self)
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Abstract. Methods exploiting problem symmetries have been very successful in several areas including constraint programming and SAT solving. We here recast a technique to enhance the performance of SMTsolvers by detecting symmetries in the input formulas and use them to prune the search space of the SMT algorithm. This technique is based on the concept of (syntactic) invariance by permutation of constants. An algorithm for solving SMT by taking advantage of such symmetries is presented. The implementation of this algorithm in the SMTsolver veriT is used to illustrate the practical benefits of this approach. It results in a significant improvement of veriT’s performances on the SMTLIB benchmarks that places it ahead of the winners of the last editions of the SMTCOMP contest in the QF UF category. 1
An approach to Symmetry Breaking in Distributed Constraint Satisfaction Problems
"... Abstract. Symmetry breaking in Distributed Constraint Satisfaction Problems (DisCSPs) has, to the best of our knowledge, not been investigated. In this paper, we show that most symmetries in DisCSPs are actually weak symmetries, and we demonstrate how weak symmetry breaking techniques can be applied ..."
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Abstract. Symmetry breaking in Distributed Constraint Satisfaction Problems (DisCSPs) has, to the best of our knowledge, not been investigated. In this paper, we show that most symmetries in DisCSPs are actually weak symmetries, and we demonstrate how weak symmetry breaking techniques can be applied to DisCSPs. 1
Orbital shrinking: a new tool for hybrid MIP/CP methods
 In: CPAIOR
, 2013
"... Abstract. Orbital shrinking is a newly developed technique in the MIP community to deal with symmetry issues, which is based on aggregation rather than on symmetry breaking. In a recent work, a hybrid MIP/CP scheme based on orbital shrinking was developed for the multiactivity shift scheduling prob ..."
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Abstract. Orbital shrinking is a newly developed technique in the MIP community to deal with symmetry issues, which is based on aggregation rather than on symmetry breaking. In a recent work, a hybrid MIP/CP scheme based on orbital shrinking was developed for the multiactivity shift scheduling problem, showing significant improvements over previous pure MIP approaches. In the present paper we show that the scheme above can be extended to a general framework for solving arbitrary symmetric MIP instances. This framework naturally provides a new way for devising hybrid MIP/CP decompositions. Finally, we specialize the above framework to the multiple knapsack problem. Computational results show that the resulting method can be orders of magnitude faster than pure MIP approaches on hard symmetric instances. 1
Groupoids and Conditional Symmetry
"... We introduce groupoids – generalisations of groups in which not all pairs of elements may be multiplied, or, equivalently, categories in which all morphisms are invertible – as the appropriate algebraic structures for dealing with conditional symmetries in Constraint Satisfaction Problems (CSPs). We ..."
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We introduce groupoids – generalisations of groups in which not all pairs of elements may be multiplied, or, equivalently, categories in which all morphisms are invertible – as the appropriate algebraic structures for dealing with conditional symmetries in Constraint Satisfaction Problems (CSPs). We formally define the Full Conditional Symmetry Groupoid associated with any CSP, giving bounds for the number of elements that this groupoid can contain. We describe conditions under which a Conditional Symmetry subGroupoid forms a group, and, for this case, present an algorithm for breaking all conditional symmetries that arise at a search node. Our algorithm is polynomialtime when there is a corresponding algorithm for the type of group involved. We prove that our algorithm is both sound and complete – neither gaining nor losing solutions. We report on an implementation of the algorithm.
Symmetries in Modal Logics∗
"... In this paper we develop the theoretical foundations to exploit symmetries in modal logics. We generalize the notion of symmetries of propositional formulas in conjunctive normal form to modal formulas using the framework provided by coinductive modal models introduced in [1]. Hence, the results a ..."
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In this paper we develop the theoretical foundations to exploit symmetries in modal logics. We generalize the notion of symmetries of propositional formulas in conjunctive normal form to modal formulas using the framework provided by coinductive modal models introduced in [1]. Hence, the results apply to a wide class of modal logics including, for example, hybrid logics. We present two graph constructions that enable the reduction of symmetry detection in modal formulas to the graph automorphism detection problem, and we evaluate the graph constructions on modal benchmarks. 1 Symmetries in Automated Theorem Proving Symmetry is a familiar notion. Intuitively, we say that an object is symmetric if “under any kind of transformation at least one property of the object is left invariant ” [2]. Symmetry has many uses. Not only can we study the symmetric properties of an object (geometric, mathematical, etc.) to understand its behavior, but we can also derive specific consequences regarding the object under study based on its symmetry properties, i.e., using a “symmetrybased argument ” [3]. In automated reasoning, many problem classes, in particular those arising from real world applications, present symmetries, and their presence is usually a source of additional complexity since we might end up looking for solutions in symmetrical subspaces of the problem’s search space. Ideally, if we can recognize that such symmetries exist, we might use them to direct a search algorithm to look for solutions only in nonsymmetric parts of the search space, thus reducing the overall difficulty [4]. ∗This work was partially supported by grants ANPCyTPICT2008306, ANPCyTPICT
Symmetry Breaking Constraints: Recent Results
, 2012
"... Symmetry is an important problem in many combinatorial problems. One way of dealing with symmetry is to add constraints that eliminate symmetric solutions. We survey recent results in this area, focusing especially on two common and useful cases: symmetry breaking constraints for row and column symm ..."
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Symmetry is an important problem in many combinatorial problems. One way of dealing with symmetry is to add constraints that eliminate symmetric solutions. We survey recent results in this area, focusing especially on two common and useful cases: symmetry breaking constraints for row and column symmetry, and symmetry breaking constraints for eliminating value symmetry.