Computing leximin-optimal solutions in constraint networks

AbstractIn many real-world multiobjective optimization problems one needs to find solutions or alternatives that provide a fair compromise between different conflicting objective functions—which could be criteria in a multicriteria context, or agent utilities in a multiagent context—while being efficient (i.e. informally, ensuring the greatest possible overall agents' satisfaction). This is typically the case in problems implying human agents, where fairness and efficiency requirements must be met. Preference handling, resource allocation problems are another examples of the need for balanced compromises between several conflicting objectives. A way to characterize good solutions in such problems is to use the leximin preorder to compare the vectors of objective values, and to select the solutions which maximize this preorder. In this article, we describe five algorithms for finding leximin-optimal solutions using constraint programming. Three of these algorithms are original. Other ones are adapted, in constraint programming settings, from existing works. The algorithms are compared experimentally on three benchmark problems

Symmetry Breaking in Constraint Satisfaction

Abstract Symmetry-breaking formulas, introduced by Crawford, Ginsberg, Luks and Roy, are supplementary conditions that are added to a given constraint-satisfaction problem. They are satisfied by exactly one element (e.g. the lexicographic leader) from each set of "symmetrical points " in the search space and can therefore be used to accelerate the search for a solution without sacrificing solvability. We study the computational complexity of generating lex-leader formulas. We show that, even for abelian groups, it may be intractable to generate all the essential clauses in the "natural " lex-leader formula. Nevertheless, we show that techniques of computational group theory allow efficient construction of small lex-leader formulas for these, and more general, groups

Automated static symmetry breaking in constraint satisfaction problems

Variable symmetries in constraint satisfaction problems can be broken by adding lexicographic ordering constraints. Existing general methods of generating such sets of ordering constraints can produce a huge number of additional constraints. This adds an unacceptable overhead to the solving process. Methods exist by which this large set of constraints can be reduced to a much smaller set automatically, but their application is also prohibitively costly. In contrast, this thesis takes a bottom up approach to generating symmetry breaking constraints. This will involve examining some commonly-occurring families of mathematical groups and deriving a general formula to produce a minimal set of ordering constraints which are sufficient to break all of the symmetry that each group describes. In some cases it is known that there exists no manageable sized sets of constraints to break all symmetries. One example of this occurs with matrix row and column symmetries. In such cases, incomplete symmetry breaking has been used to great effect. Double lex is a commonly used incomplete symmetry breaking technique for row and column symmetries. This thesis also describes another similar method which compares favourably to double lex. The general formulae investigated are used as building blocks to generate small sets of ordering constraints for more complex groups, constructed by combining smaller groups. Through the utilisation of graph automorphism tools and the groups and permutations software GAP we provide a method of defining variable symmetries in a problem as a group. Where this group can be described as the product of smaller groups, with known general formulae, we can construct a minimal set of ordering constraints for that problem automatically. In summary, this thesis provides the theoretical background necessary to apply efficient static symmetry breaking to constraint satisfaction problems. It also goes further, describing how this process can be automated to remove the necessity of having an expert CP practitioner, thus opening the field to a larger number of potential users

Symmetry Breaking in Constraint Satisfaction Problems

Abstract. Two methods performing Symmetry Breaking During Search (SBDS) are presented, as described in [1] and [2]. The first was also the the one that defined SBDS, under the name Symmetry Excluding Search, being the first attempt to use it for symmetries of arbitrary type, in contrast to previous attempts that could only handle certain symmetry types. The basic characteristic is that SBDS does not affect the search procedure, meaning that it does not force it to use paths with a certain order. The method was refined later in [2]. While in [1] mathematical proofs were given in order to obtain the final results, in [2], the use of group theory and the known properties of symmetric groups make the method more compact. Moreover, in the first case the authors had to implement their method by writing explicit code for everything, while in the second case, a combination of two systems, ECL i PS e and GAP, is used, which results in formalizing the use of symmetries, making the separation between the search procedure and symmetry breaking more clear. In addition, the method becomes now easier to use, since the programmer no longer needs to keep in mind how symmetries work. GAP provides all the information needed, leaving the programmer with the need only to implement the search procedure, using directly the results for symmetry exclusion given by GAP. Examples and experimental results are provided for both cases, indicating the significant profit in reducing the search space, and thus search time in solving Constraint Satisfaction Problems.

Automated static symmetry breaking in constraint satisfaction problems

Variable symmetries in constraint satisfaction problems can be broken by adding lexicographic ordering constraints. Existing general methods of generating such sets of ordering constraints can produce a huge number of additional constraints. This adds an unacceptable overhead to the solving process. Methods exist by which this large set of constraints can be reduced to a much smaller set automatically, but their application is also prohibitively costly. In contrast, this thesis takes a bottom up approach to generating symmetry breaking constraints. This will involve examining some commonly-occurring families of mathematical groups and deriving a general formula to produce a minimal set of ordering constraints which are sufficient to break all of the symmetry that each group describes. In some cases it is known that there exists no manageable sized sets of constraints to break all symmetries. One example of this occurs with matrix row and column symmetries. In such cases, incomplete symmetry breaking has been used to great effect. Double lex is a commonly used incomplete symmetry breaking technique for row and column symmetries. This thesis also describes another similar method which compares favourably to double lex. The general formulae investigated are used as building blocks to generate small sets of ordering constraints for more complex groups, constructed by combining smaller groups. Through the utilisation of graph automorphism tools and the groups and permutations software GAP we provide a method of defining variable symmetries in a problem as a group. Where this group can be described as the product of smaller groups, with known general formulae, we can construct a minimal set of ordering constraints for that problem automatically. In summary, this thesis provides the theoretical background necessary to apply efficient static symmetry breaking to constraint satisfaction problems. It also goes further, describing how this process can be automated to remove the necessity of having an expert CP practitioner, thus opening the field to a larger number of potential users.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Symmetry Breaking in Constraint Satisfaction with Graph-Isomorphism: Comma-Free Codes

In this paper the use of graph isomorphism is investigated within the framework of symmetry breaking in constraint satisfaction problems. A running example of Comma-free codes is used to test the methods. But the technique can be extended to other problems. In particular Symmetry Breaking via Dominance Detection (SBDD) is applied to find Comma-Free Codes. To check if a current partial solution is symmetrically equivalent to a previously found no-good, graph isomorphism is used. In particular the powerful and fast graph isomorphism package nauty is used. Experimental results show that for difficult instances SBDD+Nauty out performs lexicographic ordering

In search of a better method to break row and column symmetries

Complete row and column symmetry breaking in constraint satisfaction problems using the lex leader method is generally prohibitively costly. Double lex, which is derived from lex leader, is commonly used in practice as an incomplete symmetry-breaking method for row and column symmetries. This technique uses a row-wise ordering to construct the lex leader. For this reason, it is generally counterproductive to choose a search ordering that is not also row-wise. It seems logical that the search order should be used to pick the symmetry breaking technique, rather than the other way around. This paper surveys other possible orderings and investigates one particular ordering, snake ordering. From this we derive a corresponding incomplete set of symmetry breaking constraints, snake lex. We present experimental data comparing double lex and the snake lex, showing that snake lex is substantially faster than double lex in many cases