810 research outputs found
Lexicographically-ordered constraint satisfaction problems
We describe a simple CSP formalism for handling multi-attribute preference problems with hard constraints, one that combines hard constraints and preferences so the two are easily distinguished conceptually and for purposes of problem solving. Preferences are represented as a lexicographic order over complete assignments based on variable importance and rankings of values in each domain. Feasibility constraints are treated in the usual manner. Since the preference representation is ordinal in character, these problems can be solved with algorithms that do not require evaluations to be represented explicitly. This includes ordinary CSP algorithms, although these cannot stop searching until all solutions have been checked, with the important exception of heuristics that follow the preference order (lexical variable and value ordering). We describe relations between lexicographic CSPs and more general soft constraint formalisms and show how a full lexicographic ordering can be expressed in the latter. We discuss relations with (T)CP-nets, highlighting the advantages of the present formulation, and we discuss the use of lexicographic ordering in multiobjective optimisation. We also consider strengths and limitations of this form of representation with respect to expressiveness and usability. We then show how the simple structure of lexicographic CSPs can support specialised algorithms: a branch and bound algorithm with an implicit cost function, and an iterative algorithm that obtains optimal values for successive variables in the importance ordering, both of which can be combined with appropriate variable ordering heuristics to improve performance. We show experimentally that with these procedures a variety of problems can be solved efficiently, including some for which the basic lexically ordered search is infeasible in practice
On The Complexity and Completeness of Static Constraints for Breaking Row and Column Symmetry
We consider a common type of symmetry where we have a matrix of decision
variables with interchangeable rows and columns. A simple and efficient method
to deal with such row and column symmetry is to post symmetry breaking
constraints like DOUBLELEX and SNAKELEX. We provide a number of positive and
negative results on posting such symmetry breaking constraints. On the positive
side, we prove that we can compute in polynomial time a unique representative
of an equivalence class in a matrix model with row and column symmetry if the
number of rows (or of columns) is bounded and in a number of other special
cases. On the negative side, we show that whilst DOUBLELEX and SNAKELEX are
often effective in practice, they can leave a large number of symmetric
solutions in the worst case. In addition, we prove that propagating DOUBLELEX
completely is NP-hard. Finally we consider how to break row, column and value
symmetry, correcting a result in the literature about the safeness of combining
different symmetry breaking constraints. We end with the first experimental
study on how much symmetry is left by DOUBLELEX and SNAKELEX on some benchmark
problems.Comment: To appear in the Proceedings of the 16th International Conference on
Principles and Practice of Constraint Programming (CP 2010
Symmetry Breaking Constraints: Recent Results
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 symmetryComment: To appear in Proceedings of Twenty-Sixth Conference on Artificial
Intelligence (AAAI-12
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Transformations between HCLP and PCSP
We present a general methodology for transforming between HCLP and PCSP in both directions. HCLP and PCSP each have advantages when modelling problems, and each have advantages when implementing models and solving them. Using the work presented in this paper, the appropriate paradigm can be used for each of these steps, with a meaning-preserving transformation in between if necessary
The Complexity of Computing Optimal Assignments of Generalized Propositional Formulae
We consider the problems of finding the lexicographically minimal (or
maximal) satisfying assignment of propositional formulae for different
restricted formula classes. It turns out that for each class from our
framework, the above problem is either polynomial time solvable or complete for
OptP. We also consider the problem of deciding if in the optimal assignment the
largest variable gets value 1. We show that this problem is either in P or P^NP
complete.Comment: 17 pages, 1 figur
Solving Graph Coloring Problems with Abstraction and Symmetry
This paper introduces a general methodology, based on abstraction and
symmetry, that applies to solve hard graph edge-coloring problems and
demonstrates its use to provide further evidence that the Ramsey number
. The number is often presented as the unknown Ramsey
number with the best chances of being found "soon". Yet, its precise value has
remained unknown for more than 50 years. We illustrate our approach by showing
that: (1) there are precisely 78{,}892 Ramsey colorings; and (2)
if there exists a Ramsey coloring then it is (13,8,8) regular.
Specifically each node has 13 edges in the first color, 8 in the second, and 8
in the third. We conjecture that these two results will help provide a proof
that no Ramsey coloring exists implying that
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