This paper presents a new analysis of dynamic constraint satisfaction problems (DCSPs) with finite domains and a new approach to solving them. We first show that even very small changes in a CSP, in the form of addition of constraints or changes in constraint relations, can have profound effects on search performance. These effects are reflected in the amenability of the problem to different forms of heuristic action as well as overall quality of search. In addition, classical DCSP methods perform poorly on these problems because there are sometimes no solutions similar to the original one found. We then show that the same changes do not markedly affect the locations of the major sources of contention in the problem. A technique for iterated sampling that performs a careful assessment of this property and uses the information during subsequent search, performs well even when it only uses information based on the original problem in the DCSP sequence. The result is a new approach to solving DCSPs that is based on a robust strategy for ordering variables rather than on robust solutions.

Optimization in graphical models is an important problem which has been studied in many AI frameworks such as weighted CSP, maximum satisfiability or probabilistic networks. By identifying conditionally independent subproblems, which are solved independently and whose optimum is cached, recent Branch and Bound algorithms offer better asymptotic time complexity. But the locality of bounds induced by decomposition often hampers the practical effects of this result because subproblems are often uselessly solved to optimality. Following the Russian Doll Search (RDS) algorithm, a possible approach to overcome this weakness is to (inductively) solve a relaxation of each subproblem to strengthen bounds. The algorithm obtained generalizes both RDS and tree-decomposition based algorithms such as BTD or AND-OR Branch and Bound. We study its efficiency on different problems, closing a very hard frequency assignment instance which has been open for more than 10 years.

Open forms of global constraints allow the addition of new variables to an argument during the execution of a constraint program.  Such forms are needed for difficult constraint programming problems where problem construction and problem solving are interleaved.  However, in general, filtering that is sound for a global constraint can be unsound when the constraint is open.  This paper provides a simple characterization, called contractibility, of the constraints where filtering remains sound when the constraint is open.  With this characterization we can easily determine whether a constraint is contractible or not.  In the latter case,  we can use it to derive the strongest contractible approximation to the constraint.  We demonstrate how specific algorithms for some closed contractible constraints are easily adapted to open constraints.

Deciding consistency of constraint networks is a fundamental problem in qualitative spatial and temporal reasoning. In this paper we introduce a divide-and-conquer method that recursively partitions a given problem into smaller sub-problems in deciding consistency. We identify a key theoretical property of a qualitative calculus that ensures the soundness and completeness of this method, and show that it is satisfied by the Interval Algebra (IA) and the Point Algebra (PA). We develop a new encoding scheme for IA networks based on a combination of our divide-and-conquer method with an existing encoding of IA networks into SAT. We empirically show that our new encoding scheme scales to much larger problems and exhibits a consistent and significant improvement in efficiency over state-of-the-art solvers on the most difficult instances.

Hard and soft precedence constraints  play  a key role in many application domains.  In telecommunications, one application is the configuration of call control feature subscriptions  where the task is to sequence a set of user-selected features subject to a set of hard (catalogue) precedence constraints and a set of soft (user-selected) precedence constraints. When no such consistent sequence exists, the task is to find an optimal relaxation by discarding some  features or user precedences. For this purpose, we present the global constraint SOFTPREC. Enforcing Generalized Arc Consistency  (GAC)  on SOFTPREC is NP-complete. Therefore, we approximate  GAC based on domain pruning  rules that follow from the semantics of  SOFTPREC; this pruning is polynomial. Empirical results demonstrate  that  the search effort  required by SOFTPREC is  up to one order of magnitude less than  the previously known best CP approach for the feature subscription problem. SOFTPREC is also applicable to other  problem domains  including minimum  cutset problems  for which initial experiments confirm the interest.

Powerful consistency techniques, such as AC* and FDAC*, have been developed for Weighted Constraint Satisfaction Problems (WCSPs) to reduce the space in solution search, but are restricted to only unary and binary constraints.  On the other hand, van Hoeve et al developed efficient graph-based algorithms for handling soft constraints as classical constraint optimization problems. We prove that naively incorporating van Hoeve's method into the WCSP framework can enforce a strong form of varnothing-Inverse Consistency, which can prune infeasible values and deduce good lower bound estimates.  We further show how Van Hoeve's method can be modified so as to handle cost projection and extension to maintain the stronger AC* and FDAC* generalized for non-binary constraints.  Using the soft allDifferent constraint as a testbed, preliminary results demonstrate that our proposal gives improvements up to an order of magnitude both in terms of time and pruning.

Set variables in constraint satisfaction problems (CSPs) are typically propagated by enforcing set bounds consistency together with cardinality reasoning, which uses some inference rules involving the cardinality of a set variable to produce more prunings than set bounds propagation alone. Multiset variables are a generalization of set variables by allowing the elements to have repetitions. In this paper, we generalize cardinality reasoning for multiset variables. In addition, we propose to exploit the variety of a multiset — the number of distinct elements in it — to improve modeling expressiveness and further enhance constraint propagation. We derive a number of inference rules involving the varieties of multiset variables. The rules interact varieties with the traditional components of multiset variables (such as cardinalities) to obtain stronger propagation. We also demonstrate how to apply the rules to perform variety reasoning on some common multiset constraints. Experimental results show that performing variety reasoning on top of cardinality reasoning can effectively reduce more search space and achieve better runtime in solving multiset CSPs.

Branch and bound is an effective technique for solving constraint optimization problems (COP’s). However, its search space expands very rapidly as the domain sizes of the problem variables grow. In this paper, we present an algorithm that clusters the values of a variable’s domain into sets. Branch and bound can then branch on these sets of values rather than on individual values, thereby reducing the branching factor of its search space. The aim of our clustering algorithm is to construct a collection of sets such that branching on these sets will still allow effective bounding. In conjunction with the reduced branching factor, the size of the explored search space is thus significantly reduced. We test our method and show empirically that it can yield significant performance gains over existing stateof- the-art techniques.

Decomposition is an effective technique for solving discrete Constraint Optimization Problems (COPs) with low tree-width. On problems with high treewidth, however, existing decomposition algorithms offer little advantage over branch and bound search (B&B). In this paper we propose a method for exploiting decomposition on problems with high treewidth. Our technique involves modifying B&B to detect and exploit decomposition on a selected subset of the problem’s objectives. Decompositions over this subset, generated during search, are exploited to compute tighter bounds allowing B&B to prune more of its search space. We present a heuristic for selecting an appropriate subset of objectives—one that readily decomposes during search and yet can still provide good bounds. We demonstrate empirically that our approach can significantly improve B&B’s performance and outperform standard decomposition algorithms on a variety of high tree-width problems.

The main result of this paper is to show that the problem of instantiating a finite and path-consistent constraint network of lines in the Euclidean space is NP-complete. Indeed, we already know that reasoning with lines in the Euclidean space is NP-hard. In order to prove that this problem is NP-complete, we first establish that a particular instance  of this problem can be solved by a nondeterministic polynomial-time algorithm, and then we show that solving any finite and path-consistent constraint network of lines in the Euclidean space is at most as difficult as solving that instance.