Emmanuel Hebrard

Filtering Algorithms for the NValue Constraint

The NValue constraint counts the number of different values assigned to a vector of variables. Propagating generalized arc consistency on this constraint is NP-hard. We show that computing even the lower bound on the number of values is NP-hard. We therefore study different approximation heuristics for this problem. We introduce three new methods for computing a lower bound on the number of values. The first two are based on the maximum independent set problem and are incomparable to a previous approach based on intervals. The last method is a linear relaxation of the problem. This gives a tighter lower bound than all other methods, but at a greater asymptotic cost.

Models and strategies for variants of the job shop scheduling problem

Recently, a variety of constraint programming and Boolean satisfiability approaches to scheduling problems have been introduced. They have in common the use of relatively simple propagation mechanisms and an adaptive way to focus on the most constrained part of the problem. In some cases, these methods compare favorably to more classical constraint programming methods relying on propagation algorithms for global unary or cumulative resource constraints and dedicated search heuristics. In particular, we described an approach that combines restarting, with a generic adaptive heuristic and solution guided branching on a simple model based on a decomposition of disjunctive constraints. n nIn this paper, we introduce an adaptation of this technique for an important subclass of job shop scheduling problems (JSPs), where the objective function involves minimization of earliness/tardiness costs.We further show that our technique can be improved by adding domain specific information for one variant of the JSP (involving time lag constraints). In particular we introduce a dedicated greedy heuristic, and an improved model for the case where the maximal time lag is 0 (also referred to as no-wait JSPs).

New Approaches to Constraint Acquisition

In this chapter we present the recent results on constraint acquisition obtained by the Coconut team and their collaborators. In a first part we show how to learn constraint networks by asking the user partial queries. That is, we ask the user to classify assignments to subsets of the variables as positive or negative. We provide an algorithm, called QuAcq, that, given a negative example, finds a constraint of the target network in a number of queries logarithmic in the size of the example. In a second part, we show that using some background knowledge may improve the acquisition process a lot. We introduce the concept of generalization query based on an aggregation of variables into types. We propose a generalization algorithm together with several strategies that we incorporate in QuAcq. Finally we evaluate our algorithms on some benchmarks.

Solving Variants of the Job Shop Scheduling Problem Through Conflict-Directed Search

We introduce a simple technique for disjunctive machine scheduling problems and show that this method can match or even outperform state-of-the-art algorithms on a number of problem types. Our approach combines a number of generic search techniques such as restarts, adaptive heuristics, and solution-guided branching on a simple model based on a decomposition of disjunctive constraints and on the reification of these disjuncts. n nThis paper describes the method and its application to variants of the job shop scheduling problem JSP. We show that our method can easily be adapted to handle additional side constraints and different objective functions, often outperforming the state-of-the-art and closing a number of open problems. Moreover, we perform in-depth analysis of the various factors that make this approach efficient. We show that, while most of the factors give moderate benefits, the variable and value ordering components are key.

The Balance Constraint Family

The Balance constraint introduced by Beldiceanu ensures solutions are balanced. This is useful when, for example, there is a requirement for solutions to be fair. Balance bounds the difference B between the minimum and maximum number of occurrences of the values assigned to the variables. We show that achieving domain consistency on Balance is NP-hard. We therefore introduce a variant, AllBalance with a similar semantics that is only polynomial to propagate. We consider various forms of AllBalance and focus on AtMostallBalance which achieves what is usually the main goal, namely constraining the upper bound on B. We provide a specialized propagation algorithm, and a powerful decomposition both of which run in low polynomial time. Experimental results demonstrate the promise of these new filtering methods.

An optimal arc consistency algorithm for a particular case of sequence constraint

The AtMostSeqCard constraint is the conjunction of a cardinality constraint on a sequence of n variables and of nu2009−u2009qu2009+u20091 constraints AtMostu on each subsequence of size q. This constraint is useful in car-sequencing and crew-rostering problems. In vanxa0Hoeve et al. (Constraints 14(2):273–292, 2009), two algorithms designed for the AmongSeq constraint were adapted to this constraint with an O(2qn) and O(n3) worst case time complexity, respectively. In Maher et al. (2008), another algorithm similarly adaptable to filter the AtMostSeqCard constraint with a time complexity of O(n2) was proposed. In this paper, we introduce an algorithm for achieving arc consistency on the AtMostSeqCard constraint with an O(n) (hence optimal) worst case time complexity. Next, we show that this algorithm can be easily modified to achieve arc consistency on some extensions of this constraint. In particular, the conjunction of a set of mAtMostSeqCard constraints sharing the same scope can be filtered in O(nm). We then empirically study the efficiency of our propagator on instances of the car-sequencing and crew-rostering problems.

Soft constraints of difference and equality

In many combinatorial problems one may need to model the diversity or similarity of sets of assignments. For example, one may wish to maximise or minimise the number of distinct values in a solution. To formulate problems of this type we can use soft variants of the well known ALLDIFFERENT and ALLEQUAL constraints. We present a taxonomy of six soft global constraints, generated by combining the two latter ones and the two standard cost functions, which are either maximised or minimised. We characterise the complexity of achieving arc and bounds consistency on these constraints, resolving those cases for which NP-hardness was neither proven nor disproven. In particular, we explore in depth the constraint ensuring that at least k pairs of variables have a common value. We show that achieving arc consistency is NP-hard, however bounds consistency can be achieved in polynomial time through dynamic programming. Moreover, we show that the maximum number of pairs of equal variables can be approximated by a factor of 1/2 with a linear time greedy algorithm. Finally, we provide a fixed parameter tractable algorithm with respect to the number of values appearing in more than two distinct domains. Interestingly, this taxonomy shows that enforcing equality is harder than enforcing difference.

Explanation-Based Weighted Degree

The weighted degree heuristic is among the state of the art generic variable ordering strategies in constraint programming. However, it was often observed that when using large arity constraints, its efficiency deteriorates significantly since it loses its ability to discriminate variables. A possible answer to this drawback is to weight a conflict set rather than the entire scope of a failed constraint.

Propagation via Kernelization: The Vertex Cover Constraint

The technique of kernelization consists in extracting, from an instance of a problem, an essentially equivalent instance whose size is bounded in a parameter (k). Besides being the basis for efficient parameterized algorithms, this method also provides a wealth of information to reason about in the context of constraint programming. We study the use of kernelization for designing propagators through the example of the Vertex Cover constraint. Since the classic kernelization rules often correspond to dominance rather than consistency, we introduce the notion of “loss-less” kernel. While our preliminary experimental results show the potential of the approach, they also show some of its limits. In particular, this method is more effective for vertex covers of large and sparse graphs, as they tend to have, relatively, smaller kernels.

Scheduling scientific experiments on the rosetta/philae mission

The Rosetta/Philae mission was launched in 2004 by the European Space Agency (ESA). It is scheduled to reach the comet 67P/Churyumov-Gerasimenko in 2014 after traveling more than six billion kilometers. The Philae module will then be separated from the orbiter (Rosetta) to attempt the first ever landing on the surface of a comet. If it succeeds, it will engage a sequence of scientific exploratory experiments on the comet. n nIn this paper we describe a constraint programming model for scheduling the different experiments of the mission. A feasible plan must satisfy a number of constraints induced by energetic resources, precedence relations on activities, or incompatibility between instruments. Moreover, a very important aspect is related to the transfer (to the orbiter then to the Earth) of all the data produced by the instruments. The capacity of inboard memories and the limitation of transfers within visibility windows between lander and orbiter, make the transfer policy implemented on the landers CPU prone to data loss. We introduce a global constraint to handle data transfers. The goal of this constraint is to ensure that data-producing activities are scheduled in such a way that no data is lost. n nThanks to this constraint and to the filtering rules we propose, mission control is now able to compute feasible plans in a few seconds for scenarios where minutes were previously often required. Moreover, in many cases, data transfers are now much more accurately simulated, thus increasing the reliability of the plans.