Zivn ´y, 2011b). A dichotomy has even been discovered for classes of binary CSP instances defined by forbidding configurations of incompatibilities (Cohen et al., 2012).
One concrete example of a tractable class defined by forbidding a generic subproblem (known as a pattern) is the set of binary CSP instances satisfying the broken-triangle property (Cooper et al., 2010): a binary CSP instance on variables v 1 , . . . , vn satisfies the broken-triangle property if ∀i < j < k ∈ {1, . . . , n}, whenever the assignments a 1 = hv i , ai , a 2 = hv j , bi , a 3 = hv k , ci , a 4 = hv k , di are such that the pairs of assignments (a 1 , a 2 ), (a 1 , a 3 ), (a 2 , a 4 ) are compatible, then at least one of the pairs of assignments (a 1 , a 4 ), (a 2 , a 3 ) is also compatible. The **forbidden** subproblem is shown **in** Figure 10. For example, any binary CSP instance whose **constraint** graph is a tree satisfies the broken-triangle property for some ordering of its variables; furthermore such an ordering can be determined **in** polynomial time. However, tractability is not due to a property of the **constraint** graph, since instances satisfying the broken-triangle property exist for arbitrary **constraint** graphs. As we will see later, the broken-triangle property also inspired our development of simplification operations based on the absence of **patterns** of compatibilities and incompatibilities on particular variables and values (that we refer to as existential **patterns**).

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There are several interesting directions for further research. Can we generalise the variable or value elimination **patterns** described **in** this paper to arbitrary-arity CSP instances (perhaps using one of the possible definitions of microstructure for constraints of arbitrary arity [25] )? A partial positive answer to this question has recently been provided by arbitrary-arity versions of BTP [12] . Do the variable and value elimination **patterns** introduced **in** this paper generalise to other versions of **constraint** **satisfaction**, such as the QCSP (as is the case for the tractable class defined by BTP [19] ) or the Weighted CSP (as is the case for tractable class defined by the so-called joint-winner pattern [9] )? The research reported **in** the present paper has recently led to the discovery of sound variable and value elimination rules defined by local properties which strictly generalise the absence of **patterns** [15] . The characterisation of all such generalised variable or value elimination rules is a challenging open problem.

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a b s t r a c t
Variable or value elimination **in** a **constraint** **satisfaction** problem (CSP) can be used **in** preprocessing or during search to reduce search space size. A variable elimination rule (value elimination rule) allows the polynomial-time identification of certain variables (domain elements) whose elimination, without the introduction of extra compensatory constraints, does not affect the satisfiability of an instance. We show that there are essentially just four variable elimination rules and three value elimination rules defined by forbidding generic sub-instances, known as irreducible existential **patterns**, **in** arc- consistent CSP instances. One of the variable elimination rules is the already-known Broken Triangle Property, whereas the other three are novel. The three value elimination rules can all be seen as strict generalisations of neighbourhood substitution.

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1 What is tractability?
The idea that an algorithm is efficient if its time complexity is a polynomial function of the size of its input can be traced back to pioneering work of Cobham [45] and Edmonds [81], but the foundations of complexity theory are based on the seminal work of Cook [52] and Karp [119]. A computational decision problem (such as the **constraint** **satisfaction** problem or CSP) consists of a generic instance (**in** the case of the CSP, a set of variables, their domains and a set of constraints) together with a yes-no question (Is there an assignment of values to the variables which simultaneously satisfies all the constraints?). A problem Q is NP-hard if all **problems** P **in** NP are polynomially reducible to Q (and NP-complete if we also have Q ∈ NP). On the other hand, the class P consists of all decision **problems** that can be decided **in** polynomial time. Although P6=NP is still an open question, it is generally assumed to be true, and hence proving that a problem is NP-hard is universally accepted as a proof of computational hardness. The CSP is NP-hard since it includes as a subproblem 3SAT which is a canonical example of an NP-complete problem [52].

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b INSIGHT, University College Cork, Ireland
a b s t r a c t
Although the CSP (**constraint** **satisfaction** problem) is NP-complete, even **in** the case when all constraints are binary, certain classes of instances are tractable. We study classes of binary CSP instances defined by excluding subproblems. This approach has recently led to the discovery of novel tractable classes. The complete characterisation of all tractable classes defined by forbidding **patterns** (where a pattern is simply a compact representation of a set of subproblems) is a challenging problem. We demonstrate a dichotomy **in** the case of **forbidden** **patterns** consisting of either one or two constraints. This has allowed us to discover several new tractable classes including, for example, a novel generalisation of 2SAT. We then extend this dichotomy to existential **patterns** which are only **forbidden** on specific domain values.

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Given that no dichotomy can exist for **constraint** languages over infinite domains, it is natural to restrict attention to classes of constraints which commonly occur **in** real ap- plications. A temporal relation is a relation R ⊆ Qk , for some finite k, with a first-order definition **in** (Q; <), the ordered rational numbers. A temporal **constraint** language is a set of temporal relations. Bodirsky and K´ara showed that there are exactly nine tractable temporal **constraint** languages [20]. Allen’s interval algebra consists of binary relations between intervals which are disjunctions of 13 basic interval relations (such as before, meets, includes, overlaps, starts, finishes or equals) [3]. For the problem of deciding whether there exist intervals on the real line satisfying a set of relations, a dichotomy has been given for all tractable subalgebras of Allen’s algebra [126,127]. **In** the Region Containment Calculus (RCC-5) used **in** spatial reasoning, variables denote non-empty regions and the basic relations **in** the calculus express containment, disjointness, over- lap or equality of pairs of regions. A complete classification of all tractable fragments of the region connection calculus RCC-5 has been given [117].

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A particular data structure named a Quad Tree allows a better representation of solution space of binary continuous constraints Cðx 1 ; x 2 Þ, than classical continuous consistencies. The generation and integration
of this data structure do not raise any particular problem for continuous constraints defined by only one mathematical formula [Sam, D., 1995. **Constraint** consistency techniques for continuous domains. Ph.D. Thesis, E´cole Polytechnique Fe´de´rale de Lausanne]. **In** this paper, we propose to extend the method of generating Quad Trees **in** order to take into account, **in** CSPs, binary continuous constraints defined by a piecewise **constraint**, i.e. a set of functions defined on intervals. The first section presents the industrial requirements which led us to take into account this type of **constraint** **in** CSP. The second section recalls the principles of the Quad Tree. The last section describes our contributions relevant to Quad Tree extensions dealing with piecewise constraints.

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IRIN, Universite & de Nantes, 2 Rue de la Houssinie%re, BP 92208, F-44322 Nantes Cedex 3, France
Abstract
The aim of this paper is to provide a new model of fuzzy or vague properties according to linguistic notions within the scope of declarative modeling and **constraint** **satisfaction** **problems** (CSPs). Some generic modi"ers and generic fuzzy operators are applied on these properties. This model allows us to build a CSP according a pseudo-natural scene description. Solving this CSP provides possible scenes. Moreover, we propose a linguistic interpretation of negative properties instead of the classical logical negation. This negation process selects plausible properties as linguistic negation. Then, it sorts them to provide "rst the ones that have the best chance to be selected by the user. 2000 Elsevier Science Ltd. All rights reserved.

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This follows from the fact that a set of polymorphisms satisfying M is a polyno- mially verifiable certificate for both satisfiable and unsatisfiable instances. Recall that M has a uniform algorithm if and only if CSP(M) ∈ P; Observation 1 tells us that CSP(M) is not NP-hard unless NP = coNP [76]. The latter statement is weaker than the former, but not by a large margin. One could also be tempted to say that NP ∩ coNP **problems** not believed to be **in** P are few **in** number, but keep **in** mind that CSP(M) is a promise problem and constructing very hard NP ∩ coNP promise **problems** is easy by manipulating the promise. For example, given two SAT instances (I 1 , I 2 ) deciding if I 1 is satisfiable is **in** NP ∩ coNP if we have the promise that exactly one instance among (I 1 , I 2 ) is satisfiable, but this problem is unlikely to be **in** P . However, the certificates of those hard NP ∩ coNP promise **problems** often rely crucially on the promise to be useful, while the certificates of CSP(M) (polymorphisms) do not. Overall, Observation 1 should be taken as mild evidence that semiuniformity might imply uniformity, at least **in** some cases. When the complexity of the meta-problem is taken into account as well, we obtain a picture where an idempotent strong linear Mal’tsev condition M with a semiuniform algorithm has a uniform algorithm if and only if the search version of the meta-problem is polynomial-time (a variant of the meta-problem where we have not only to decide if Γ satisfies M, but also produce the polymorphisms that witness it). Because a polynomial-time algorithm for the meta-problem will almost always produce the polymorphisms, this means that replacing the require- ment of uniformity with semiuniformity **in** Proposition 8 is unlikely to provide a generalization **in** the absolute sense but instead make its applicability easier.

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It appears that, from a theoretical point of view at least, basic ordered difference decision diagram are disappointing: they do not satisfy any of the considered requests. And this despite the fact than an order on the variables is required: **in** the DDD framework, this assumption does not provide any good property, contrarily to what happens **in** the classical decision diagram one. The language of path-feasible DDD (and its sub languages) is more interesting from the perspective of queries. Unsurprisingly, compiling a temporal constraints **satisfaction** problem (a TCSP or a DTP instance) as a path- feasible decision diagram is hard, and the compiled form can be exponentially more space-consuming than the original one. The bad news is that even the most basic transformation, namely conditioning, is not satisfied by any kind of path- feasible DDD.

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We have implemented the clique generation and clique **constraint** propaga- tion inside toulbar2 (version 0.9.8), an open-source WCSP solver **in** C++. Among the various benchmarks from [19] (available **in** LP, WCNF, WCSP, UAI, and MiniZinc formats) where cplex reports that clique cuts applied, we chose four problem categories, combinatorial auctions Auction/path, Auction/sched, maximum clique MaxClique, and satellite management SPOT5, a total of 252 **in**- stances, having binary **forbidden** tuples and initial unary cost functions such that the unary-to-clique transform increases the lower bound **in** preprocessing. The first three categories have Boolean domains, whereas SPOT5 has maximum do- main size of 4. For each category, we report **in** Tab.1 the mean value of the size of the problem, the number of cliques found, the number of selected cliques among them and their arity, and the CPU time to find and select the cliques. We limit the maximum number of cliques found to 10,000 **in** order to control the compu- tation time. The largest CPU time was 11.61 seconds for MaxClique/c-fat200-5 (n = 200 variables, e = 11, 627 cost functions, and 10,000 selected cliques of arity 3). The arity of selected cliques varies from 3 to 67 (MaxClique/san1000). We compare solving time to find and prove optimality for toulbar2 exploit- ing cliques (denoted as toulbar2 clq ) against the original code without cliques

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A variable elimination rule allows the polynomial- time identiﬁcation of certain variables whose elim- ination does not affect the satisﬁability of an **in**- stance. Variable elimination **in** the **constraint** sat- isfaction problem (CSP) can be used **in** prepro- cessing or during search to reduce search space size. We show that there are essentially just four variable elimination rules deﬁned by forbidding generic sub-instances, known as irreducible pat- terns, **in** arc-consistent CSP instances. One of these rules is the Broken Triangle Property, whereas the other three are novel.

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version of the **problems** and state a different dichotomy theorem. However, the different classes arising from this classification are not known to be distinct.
1. Introduction
A common way to solve a constrained problem **in** industry consists **in** reducing it to a **satisfaction** problem over propositional logic and using a SAT solver. The generality of the framework and its multiple applications make it a natural subject of interest for the scientific community and **constraint** **satisfaction** **problems** remains an active field of research.

“literals” **in** a single **constraint** may relate different variables. The difficulty is that TCSP and DTP **problems** can not be solved online with the guarantee that the response will be given **in** polynomial time: checking the consistency of this type of network defines a NP-complete problem [2], [3]. Hence, the idea of a preprocessing, a “compilation” of the original problem by its translation into a form that allows an efficient treatment of the queries. It is an emerging idea **in** different areas of IA, like **constraint**-based reasoning [4], [5], product configuration [6], planning under uncertainty [7] or automated reasoning [8]. Technically, this compilation is performed offline, before the online query phase: this relaxes the constraints on its temporal complexity 1 .

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δ
), thus obtaining the best possible inapproximability factor result for this problem. We prove **in** this paper that all the cases that were considered by Creigou [Cre95] and Khanna and Sudan [KS96] **in** the dichotomy theorem become APX -hard and also that most of the trivial maximization **constraint** **satisfaction** **problems** have their balanced version APX - hard. **In** particular, using an inapproximability result for Densest k Subgraph established recently by Khot [Kho04], we prove that the balanced version of Min Monotone-E2Sat has no polynomial time approximation scheme, if NP 6⊆ ∩ δ>0 BTIME (2 n

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Proposition 5 . For each of them found to be a solution to I i − 1 , we recurse with I i − 1 . This requires O (n) time per step, since
again there are at most n − 1 constraints to be checked (those involving x i ) and these have been precomputed. Finally, since at each step either s a or s b is guaranteed to be a solution to I i − 1 , we indeed generate solutions to I with delay O (mn) . ✷
The weaker operation of neighbourhood substitution has the property that two different convergent sequences of elim- inations by neighbourhood substitution necessarily produce isomorphic instances I m 1 , I m 2 [22] . This is not the case for BTP-merging. Firstly, and perhaps rather surprisingly, BTP-merging can have as a side-effect to eliminate broken triangles. This is illustrated **in** the 3-variable instance shown **in** Fig. 2 . **In** order to avoid cluttering up figures with broken lines linking each pair of incompatible assignments, **in** all figures illustrating binary CSP instances, we use the convention that those pairs of assignments which are not explicitly linked with a solid line are incompatible. The instance **in** Fig. 2 (a) contains a broken triangle on values a ′ , b ′ for variable z, but after BTP-merging of values a, b ∈ D(x) into a new value c, as shown **in** Fig. 2 (b),

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1. Introduction
**Constraint** Programming (CP) is widely used to express and solve combinatorial **problems**. Once a problem is modeled as a **constraint** network, efficient solving techniques generate a solution satisfying the constraints, if such a solution exists. However, there are situations where the user has strong opinions about the way to build good solutions to the problem but some of the desirable/undesirable combinations will become clear only once some of the variables are assigned. **In** this case, the **constraint** solver should be there to assist the user **in** the solution design and to ensure her choices remain **in** the feasible space, removing the combinatorial complexity from her shoulders. See the Synthia system for protein design as an early example of using CP to interactively solve a problem [2] . Another well known example of such an interactive solving of **constraint**-based models is product configuration [3,4] . The person modeling the product as a **constraint** network for the company knows its technical and marketing requirements. She models the feasibility, availability and/or marketing constraints about the product. This **constraint** network captures the catalog of possible products, which may contain billions of solutions, but **in** an intentional and compact way. Nevertheless, the modeler does not know the constraints or preferences of the customer(s). This is the customer who will look for solutions, with her own constraints and preferences on the price, the color, or any other configurable feature.

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methods [ 30 ] learn, for each label, a (linear) utility function from which the rank- ing is deduced. Those latter approaches are close to other proposals [ 18 ] that perform a label-wise decomposition.
**In** ranking **problems**, it may also be interesting [ 9 , 18 ] to predict partial rather than complete rankings, abstaining to make a precise prediction **in** presence of too little information. Such predictions can be seen as extensions of the reject option [ 4 ] or of partial predictions [ 11 ]. They can prevent harmful decisions based on incorrect predictions, and have been applied for different decomposi- tion schemes, be it pairwise [ 10 ] or label-wise [ 18 ], always producing cautious predictions **in** the form of partial order relations.

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CRIL-CNRS, University of Artois, Lens, France bessiere@lirmm.fr, fargier@irit.fr, lecoutre@cril.fr
Abstract. Some applications require the interactive resolution of a **constraint** problem by a human user. **In** such cases, it is highly desirable that the person who interactively solves the problem is not given the choice to select values that do not lead to solutions. We call this property global inverse consistency. Existing systems simulate this either by maintaining arc consistency after each assignment performed by the user or by compiling offline the problem as a multi-valued de- cision diagram. **In** this paper, we define several questions related to global inverse consistency and analyse their complexity. Despite their theoretical intractability, we propose several algorithms for enforcing global inverse consistency and we show that the best version is efficient enough to be used **in** an interactive setting on several configuration and design **problems**. We finally extend our contribution to the inverse consistency of tuples.

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3.2 Integration **in** tree search methods
The heuristic presented previously can be grafted into tree search methods, such as discrepancy search or chronological backtracking. For discrepancy search and non-binary trees, two modes can be used to count discrepancies [10, 27]. First, the binary way: exploring the branch associ- ated with the best value, according to a value ordering heuristic, involves no discrepancy, while exploring the remaining branches implies a single discrepancy. Second, the non-binary way: the values are ranked according to a value ordering heuristic such that the best value has rank 1; exploring the branch associated with a value of rank k > 1 leads to make k − 1 discrepancies. **In** the following of this section, the heuristic is integrated into the YIELDS method proposed **in** [20] with a binary counting of discrepancies. **In** the YIELDS method, the Wvar_YIELDS_Probe algo- rithm (see Algorithm 1) is iterated either until a solution is found or until CurrentM axDiscr reached the maximum number of allowed discrepancies or until an inconsistency is detected.

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