Sharp thresholds for constraint satisfaction problems and homomorphisms

The order in which the variables are assigned can have an enormous impact on the time taken by a backtracking search algorithm to solve a constraint satisfaction problem (CSP). The Brelaz heuristic is a dynamic variable ordering heuristic which has been shown to give good results for some classes of binary CSPs when the constraint graph is not complete. Its advantage over the simpler smallest-domain heuristic is that it uses information about the constraint graph. This paper uses theoretical work by Nudel to assess the performance of the Brelaz heuristic. Nudel’s work gives the expected number of nodes at each level of the search tree when using the forward checking algorithm to find all solutions to a CSP, given a specified order of the variables. From this, optimal static orderings are found for a sample of small binary CSPs. The optimal orderings are used to learn rules for a static ordering heuristic, which are converted into modifications to the Brelaz heuristic. The improved heuristic is shown to halve the mean search cost of solving sparse random binary CSPs with 50 variables, at the phase transition. However, our modifications, and the Brelaz heuristic itself, are mainly in the form of improved tie-breakers for the smallest-domain heuristic, which the results suggest is still the basis of good heuristics for this class of problem.

Finding a solution to a Constraint Satisfaction Problem (CSP) is known to be an NP-hard task. This has motivatedthe multitude of works that have been devoted to developing techniques that simplify CSP instances before or duringtheir resolution.The present work proposes rigidly enforced schemes for simplifying binary CSPs that allow the narrowing of valuedomains, either via value merging or via value suppression. The proposed schemes can be viewed as parametrizedgeneralizations of two widely studied CSP simplification techniques, namely, value merging and neighbourhoodsubstitutability. Besides, we show that both schemes may be strengthened in order to allow variable elimination,which may result in more significant simplifications. This work contributes also to the theory of tractable CSPs byidentifying a new tractable class of binary CSP.

A new branch-and-filter exact algorithm for binary constraint satisfaction problems

Evolutionary algorithms (EAs) for solving constraint satisfaction problems (CSPs) can be roughly divided into two classes: EAs with adaptive fitness functions and heuristic-based EAs. A.E. Eiben et al. (1998) compared effective EAs of the first class experimentally using a large set of benchmark instances consisting of randomly-generated binary CSPs. In this paper, we complete this comparison by performing the same experiments using three of the most effective heuristic-based EAs. The results of our experiments indicate that the three heuristic-based EAs have similar performances on random binary CSPs. Comparing these results with those of A.E. Eiben et al., we are able to identify the best EA for binary CSPs as the algorithm introduced by G. Dozier et al. (1994), which uses a heuristic as well as an adaptive fitness function.

The aim of this paper is to establish a connection between the propositional logic and the constraint based reasoning frameworks. This work is based on a translation of the satisfiability problem (SAT) into the binary constraint-satisfaction problem (CSP). The structure of the SAT problem and its associated CSP are then exploited together for characterizing tractable SAT problems, increasing the effectiveness of the classical reduction rules: unit clause and monotone literal rules, and expressing the arc and path consistency concepts with logical inference rules. This study leads to compare the behaviors of the DP and MAC procedures for solving respectively a SAT instance and its binary CSP expression.

Constraint satisfaction arises in many domains in different forms. Search and inference compete for solving constraint satisfaction problems (CSPs) but the most successful approaches are those which benefit from both techniques. Based on this idea, this article introduces a new scheme for solving the general Max-CSP problem. The new approach exploits the simplicity and efficiency of a modified Particle Swarm Optimization and the advantage of adaptable inference levels offered by the Mini-Bucket Elimination algorithm. Experiments conducted on binary CSPs using different levels of inference are illustrative for the inference/search trade-off. Comparative studies highlight the differences between our stochastic population-based method and the systematic search performed by a Branch and Bound algorithm.

We investigate rules which allow variable elimination in binary CSP (constraint satisfaction problem) instances while conserving satisfiability. We study variable-elimination rules based on the language of forbidden patterns enriched with counting and quantification over variables and values. We propose new rules and compare them, both theoretically and experimentally. We give optimised algorithms to apply these rules and show that each define a novel tractable class. Using our variable-elimination rules in preprocessing allowed us to solve more benchmark problems than without.

This paper introduces a genetic local search algorithm for bi­ nary constraint satisfaction problems. The core of the algo­ rithm consists of an ad-hoc optimization procedure followed by the application of blind genetic operators. A standard set of benchmark instances is used in order to assess the performance of the algorithm. The results indicate that this apparently naive hybridation of a genetic algorithm with lo­ cal search yields a rather powerful heuristic algorithm for random binary constraint satisfaction problems.

The exponential-time hypothesis (ETH) states that 3-SAT is not solvable in subexponential time, i.e. not solvable in O(c^n) time for arbitrary c > 1, where n denotes the number of variables. Problems like k-SAT can be viewed as special cases of the constraint satisfaction problem (CSP), which is the problem of determining whether a set of constraints is satisfiable. In this paper we study the worst-case time complexity of NP-complete CSPs. Our main interest is in the CSP problem parameterized by a constraint language Gamma (CSP(Gamma)), and how the choice of Gamma affects the time complexity. It is believed that CSP(Gamma) is either tractable or NP-complete, and the algebraic CSP dichotomy conjecture gives a sharp delineation of these two classes based on algebraic properties of constraint languages. Under this conjecture and the ETH, we first rule out the existence of subexponential algorithms for finite domain NP-complete CSP(Gamma) problems. This result also extends to certain infinite-domain CSPs and structurally restricted CSP(Gamma) problems. We then begin a study of the complexity of NP-complete CSPs where one is allowed to arbitrarily restrict the values of individual variables, which is a very well-studied subclass of CSPs. For such CSPs with finite domain D, we identify a relation SD such that (1) CSP({SD}) is NP-complete and (2) if CSP(Gamma) over D is NP-complete and solvable in O(c^n) time, then CSP({SD}) is solvable in O(c^n) time, too. Hence, the time complexity of CSP({SD}) is a lower bound for all CSPs of this particular kind. We also prove that the complexity of CSP({SD}) is decreasing when |D| increases, unless the ETH is false. This implies, for instance, that for every c>1 there exists a finite-domain Gamma such that CSP(Gamma) is NP complete and solvable in O(c^n) time.

This paper provides a technique for solving general constraint satisfaction problems (CSPs) with continuous variables. Constraints are represented by a hierarchical binary decomposition of the space of feasible values. We propose algorithms for path- and higher degrees of consistency based on logical operations defined on this representation and demonstrate that the algorithms terminate in polynomial time. We show that, in analogy to convex temporal problems and discrete row-convex problems, convexity properties of the solution spaces can be exploited to compute minimal and decomposable networks using path consistency algorithms. Based on these properties, we also show that a certain class of non binary CSPs can be solved using strong 5-consistency.

Constraint satisfaction problems (CSPs) are one of the central topics in theoretical computer science, in particular, in the area of artificial intelligence. Their computational complexity is due to relatively recent results from areas of mathematics, including finite-model-theory, algebra and graph homomorphisms. The main conjecture by Feder and Vardi states that any CSP over a finite relational template is either in P or is NP-complete. Further, it amounts to showing that every non NP-complete CSP can be expressed as an extension of first-order logic. A finite template is Mal'tsev, a compatible algebraic operation, which is closely related to an affine space over a finite field. The so-called Bulatov-Dalmau algorithm, a natural generalization of the Gaussian elimination on vector spaces, shows such CSPs are tractable. In this work, we prove that CSPs described over a finite template Mal'tsev are expressible in logic LFP+rnk, providing a logical proof that such CSPs are tractable.

Feder and Vardi have conjectured that all constraint satisfaction problems to a fixed structure (constraint language) are polynomial or NP-complete. This so-called dichotomy conjecture remains open, although it has been proved in a number of special cases. Most recently, Bulatov has verified the conjecture for conservative structures, i.e., structures which contain all possible unary relations. We explore three different implications of Bulatov's result. First, the above dichotomy can be extended to so-called inclusive structures, corresponding to conservative constraint satisfaction problems in which each variable comes with its own domain. (This has also been independently observed by Bulatov.) We prove a more general version, extending the dichotomy to so-called three-inclusive structures, i.e., structures which contain, with any unary relation R, all unary relations $R'$ for subsets $R' \subseteq R$ with at most three elements. For the constraint satisfaction problems in this generalization we must restrict the instances to so-called 1-full structures, in which each variable is involved in a unary constraint. This leads to our second focus, which is on restrictions to more general kinds of "full" input structures. For any set W of positive integers, we consider a restriction to W-full input structures, i.e., structures in which, for each $w \in W$, any w variables are involved in a w-ary constraint. We identify a class of structures (the so-called W-set-full structures) for which the restriction to W-full input structures does not change the complexity of the constraint satisfaction problem, and hence the family of these restricted problems also exhibits dichotomy. The general family of three-inclusive constraint satisfaction problems restricted to W-full input structures contains examples which we cannot seem to prove either polynomial or NP-complete. Nevertheless, we are able to use our result on the dichotomy for three-inclusive constraint satisfaction problems, to deduce the fact that all three-inclusive constraint satisfaction problems restricted to W-full input structures are NP-complete or "quasi-polynomial" (of order $n^{O(\log n)}$). Our third focus deals with bounding the number of occurrences of a variable, which we call the degree. We conjecture that the complexity classification of three-inclusive constraint satisfaction problems extends to the case where all degrees are bounded by three. Using previous results, we are able to verify this conjecture in a number of special cases. Conservative, inclusive, and three-inclusive constraint satisfaction problems can be viewed as problems in which each variable is restricted to a "list" of allowed values. This point of view of lists is frequently encountered in the study of graph colorings, graph homomorphisms, and graph partitions. Our results presented here, in all three areas, were strongly motivated by these results on graphs.

Relational learning and Inductive Logic Programming (ILP) commonly use as covering test the ?-subsumption test defined by Plotkin. Based on a reformulation of ?-subsumption as a binary constraint satisfaction problem, this paper describes a novel ?-subsumption algorithm named Django,1 which combines well-known CSP procedures and ?-subsumption-specific data structures. Django is validated using the stochastic complexity framework developed in CSPs, and imported in ILP by Giordana et Saitta. Principled and extensive experiments within this framework show that Django improves on earlier ?-subsumption algorithms by several orders of magnitude, and that different procedures are better at different regions of the stochastic complexity landscape. These experiments allow for building a control layer over Django, termed Meta-Django, which determines the best procedures to use depending on the order parameters of the ?-subsumption problem instance. The performance gains and good scalability of Django and Meta-Django are finally demonstrated on a real-world ILP task (emulating the search for frequent clauses in the mutagenesis domain) though the smaller size of the problems results in smaller gain factors (ranging from 2.5 to 30).

The way the graph structure of the constraints influences the complexity of constraint satisfaction problems (CSP) is well understood for bounded-arity constraints. The situation is less clear if there is no bound on the arities. In this case the answer depends also on how the constraints are represented in the input. We study this question for the truth table representation of constraints. We introduce a new hypergraph measure {\em adaptive width} and show that CSP with truth tables is polynomial-time solvable if restricted to a class of hypergraphs with bounded adaptive width. Conversely, assuming a conjecture on the complexity of binary CSP, there is no other polynomial-time solvable case.

The Constraint Satisfaction Problem (CSP) is a central and generic computational problem which provides a common framework for many theoretical and practical applications. A central line of research is concerned with the identification of classes of instances for which CSP can be solved in polynomial time; such classes are often called “islands of tractability.” A prominent way of defining islands of tractability for CSP is to restrict the relations that may occur in the constraints to a fixed set, called a constraint language , whereas a constraint language is conservative if it contains all unary relations. Schaefer’s famous Dichotomy Theorem (STOC 1978) identifies all islands of tractability in terms of tractable constraint languages over a Boolean domain of values. Since then, many extensions and generalizations of this result have been obtained. Recently, Bulatov (TOCL 2011, JACM 2013) gave a full characterization of all islands of tractability for CSP and the counting version #CSP that are defined in terms of conservative constraint languages. This article addresses the general limit of the mentioned tractability results for CSP and #CSP, that they only apply to instances where all constraints belong to a single tractable language (in general, the union of two tractable languages is not tractable). We show that we can overcome this limitation as long as we keep some control of how constraints over the various considered tractable languages interact with each other. For this purpose, we utilize the notion of a strong backdoor of a CSP instance, as introduced by Williams et al. (IJCAI 2003), which is a set of variables that when instantiated, moves the instance to an island of tractability, that is, to a tractable class of instances. We consider strong backdoors into scattered classes , consisting of CSP instances where each connected component belongs entirely to some class from a list of tractable classes. Figuratively speaking, a scattered class constitutes an archipelago of tractability . The main difficulty lies in finding a strong backdoor of given size k ; once it is found, we can try all possible instantiations of the backdoor variables and apply the polynomial time algorithms associated with the islands of tractability on the list component-wise. Our main result is an algorithm that, given a CSP instance with n variables, finds in time f ( k ) n O (1) a strong backdoor into a scattered class (associated with a list of finite conservative constraint languages) of size k or correctly decides that there is not such a backdoor. This also gives the running time for solving (#)CSP, provided that (#)CSP is polynomial-time tractable for the considered constraint languages. Our result makes significant progress towards the main goal of the backdoor-based approach to CSPs—the identification of maximal base classes for which small backdoors can be detected efficiently.