Reconstruction and Clustering in Random Constraint Satisfaction Problems

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.

In many practical settings it is useful to find a small unsatisfiable subset of a given unsatisfiable set of constraints. We study this problem from a parameterized complexity perspective, taking the size of the unsatisfiable subset as the natural parameter where the set of constraints is either (i) given a set of clauses, i.e., a formula in conjunctive normal Form (CNF), or (ii) as an instance of the Constraint Satisfaction Problem (CSP). In general, the problem is fixed-parameter in tractable. For an instance of the propositional satisfiability problem (SAT), it was known to be W[1]-complete. We establish A[2]-completeness for CSP instances, where A[2]-hardness prevails already for the Boolean case. With these fixed-parameter intractability results for the general case in mind, we consider various restricted classes of inputs and draw a detailed complexity landscape. It turns out that often Boolean CSP and CNF formulas behave similarly, but we also identify notable exceptions to this rule. The main part of this article is dedicated to classes of inputs that are induced by Boolean constraint languages that Schaefer [1978] identified as the maximal constraint languages with a tractable satisfiability problem. We show that for the CSP setting, the problem of finding small unsatisfiable subsets remains fixed-parameter intractable for all Schaefer languages for which the problem is non-trivial. We show that this is also the case for CNF formulas with the exception of the class of bijunctive (Krom) formulas, which allows for an identification of a small unsatisfiable subset in polynomial time. In addition, we consider various restricted classes of inputs with bounds on the maximum number of times that a variable occurs (the degree), bounds on the arity of constraints, and bounds on the domain size. For the case of CNF formulas, we show that restricting the degree is enough to obtain fixed-parameter tractability, whereas for the case of CSP instances, one needs to restrict the degree, the arity, and the domain size simultaneously to establish fixed-parameter tractability. Finally, we relate the problem of finding small unsatisfiable subsets of a set of constraints to the problem of identifying whether a given variable-value assignment is entailed or forbidden already by a small subset of constraints. Moreover, we use the connection between the two problems to establish similar parameterized complexity results also for the latter problem.

In this paper we propose a method for integrating constraint propagation algorithms into an optimization procedure for vertex coloring with the goal of finding improved lower bounds. The key point we address is how to get instances of Constraint Satisfaction Problems (CSPs) from a graph coloring problem in order to give rise to new lower bounds outperforming the maximum clique bound. More precisely, the algorithms presented have the common goal of finding CSPs in the graph for which infeasibility can be proven. This is achieved by means of constraint propagation techniques which allow the algorithms to eliminate inconsistencies in the CSPs by updating domains dynamically and rendering such infeasibilities explicit. At the end of this process we use the largest CSP for which it has not been possible to prove infeasibility as an input for an algorithm which enlarges such CSP to get a feasible coloring. We experimented with a set of middle-high density graphs with quite a large difference between the maximum clique and the chromatic number.

Constraint satisfaction problems are of special interest for the artificial intelligence and operations research community due to their many applications. Although heuristics involved in solving these problems have largely been studied in the past, little is known about the relation between instances and the respective performance of the heuristics used to solve them. This paper focuses on both the exploration of the instance space to identify relations between instances and good performing heuristics and how to use such relations to improve the search. Firstly, the document describes a methodology to explore the instance space of constraint satisfaction problems and evaluate the corresponding performance of six variable ordering heuristics for such instances in order to find regions on the instance space where some heuristics outperform the others. Analyzing such regions favors the understanding of how these heuristics work and contribute to their improvement. Secondly, we use the information gathered from the first stage to predict the most suitable heuristic to use according to the features of the instance currently being solved. This approach proved to be competitive when compared against the heuristics applied in isolation on both randomly generated and structured instances of constraint satisfaction problems.

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. This paper 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 isn't 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 \emph{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, i.e., to a tractable class of instances. In this paper, we consider strong backdoors into \emph{scattered classes}, consisting of CSP instances where each connected component belongs entirely to some class from a list of tractable classes. 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 isn't such a backdoor.

The Unique Games Conjecture (UGC) has pinned down the approximability of all constraint satisfaction problems (CSPs), showing that a natural semidefinite programming relaxation offers the optimal worst-case approximation ratio for any CSP. This elegant picture, however, does not apply for CSP instances that are perfectly satisfiable, due to the imperfect completeness inherent in the UGC. For the important case when the input CSP instance admits a satisfying assignment, it therefore remains wide open to understand how well it can be approximated. This work is motivated by the pursuit of a better understanding of the inapproximability of perfectly satisfiable instances of CSPs. Our main conceptual contribution is the formulation of a (hypergraph) version of Label Cover which we call label cover. Assuming a conjecture concerning the inapproximability of V label cover on perfectly satisfiable instances, we prove the following implications: * There is an absolute constant c0 such that for k >= 3, given a satisfiable instance of Boolean k-CSP, it is hard to find an assignment satisfying more than c0 k^2/2^k fraction of the constraints. * Given a k-uniform hypergraph, k >= 2, for all epsilon > 0, it is hard to tell if it is q-strongly colorable or has no independent set with an epsilon fraction of vertices, where q = ceiling[k + sqrt(k) - 0.5]. * Given a k-uniform hypergraph, k >= 3, for all epsilon > 0, it is hard to tell if it is (k-1)-rainbow colorable or has no independent set with an epsilon fraction of vertices. We further supplement the above results with a proof that an ``almost Unique'' version of Label Cover can be approximated within a constant factor on satisfiable instances.

Constraint programming is a successful technology for solving combinatorial problems modeled as constraint satisfaction problems (CSPs). An important extension of constraint technology involves problems that undergo changes that may invalidate the current solution. These problems are called Dynamic Constraint Satisfaction Problems (DynCSP). Many works on dynamic problems sought methods for finding new solutions. In this paper, we focus our attention on studying the robustness of solutions in DynCSPs. Thus, most robust solutions will be able to absorb changes in the constraints of the problem. To this end, we label each constraint with two parameters that measure the degree of dynamism and the quantity of change. Thus, we randomly generate a set of more restricted CSPs by using these labels. The solutions that satisfy more random CSPs will have a higher probability of remain valid under constraints changes of the original CSP.

In this paper, we propose mechanisms to improve instantiation heuristics by incorporating weighted factors on variables. The proposed weight-based heuristics are evaluated on several tree search methods such as chronological backtracking and discrepancy-based search for both constraint satisfaction and optimization problems. Experiments are carried out on random constraint satisfaction problems, car sequencing problems, and jobshop scheduling with time-lags, considering various parameter settings and variants of the methods.The results show that weighting mechanisms reduce the tree size and then speed up the solving time, especially for the discrepancy-based search method.

It is widely believed that for many optimization problems, no algorithm is substantially more efficient than exhaustive search. This means that finding optimal solutions for many practical problems is completely beyond any current or projected computational capacity. To understand the origin of this extreme 'hardness', computer scientists, mathematicians and physicists have been investigating for two decades a connection between computational complexity and phase transitions in random instances of constraint satisfaction problems. Here we present a mathematically rigorous method for locating such phase transitions. Our method works by analysing the distribution of distances between pairs of solutions as constraints are added. By identifying critical behaviour in the evolution of this distribution, we can pinpoint the threshold location for a number of problems, including the two most-studied ones: random k-SAT and random graph colouring. Our results prove that the heuristic predictions of statistical physics in this context are essentially correct. Moreover, we establish that random instances of constraint satisfaction problems have solutions well beyond the reach of any analysed algorithm.

A constraint satisfaction problem (CSP) is, in its most general form, an NP-complete problem. One of the several classes of tractable problems that exist contains all the problems with a restricted structure of the constraint scopes. This paper studies community structure, a particular type of restricted structure underpinning a class of tractable SAT problems with potentially similar relevance to CSPs. Using the modularity, it explores the community structure of a wide variety of problems with both academic and industrial relevance. Its impact on the search times of several general solvers, as well as one that uses tree-decomposition, is also analysed to determine whether constraint solvers exploit this type of structure. Nearly all CSP instances have a strong community structure, and those belonging to the same class have comparable modularity values. For the general solvers, strong correlations between the community structure and the search times are not apparent. A more definite correlation exists between the modularity and the search times of the tree-decomposition, suggesting that it might, in part, be able to take advantage of the community structure. However, combined with the relatively strong correlation between the modularity and the tree-width, it could also indicate a similarity between these two measures.

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.

Random constraint satisfaction problems (CSP) have been studied extensively using statistical physics techniques. They provide a benchmark to study average case scenarios instead of the worst case one. The interplay between statistical physics of disordered systems and computer science has brought new light into the realm of computational complexity theory, by introducing the notion of clustering of solutions, related to replica symmetry breaking. However, the class of problems in which clustering has been studied often involve discrete degrees of freedom: standard random CSPs are random ( aka disordered Ising models) or random coloring problems ( aka disordered Potts models). In this work we consider instead problems that involve continuous degrees of freedom. The simplest prototype of these problems is the perceptron. Here we discuss in detail the full phase diagram of the model. In the regions of parameter space where the problem is non-convex, leading to multiple disconnected clusters of solutions, the solution is critical at the SAT/UNSAT threshold and lies in the same universality class of the jamming transition of soft spheres. We show how the critical behavior at the satisfiability threshold emerges, and we compute the critical exponents associated to the approach to the transition from both the SAT and UNSAT phase. We conjecture that there is a large universality class of non-convex continuous CSPs whose SAT-UNSAT threshold is described by the same scaling solution.

The use of consistency techniques is the main feature of any Constraint Satisfaction Problems(CSPs)solver. Arc Consistency (AC) applied to preprocessing can reduce the search space by removing the values which are arc inconsistent. Singleton Arc Consistency (SAC) is a stronger level of consistency than AC. Arc consistency technique with heuristic strategy has been proved to be an efficient technique to improve the efficiency of algorithms. In this paper, we first study the well-known SAC-3 algorithm indetail, and propose two algorithms with heuristic strategies for the drawbacks in SAC-3, SAC-3-FFPand SAC-3-MINIsup. Then, we give the correctness proof and complexity analysis. In the end, we implement the two algorithms in some random CSPs and benchmarks during the preprocessing step. Experimental results show that the two algorithms perform better than SAC-3 for most test cases.

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.

In this paper we propose a new type of random CSP model, called Model RB, which is a revision to the standard Model B. It is proved that phase transitions from a region where almost all problems ar...