Extended analysis of intelligent backtracking algorithms for the maximal constraint satisfaction problem

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.

We study two natural extensions of Constraint Satisfaction Problems (CSPs). Balance-Max-CSP requires that in any feasible assignment each element in the domain is used an equal number of times. An instance of Hard-Max-CSP consists of soft constraints and hard constraints, and the goal is to maximize the weight of satisfied soft constraints while satisfying all the hard constraints. These two extensions contain many fundamental problems not captured by CSPs, and challenge traditional theories about CSPs in a more general framework. Max-2-SAT and Max-Horn-SAT are the only two nontrivial classes of Boolean CSPs that admit a robust satisfibiality algorithm, i.e., an algorithm that finds an assignment satisfying at least (1 ? g(?)) fraction of constraints given a (1 ? ?)-satisfiable instance, where g(?) ? 0 as ? ? 0, and g(0) = 0. We prove the inapproximability of these problems with balance or hard constraints, showing that each variant changes the nature of the problems significantly (in different ways). For instance, deciding whether an instance of 2-SAT admits a balanced assignment is NP-hard, and for Max-2-SAT with hard constraints, it is hard to find a constant-factor approximation even on (1 ? ?)-satisfiable instances (in particular, the version with hard constraints does not admit a robust satisfiability algorithm). We also study hardness results for a certain CSP over a larger domain capturing ordering constraints: we show that hard constraints rule out constant-factor approximation algorithms. All our hardness results are almost optimal -- they completely rule out algorithms with certain properties, or can be matched by simple extensions to existing algorithms.

Handbook of Constraint Programming

This paper presents a VLSI design for a competitive neural network model, known as GENET (Wang and Tsang 1991), for solving Constraint Satisfaction Problems (CSP). The CSP is a mathematical abstraction of the problems in many AI application domains. In essence, a CSP can be defined as a triple (Z, D, C), where Z is a finite set of variables, D is a mapping from every variable to a domain, which is a finite set of arbitrary objects, and C is a set of constraints. Each constraint in C restricts the values that can be simultaneously assigned to a number of variables in Z. If the constraints in C involve up to but no more than n variables it is called an n-ary CSP. The task is to assign one value per variable satisfying all the constraints in C (Mackworth 1977). In addition, associated with the variable assignments might be costs and utilities. This turns CSPs into optimization problems, demanding CSP solvers to find a set of variable assignments that would produce a maximum total utility at a minimal cost. Furthermore, some CSPs might be over-constrained, i.e. not all the constraints in C can be satisfied simultaneously. In this case, the set of assignments to a maximum number of variables without violating any constraints in C, or the set of assignments to all the variables which violates a minimal number of constraints might be sought for.

Over-constrained problems are ubiquitous in real-world applications. In constraint programming, over-constrained problems can be modeled and solved using soft constraints. Soft constraints, as opposed to hard constraints, are allowed to be violated, and the goal is to find a solution that minimizes the total amount of violation. In this chapter, an overview of recent developments in solution methods for over-constrained problems using constraint programming is presented, with an emphasis on soft global constraints.

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.

Genetic based discrete particle swarm optimization for Elderly Day Care Center timetabling

Conventional techniques for the constraint satisfaction problem (CSP) have had considerable success in their applications. However, there are many areas in which the performance of the basic approaches may be improved. These include heuristic ordering of certain tasks performed by the CSP solver, hybrids which combine compatible solution techniques and graph based methods which exploit the structure of the constraint graph representation of a CSP. Also, conventional constraint satisfaction techniques only address problems with hard constraints (i.e. each of which are completely satisfied or completely violated, and all of which must be satisfied by a valid solution). Many real applications require a more flexible approach which relaxes somewhat these rigid requirements. To address these issues various approaches have been developed. This paper attempts a systematic review of them.

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.

Problems in planning IT services can be regarded as constraint satisfaction problems (CSP). A CSP is defined by a set of variables, a set of values for variables, and a set of constraints over variables. A solution to the CSP is an assignment of variables (i.e., a value for each variable) such that all constraints are satisfied.

The work describes a scheduling system based on an opportunistic framework. It exploits the potentiality of a predictive module that provides a-priori information about the conflicts that may arise in a schedule, managing a constraint based approach in problem solving. The system SOS has been designed and realized to apply in real productive environments this theoretical approach. Requirements of efficiency and flexibility of use have been particularly considered. GUIDELINES AND SCHEDULING STRATEGIES IN SOS In this paper we will present an approach to scheduling problems which is focused on the requirements of real manufacturing environments. Therefore we considered aspects that are tied to the managerial perspective of the problem. The SOS system is designed both for computing solutions in scheduling problems and for supporting manager in obtaining schedules that reflect the state of shop-floor and organizational problems in production. Two aspects have been considered in these perspective: • it develops a temporal predictive method [7] to support a preventive analysis of activity interactions and to detect bottlenecks. It exploits an a-priori analysis of the effects of organizational constraints on the scheduling problem, evaluating alternatives as quick as possible. Transactions on Information and Communications Technologies vol 1, © 1993 WIT Press, www.witpress.com, ISSN 1743-3517 628 Artificial Intelligence in Engineering • If some unexpected event happens on the job shop, it works as a support tool for the evaluation of different reaction decisions (reactive scheduling [4]). The contribution of Al-based techniques is focused especially on the analysis of the effects that scheduling variables have on solutions and on the definition of adequate reasoning patterns. We based our approach on opportunistic reasoning [5]. This term characterizes a problem solving process in which the focus of attention is constantly directed to those choices that appear the most promising for reaching a solution. This policy is expected to reduce the search space (and therefore the computational complexity), focusing the reasoning on areas that are considered as the more constrained (island driving). The capability to provide a feasible solution in a sufficiently short time is fundamental for manufacturing environments, but we must remember that the existence of at least one feasible solution is not always guaranteed. The evaluation of effects of constraints becomes fundamental to identify a solution. Production managers want to know how the decision to change or to relaxing the value of some constraints (concerning for example orders due-dates or priorities) affects the quality of final schedules and the global productive efficiency. This feature is particularly important when a feasible solution does not exist; in this case, from a managerial viewpoint, it is required to the system to support the user in building a feasible solution. From a mathematical perspective we considered scheduling as a Constraint Satisfaction Problem (CSP) [6], concerned with the assignment of values to variables subject to a set of constraints. The solution of the CSP (a feasible set of values assumed by the variables) depends on the constraints defined on the problem: by changing the constraints, the solution will change, too. In solving a CSP, two decisions have to be made at each cycle, i.e. which variable to instantiate next and which value to assign to that variable. This formalization highlights the importance associated to constraints analysis: • they allow the quick computation of a feasible solution, without exploring the entire problem domain. • Through their analysis we can obtain information about how they can be modified opportunistically during scheduling. Transactions on Information and Communications Technologies vol 1, © 1993 WIT Press, www.witpress.com, ISSN 1743-3517 Artificial Intelligence in Engineering 629 We distinguished between two classes of constraints [3], considering the possibility for production managers to change their values in a very short time horizon: Restrictions: they must be always respected and cannot be easily and quickly changed; they can be furtherly divided into causal constraints (they represent conditions to be satisfied before starting an activity), physical constraints (each equipment has specific capabilities that restrict the type and the speed of operations it can accomplish), and resources unavailability. Preferences: restrictions leave scheduling problem underconstrained. Preferences represent the set of managerial constraints and the definition of their values is subject to a decision process. Some examples are: defining orders due date and priority, meeting due-dates, minimizing WIP, maximizing resource utilization, the definition of operational preferences (an order may follow alternative production paths in shop floor), etc. The role of constraints can be underlined considering a second classification, which concerns the links among activities. In a manufacturing environment orders (made of many activities) are processed contemporaneously; moreover many of them may require the same productive resources. These connections cause two kinds of constraints [9]: • intra-order: they represent explicit links existing among activities belonging to an order (for example temporal relations [1]. These constraints are explicitly defined in the problem. • Inter-order: they represent implicit links existing among activities belonging to different orders. For example activities belonging to different orders may require contemporaneously the same productive resource. These constraints remain implicit after the problem definition. THE SOS SCHEDULING PROCESS The goal of our system is to find solutions that satisfies a given set of constraints as quick as possible. If a solution does not exist, the system relaxes only preferences, minimizing the overall effects on final schedule. The scheduling policies followed in SOS can be summarized as follows: 1. Scheduling is made allocating one activities at a time, consider at each step the effects of intraand inter-order constraints; Transactions on Information and Communications Technologies vol 1, © 1993 WIT Press, www.witpress.com, ISSN 1743-3517 630 Artificial Intelligence in Engineering 2. Most constrained and least-impact policies are implemented at each step. First the most-constrained policy selects dynamically on which agent must be focused scheduling attention. Then, the least impact policy chooses for that agent a value whose impact on the rest of the non-scheduled agents is as small as possible. The strategic goal is the identification of critical activities that heavily rely on the possession of highly contended temporal intervals or resources because of intra-order and inter-order interactions (look-ahead strategy). These two policies are based on numeric indexes which account for the particular structure of a problem. They give a measure of the interaction among activities and resources in terms of variable looseness (it is a measure of constraint degree) and value goodness (it is a measure of the impact of a scheduling decision on non-scheduled activities) [9]. Finally -we decided to avoid the utilization of backtracking in SOS systems because of the particular field of application that we identified. In real problems, we have no a priori guarantees concerning the existence of a solution: in this case backtracking is completely useless. Moreover, in order to meet our basic requirements, we prefer to look for a satisfying solution in a short time, instead than to wait for the best solution (which may not exist). V)PI ( Predictive pre-analysis 1 constraints modification / jtraints modification W Result and jus sy ' xi' :•:':':•:•::•:: •;::-:•: j @ Sch, Pr X^x:: :.:••::.::.-::: s display jtification stem

The Dynamic Distributed Double Guided Genetic Algorithm (D3G2A) deals with Maximal Constraint Satisfaction Problems. The approach consists in creating agents cooperating together to solve problems. This paper aims to improve the D3G2A. The main purpose is to balance agent loads this distributed approach. The proposed approach will redistribute the load of Species agents more equally in order to improve the CPU time. This improvement allows not only reduction of the number of Species agents but also decreases communications agents cost. In this regard, a sub-population is composed of chromosomes violating a number of constraints in the same interval. Secondly, another proposed approach will redistribute the work load. This improvement allows diminution of inactive Species agents and it results in a balanced workloads. In fact, by analogy with social animal flocks, Species agents cooperate together to do all tasks in a reduced CPU time. Several comparisons are made about new approaches with the old version of the D3G2A. Results are promising.

Constraint programming for solving various assembly line balancing problems

Constraint programming offers a declarative approach to solving problems modeled as constraint satisfaction problems (CSPs). However, the precise specification of a set of constraints is sometimes not available, but may have to be learned, for instance, from a set of examples of its solutions and non-solutions. In general, one may wish to learn generalized CSPs involving classical, fuzzy, weighted or probabilistic constraints, for example. This paper introduces a unifying framework for CSP learning. The framework is generic in that it can be instantiated to obtain specific formulations for learning classical, fuzzy, weighted or probabilistic CSPs. In particular, a new formulation for classical CSP learning, which minimizes the number of examples violated by candidate CSPs, is obtained by instantiating the framework. This formulation is equivalent to a simple pseudo-boolean optimization problem, thus being efficiently solvable using many optimization tools.

Constraint Satisfaction Problems (CSPs) provide an appropriate framework to formulate many real-world applications including scheduling, planning and resource allocation. However, the CSP description can change due to the evolving environment. The latter points to the fact that constraints might be subject to change over time and this can affect the feasibility of the solutions found so far. These changes can be captured with the Dynamic CSP (DCSP) formalism that has been proposed and investigated in the literature. More formally, a DCSP can be seen as a series of static CSPs, each resulting from a restriction or a relaxation of some constraints in the previous CSP constraint set. This paper focuses on constraint restriction (constraint addition) and the goal is to obtain the most similar solution to the previous one that satisfies the old and new constraints. In this regard, we propose a new method based on the Particle Swarm Optimization algorithm to solve these DCSPs with minimal perturbation. To evaluate the efficiency of the proposed method, we conducted extensive experiments on randomly generated DCSP instances generated by the model RB. The results achieved clearly demonstrate the efficiency of the proposed algorithm over other known exact and approximation techniques used in the literature for solving these problems.