A Structural Approach to 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.

Variable and value ordering heuristics for the job shop scheduling constraint satisfaction problem

Handbook of Constraint Programming

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.

Quantified Constraint Satisfaction Problems (QCSP) are a generalization of Constraint Satisfaction Problems (CSP) in which variables may be quantified existentially and universally. QCSP offers a natural framework to express PSPACE problems as finite two-player games or planning under uncertainty. State-of-the-art QCSP solvers have an important drawback: they explore much larger combinatorial spaces than the natural search space of the original problem since they are unable to recognize that some sub-problems are necessarily true. We introduce a new tool, inspired by the cut rule of Prolog as a tool under responsibility of the designer of the QCSP, to prune those parts of the search space which are by construction known to be useless. We use this new tool to restore on one hand the annihilator property of true for disjunction in QCSP solver and, on the other hand, to prune the search space in two-player games. It is a simple solution to use efficiently QCSP to design finite two-player games without restricting the QCSP language. This tool does not need to modify the QCSP solver but has only one requirement: be able to tell the QCSP solver that the current QCSP is solved. Our QCSP solver built over Ge Code, a CSP library, obtained very good results compared to state-of-the-art QCSP solvers.

An initial study of time complexity in infinite-domain constraint satisfaction

Constraints are commonly used in both simulation and formal verification in order to specify expected input conditions and state transitions. Constraint solving is a process to determine input vectors which satisfy the set of constraints during constrained random simulation. Even though constraints are used in formal property checking to restrict the search space, constraint solving has never had direct application to formal property checking. There are often many simple, yet powerful, invariants that can be learned from constraint solving during constrained random simulation. These invariants are shown in this paper to significantly simplify the formal verification problem. We use approximate constraint solving to compute an approximate set of valid input vectors. The approximate set of valid input vectors are a strict superset of the set of all legal input vectors. We use BDD techniques to compute these input vectors during constrained random simulation, then process the resulting BDDs for learning invariants which can be used during formal property checking. This paper presents efficient BDD algorithms to learn invariants from the BDDs generated from approximate constraint solving. We also present how these learned invariants can be applied to the formal property checking. Experimental results show that invariants learned during constraint solving can significantly improve the performance of formal property checking with many industrial designs.

Constraints are commonly used in both simulation and formal verification in order to specify expected input conditions and state transitions. Constraint solving is a process to determine input vectors which satisfy the set of constraints during constrained random simulation. Even though constraints are used in formal property checking to restrict the search space, constraint solving has never had direct application to formal property checking. There are often many simple, yet powerful, invariants that can be learned from constraint solving during constrained random simulation. These invariants are shown in this paper to significantly simplify the formal verification problem. We use approximate constraint solving to compute an approximate set of valid input vectors. The approximate set of valid input vectors are a strict superset of the set of all legal input vectors. We use BDD techniques to compute these input vectors during constrained random simulation, then process the resulting BDDs for learning invariants which can be used during formal property checking. This paper presents efficient BDD algorithms to learn invariants from the BDDs generated from approximate constraint solving. We also present how these learned invariants can be applied to the formal property checking. Experimental results show that invariants learned during constraint solving can significantly improve the performance of formal property checking with many industrial designs.

Chapter 5 - Basic search strategies for solving CSPs

Chapter 7 - Exploitation of problem-specific features

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

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.

This work presents a constraint satisfaction problem (CSP) model for the planning and scheduling of disassembly and assembly tasks when repairing or substituting faulty parts. The problem involves not only the ordering of assembly and disassembly tasks, but also the selection of them from a set of alternatives. The goal of the plan is the minimization of the total repairing time, and the model considers, apart from the durations and resources used for the assembly and disassembly tasks, the necessary delays due to the change of configuration in the machines, and to the transportation of intermediate subassemblies between different machines. The problem considers that sub-assemblies that do not contain the faulty part are nor further disassembled, but allows non-reversible and parallel repair plans. The set of all feasible repair plans are represented by an extended And/Or graph. This extended representation embodies all of the constraints of the problem, such as temporal and resource constraints and those related to the selection of tasks for obtaining a correct plan.

In a multi-objective combinatorial optimization (MOCO) problem, multiple objectives must be optimized simultaneously. In past years, several constraint-based algorithms have been proposed for finding Pareto-optimal solutions to MOCO problems that rely on repeated calls to a constraint solver. Understanding the properties of these algorithms and analyzing their performance is an important problem. Previous work has focused on empirical evaluations on benchmark instances. Such evaluations, while important, have their limitations. Our paper adopts a different, purely theoretical approach, which is based on characterizing the search space into subspaces and analyzing the worst-case performance of a MOCO algorithm in terms of the expected number of calls to the underlying constraint solver. We apply the approach to two important constraint-based MOCO algorithms. Our analysis reveals a deep connection between the search mechanism of a constraint solver and the exploration of the search space of a MOCO problem.

HTSA: A novel hybrid task scheduling algorithm for heterogeneous cloud computing environment