Solving Infinite-Domain CSPs Using the Patchwork Property

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

This chapter describes four interdisciplinary fields originated and defined by Ashu M. G. Solo in 2011 called political engineering, public policy engineering, computational politics, and computational public policy. Political engineering is the application of engineering, computer science, mathematics, or natural science to solving problems in politics. Computational politics is the application of computer science or mathematics to solving problems in politics. Political engineering and computational politics include, but are not limited to, principles and methods for political decision-making, analysis, modeling, optimization, forecasting, simulation, and expression. Public policy engineering is the application of engineering, computer science, mathematics, or natural science to solving problems in public policy. Computational public policy is the application of computer science or mathematics to solving problems in public policy. Public policy engineering and computational public policy include, but are not limited to, principles and methods for public policy formulation, decision-making, analysis, modeling, optimization, forecasting, and simulation. The chapter describes the scope of research and development in these fields, provides examples of research and development in these fields, and provides possible university curricula for academic programs in these fields.

The problem of finding small unsatisfiable subsets of a set of constraints is important for various applications in computer science and artificial intelligence. We study the problem of identifying whether a given instance to the constraint satisfaction problem (CSP) has an unsatisfiable subset of size at most k from a parameterized complexity point of view. We show that the problem of finding small unsatisfiable subsets of a CSP instance is harder than the corresponding problem for CNF formulas. Moreover, we show that the problem is not fixed-parameter tractable when restricting the problem to any maximal tractable Boolean constraint language (for which the problem is nontrivial). We show that the problem is hard even when the maximum number of occurrences of any variable is bounded by a constant, a restriction which leads to fixed-parameter tractability for the case of CNF formulas. Finally, we relate the problem of finding small unsatisfiable subsets to the problem of identifying variable assignments that are enforced already by a small number of constraints (backbones), or that are ruled out already by a small number of constraints (anti-backbones).

The tractability conjecture for constraint satisfaction problems (CSPs) describes the constraint languages over a finite domain whose CSP can be solved in polynomial-time. The precise formulation of the conjecture uses basic notions from universal algebra. In this talk, we give a short introduction to the universal-algebraic approach to the study of the complexity of CSPs. Finally, we discuss attempts to generalise the tractability conjecture to large classes of constraint languages over infinite domains, in particular for constraint languages that arise in qualitative temporal and spatial reasoning.

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.

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Handbook of Constraint Programming

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.

One significant tool for addressing the concerns associated with the use of one of the applications in computer science is the law. Numerous laws and rules governing the use of computer science are designed to safeguard users, customers, and society at large. Law uses norms created by governmental and social institutions to control behavior. Programming is used to explore digital information in computer science. Computers got more potent and smaller as they evolved through generations, utilizing various technologies such as integrated circuits. Cyber law regulates online behavior and deals with matters pertaining to intellectual property, privacy, domain names, and other legal concerns. A few of the many concerns that the law must address in order to mitigate the electronic crimes are protecting privacy and security, upholding justice, limiting civil and criminal liability, ensuring safety, and controlling the application of AI in the workplace. Guidelines that people and organizations have to follow when utilizing computer science applications, making sure that these requirements are met in full. Additionally, as companies must commit to recording AI usage procedures and making clear the use of data and algorithms, the regulation fosters accountability and openness. By doing this, the possibility of prejudice and mistakes when utilizing AI is decreased. This survey provides a self-contained introduction to cybercrimes and types of cybercrimes. We also present ways to combat cybercrime and limit its spread.

Constraint satisfaction problems (CSPs) for first-order reducts of finitely bounded homogeneous structures form a large class of computational problems that might exhibit a complexity dichotomy, P versus NP-complete. A powerful method to obtain polynomial-time tractability results for such CSPs is a certain reduction to polynomial-time tractable finite-domain CSPs defined over k -types, for a sufficiently large k . We give sufficient conditions when this method can be applied and apply these conditions to obtain a new complexity dichotomy for CSPs of first-order expansions of the basic relations of the well-studied spatial reasoning formalism RCC5. We also classify which of these CSPs can be expressed in Datalog. Our method relies on Ramsey theory; we prove that RCC5 has a Ramsey order expansion.

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.

We discuss a coevolutionary genetic algorithm for constraint satisfaction. Our basic idea is to explore effective genetic information in the population, i.e., schemata, and to exploit the genetic information in order to guide the population to better solutions. Our coevolutionary genetic algorithm (CGA) consists of two GA populations; the first GA, called searches for the solutions in a given environment (problem), and the second GA, called P-GA, searches for effective genetic information involved in the H-GA, namely, good schemata. Thus, each individual in P-GA consists of alleles in H-GA or don't care symbol representing a schema in the H-GA. These GA populations separately evolve in each genetic space at different abstraction levels and affect with each other by two genetic operators: superposition and transcription. We then applied our CGA to constraint satisfaction problems (CSPs) incorporating a new stochastic repair operator for P-GA to raise the consistency of schemata with the (local) constraint conditions in CSPs. We carried out two experiments: First, we examined the performance of CGA on various general CSPs that are generated randomly for a wide variety of density and tightness of constraint conditions in the CSPs that are the basic measures of characterizing CSPs. Next, we examined structured CSPs involving latent cluster structures among the variables in the CSPs. For these experiments, computer simulations confirmed us the effectiveness of our CGA.

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.

An Iterative local-search framework for solving constraint satisfaction problem

We perform a comprehensive study of mappings between constraint satisfaction problems (CSPs) and propositional satisfiability (SAT). We analyse four different mappings of SAT problems into CSPs, and two of CSPs into SAT problems. For each mapping, we compare the impact of achieving arc-consistency on the CSP with unit propagation on the SAT problem. We then extend these results to CSP algorithms that maintain (some level of) arc-consistency during search like FC and MAC, and to the Davis-Putnam procedure (which performs unit propagation at each search node). Because of differences in the branching structure of their search, a result showing the dominance of achieving arc-consistency on the CSP over unit propagation on the SAT problem does not necessarily translate to the dominance of MAC over the Davis-Putnam procedure. These results provide insight into the relationship between propositional satisfiability and constraint satisfaction.