Constraint Satisfiability Problem Research Articles

Finding Solutions to the Integer Case Constraint Satisfiability Problem Using Grover’s Algorithm

Constraint satisfiability problems, crucial to several applications, are solved on a quantum computer using Grover's search algorithm, leading to a quadratic improvement over the classical case. The solutions are obtained with high probability for several cases and are illustrated for the cases involving two variables for both 3- and 4-bit numbers. Methods are defined for inequality comparisons, and these are combined according to the form of the satisfiability formula, to form the oracle for the algorithm. The circuit is constructed using IBM Qiskit and is verified on an IBM simulator. It is further executed on one of the Noisy Intermediate-Scale Quantum (NISQ) processors from IBM on the cloud. Noise levels in the processor at present are found to be too high for successful execution. Running the algorithm on the simulator with a custom noise model lets us identify the noise threshold for successful execution.

Algorithms for Workflow Satisfiability Problem with User-independent Constraints

The Workflow Satisfiability Problem (WSP) is a problem of interest in access control of information security. In its simplest form, the problem coincides with the Constraint Satisfiability Problem, where the number of variables is usually much smaller than the number of values. Wang and Li (ACM Trans. Inf. Syst. Secur. 2010) were the first to study the WSP as a problem parameterized by the number of variables. Their paper initiated very fruitful research surveyed by Cohen, Crampton, Gutin and Wahlström (2017). In this paper, we overview more recent WSP algorithmic developments and discuss computational experiments performed on two new testbeds of WSP instances. These WSP instances are closer to real-world ones than those by Karapetyan et al. (JAIR 2019). One of the two testbeds is generated using a novel iterative approach for obtaining computationally hard WSP instances.

A Conflict-Driven Solving Procedure for Poly-Power Constraints

This paper studies the satisfiability problem of poly-power constraints (conjunctions of poly-power equations and inequalities), in which poly-powers are univariate nonlinear functions that extend integer exponents of polynomials to real algebraic exponents. To solve the poly-power constraint, we present a sound and complete procedure that incorporates conflict-driven learning with the exclusion algorithm for isolating positive roots of poly-powers. Furthermore, we introduce a kind of optimal interval-splitting, based on the Stern–Brocot tree and on binary rational numbers respectively, so that the operands occurring in the execution are chosen to be as simple as possible. The solving procedure, thereby, turns out to be promisingly efficient on randomly generated examples.

Incompressibility of $$H$$ H -Free Edge Modification Problems

Given a fixed graph $$H$$H, the $$H$$H-Free Edge Deletion (resp., Completion, Editing) problem asks whether it is possible to delete from (resp., add to, delete from or add to) the input graph at most $$k$$k edges so that the resulting graph is $$H$$H-free, i.e., contains no induced subgraph isomorphic to $$H$$H. These $$H$$H-free edge modification problems are well known to be fixed-parameter tractable for every fixed $$H$$H. In this paper we study the incompressibility, i.e., nonexistence of polynomial kernels, for these $$H$$H-free edge modification problems in terms of the structure of $$H$$H, and completely characterize their nonexistence for $$H$$H being paths, cycles or 3-connected graphs. We also give a sufficient condition for the nonexistence of polynomial kernels for $${\mathcal {F}}$$F-Free Edge Deletion problems, where $${\mathcal {F}}$$F is a finite set of forbidden induced subgraphs. As an effective tool, we have introduced an incompressible constraint satisfiability problem Propagational-$$f$$f Satisfiability to express common propagational behaviors of events, and we expect the problem to be useful in studying the nonexistence of polynomial kernels in general.

On the survey-propagation equations in random constraint satisfiability problems

In this paper we study the existence of a solution of the survey-propagation equations for a given instance of a random problem in the framework of constraint satisfiability. We consider the concrete examples of K-satisfiability and coloring. We conjecture that when the number of variables goes to infinity, the solution of the survey-propagation equations for a given instance can be obtained by finding the (supposed unique) solution of the corresponding equations on an infinite tree. We conjecture that the survey-propagation equations on a random infinite tree have a unique solution in the suitable range of parameters. We also present analytic arguments that indicate that the survey-propagation equations do have solutions in the satisfiable phase. For simplicity of notation the argument is presented in the case of the coloring problem. The same results extend to other optimization problems where exist configurations that have cost zero, i.e., in the satisfiable phase. On a random graph the solutions of the belief-propagation equations are associated with the existence of many well separated clusters of configurations (clustering states). We argue that on a random graph the belief-propagation equations have solutions almost everywhere: the statement may be sharpened by introducing the concept of quasisolution of the belief-propagation equations.

Complexity of Clausal Constraints Over Chains

We investigate the complexity of the satisfiability problem of constraints over finite totally ordered domains. In our context, a clausal constraint is a disjunction of inequalities of the form x≥d and x≤d. We classify the complexity of constraints based on clausal patterns. A pattern abstracts away from variables and contains only information about the domain elements and the type of inequalities occurring in a constraint. Every finite set of patterns gives rise to a (clausal) constraint satisfaction problem in which all constraints in instances must have an allowed pattern. We prove that every such problem is either polynomially decidable or NP-complete, and give a polynomial-time algorithm for recognizing the tractable cases. Some of these tractable cases are new and have not been previously identified in the literature.

Towards Inferring Protein Interactions: Challenges and Solutions

Discovering interacting proteins has been an essential part of functional genomics. However, existing experimental techniques only uncover a small portion of any interactome. Furthermore, these data often have a very high false rate. By conceptualizing the interactions at domain level, we provide a more abstract representation of interactome, which also facilitates the discovery of unobserved protein-protein interactions. Although several domain-based approaches have been proposed to predict protein-protein interactions, they usually assume that domain interactions are independent on each other for the convenience of computational modeling. A new framework to predict protein interactions is proposed in this paper, where no assumption is made about domain interactions. Protein interactions may be the result of multiple domain interactions which are dependent on each other. A conjunctive norm form representation is used to capture the relationships between protein interactions and domain interactions. The problem of interaction inference is then modeled as a constraint satisfiability problem and solved via linear programing. Experimental results on a combined yeast data set have demonstrated the robustness and the accuracy of the proposed algorithm. Moreover, we also map some predicted interacting domains to three-dimensional structures of protein complexes to show the validity of our predictions.

A GRASP algorithm to solve the unicost set covering problem

The set covering problem (SCP) is a well-known combinatorial optimization problem. This paper presents a GRASP algorithm to solve a special SCP case known in the literature as the unicost set covering problem. The algorithm incorporates a local improvement procedure based on the heuristics to solve binary constraint satisfiability problems (SAT). The quality of the proposed algorithm is tested on a set of reference instances, comparing the obtained results with those found in the literature. Our algorithm improves the best-known solutions for many of these instances.

Complexity of Regular Language Matching and Other Decidable Cases of the Satisfiability Problem for Constraints Between Regular Open Terms

In this paper we study the satisfiability problem for language equations and constraints between regular open terms. We prove that: the constraint problem is PSPACE-complete, the regular language matching problem is EXPSPACE-complete even if we ask about satisfiability by regular languages, satisfiability of word-like language equations is in PSPACE, the finite solution problem is EXPSPACE-complete.

Constraint Solving for Term Orderings Compatible with Abelian Semigroups, Monoids and Groups

It is crucial for the performance of ordered resolution or paramodulation-based deduction systems that they incorporate specialized techniques to work efficiently with standard algebraic theories E. Essential ingredients for this purpose are term orderings that are E-compatible, for the given E, and algorithms deciding constraint satisfiability for such orderings. Here we introduce a uniform technique providing the first such algorithms for some orderings for abelian semigroups, abelian monoids and abelian groups, which we believe will lead to reasonably efficient techniques for practice. Our algorithms are in NP, and hence optimal, since in addition we show that, for any well-founded E-compatible ordering for these E, the constraint satisfiability problem is NP-hard even for conjunctions of inequations.

Querying temporal and spatial constraint networks in PTIME

We start with the assumption that temporal and spatial knowledge usually captured by constraint networks can be represented and queried more effectively by using the scheme of indefinite constraint databases. Because query evaluation in this scheme is in general a hard computational problem, we seek tractable instances of query evaluation. We assume that we have a class of constraints C with some reasonable computational and closure properties (the computational properties of interest are that the satisfiability problem and an appropriate version of the variable elimination problem for C should be solvable in PTIME). Under this assumption, we exhibit general classes of indefinite constraint databases and first-order modal queries for which query evaluation can be done with PTIME data complexity. We then search for tractable instances of C among the subclasses of Horn disjunctive linear constraints over the rationals. From previous research we know that the satisfiability problem for Horn disjunctive linear constraints is solvable in PTIME, but not the variable elimination problem. Thus we try to discover subclasses of Horn disjunctive linear constraints with tractable variable elimination problems. The class of UTVPI ≠ constraints is the largest class that we show to have this property. Finally, we restate our general tractability results with C ranging over the newly discovered tractable classes. Interesting tractable query answering problems for indefinite temporal and spatial constraint databases are identified in this way. We close our complexity analysis by precisely outlining the frontier between tractable and possibly intractable query answering problems.

Interactive scheduling as a constraint satisfiability problem

A constraint satisfiability problem consists of a set of variables, their associated domains (i.e., the set of values the variable can take) and a set of constraints on these variables. A solution to the CSP is an instantiation (or labeling) of all the variables which does not violate any of the constraints. Since constraint satisfiability problems are, in general, NP-complete, it is of interest to compare the effectiveness and efficiency of heuristic algorithms as applied, in particular, to our application. Our research effort attempts to determine which algorithms perform best in solving the student scheduling problem (SSP) and under what conditions. We also investigate the probabilistic techniques of Nudel for finding a near-optimal instantiation order for search algorithms, and develop our own modifications which can yield a significant improvement in efficiency for the SSP. Experimental results have been collected and are reported here. Our system was developed for and used at Bar-Ilan University during the registration period, being available for students to construct their timetables.