The complexity and distribution of computationally useful problems

This paper is devoted to the study of complexity of finding a special covering for a set, as well as to obtaining some important applications of special decomposition. We formulate the problem of existence of a special covering as a decision problem. To determine the complexity class in which this problem is located, we study the relationship between this problem and the Boolean satisfiability problem, treating them as formal languages. We prove that these problems are polynomially equivalent, which means that the problem of existence of a special covering for a set is an -complete problem. In this article we also introduce a new concept ‘Replaceable Subsets’. The properties of such subsets are used to fill in the missing elements needed to obtain a special set covering. It is proved that when searching for missing elements to fill a special covering, the order in which these elements are considered does not matter. This result is of great importance in the search for satisfiability of Boolean functions.

Efficient solutions to nonpolynomial (NP)-complete problems would significantly benefit both science and industry. However, such problems are intractable on digital computers based on the von Neumann architecture, thus creating the need for alternative solutions to tackle such problems. Recently, a deterministic, continuous-time dynamical system (CTDS) was proposed [1] to solve a representative NP-complete problem, Boolean Satisfiability (SAT). This solver shows polynomial analog time-complexity on even the hardest benchmark $k$ -SAT ( $k \geq 3$ ) formulas, but at an energy cost through exponentially driven auxiliary variables. This paper presents a novel analog hardware SAT solver, AC-SAT , implementing the CTDS via incorporating novel, analog circuit design ideas. AC-SAT is intended to be used as a coprocessor and is programmable for handling different problem specifications. It is especially effective for solving hard $k$ -SAT problem instances that are challenging for algorithms running on digital machines. Furthermore, with its modular design, AC-SAT can readily be extended to solve larger size problems, while the size of the circuit grows linearly with the product of the number of variables and the number of clauses. The circuit is designed and simulated based on a 32-nm CMOS technology. Simulation Program with Integrated Circuit Emphasis (SPICE) simulation results show speedup factors of ~104 on even the hardest 3-SAT problems, when compared with a state-of-the-art SAT solver on digital computers. As an example, for hard problems with $N=50$ variables and $M=212$ clauses, solutions are found within from a few nanoseconds to a few hundred nanoseconds.

Different search strategies, like Boolean satisfiability (SAT), constraint programming (CP), and integer programming (IP) have been developed to solve classes of combinatorial search problems like the quasi-group completion problem (QCP). This paper proposes a new effective algorithm developed for solving the classic QCP — the Latin-square completion problem. The method is demonstrated by solving some typical Latin-square completion problems and ways to improve upon the algorithm are discussed.

This dissertation is to illustrate the current state of the art of DNA computing achievements, especially of new approaches to solve theoretical 3-SAT problems and the hitting-set problem. Beginning with Adleman’s breakthrough which is an molecular algorithm for the solution of a NP-complete, combinatorial problem, the directed Hamiltonian path problem (HPP). Today, many researchers all over the world concentrate on proposing new methods to solve engineering or application problems with a DNA computing approach. Satisfiability problem is given a Boolean formula, and decide if a satisfying truth assignment exists. ( ) ( ) … ( ) ( ) is an example of Boolean formula. k-SAT problem means that each clause has exactly k literals. Not-All-Equal (NAE) 3-SAT problem and One-In-Three (1IN3) 3-SAT problem are both NP-complete problems. In this dissertation, we present molecular solutions to find all true assignments (3-SAT problem) and furthermore find Not-All-Equal (NAE) solutions and One-In-Three (1IN3) solutions in DNA-based Supercomputing. Hitting-set problem assume that there exists a collection C of subsets of a finite set S, and a positive integer K  |S|, and we need to know if there is a subset  S with | |  K such that contains at least one element of each subset in C. In other words, is the subset that intersects every subset in C and is called the hitting-set. In this dissertation, a DNA-based algorithm is proposed to solve the small hitting-set problem. A small hitting-set is a hitting-set with the smallest K value, i.e., the hitting-set with the smallest number of elements. Furthermore, an algorithm is introduced to find the number of ones from 2n combinations and minimum numbers of ones represents the small hitting-set since K is expected to be as small as possible. The complexity of all the presented DNA-based algorithms is also discussed. We describe time complexity and volume complexity of three Algorithms, numbers of test tube used and the longest library strand in solution space of all three Algorithms. Finally, the simulated experiment is applied to verify correctness of the proposed DNA-based algorithm for solving the One-In-Three (1IN3) 3-SAT problem, and simulation of Not-All-Equal (NAE) 3-SAT problem is similar. Also, another simulated experiment is applied to our proposed DNA-based algorithm 6-2, in order to solve the well-known hitting-set problem. This research has been motivated by the benefit and the application of DNA computing and gives new methods to solve two 3-SAT problems and the hitting-set problem which are NP-complete.

Quantum computing has grown extensively, especially in system design and development, and the current research focus has gradually evolved from validating quantum advantage to practical applications. In particular, nondeterministic‐polynomial‐time (NP)‐complete problems are central in numerous important application areas. Still, in practice, it is difficult to solved efficiently with conventional computers, limited by the exponential jump in hardness. Here, a quantum photonic microprocessor based on Gaussian boson sampling (GBS) that offers dynamic programmability to solve various graph‐related NP‐complete problems is demonstrated. The system with optical, electrical, and thermal packaging implements a GBS with 16 modes of single‐mode squeezed vacuum states, a universal programmable 16‐mode interferometer, and a single photon readout on all outputs with high accuracy, generality, and controllability. The developed system is applied to demonstrate applications in solving NP‐complete problems, manifesting the ability of photonic quantum computing to realize practical applications for conventionally intractable computations. The GBS‐based quantum photonic microprocessor is applied to solve task assignment, Boolean satisfiability, graph clique, max cut, and vertex cover. These demonstrations suggest an excellent benchmarking platform, paving the way toward large‐scale combinatorial optimization.

In this chapter, we present applications of regions and elementary (or quasielementary) net synthesis for the design of speed independent circuits. Asynchronous circuit design is a wide and complex field that encompasses other topics than Petri nets. We refer the reader to the book [46] for a complete presentation of this field. We will limit ourselves to assessing the roles played by regions and net synthesis for solving the Complete State Coding problem (CSC). This crucial problem motivated the implementation of a net synthesis procedure in Petrify, a general design tool for asynchronous circuits presented in [46]. It must be acknowledged that different methods based on net unfoldings and SAT solvers are now preferred over net synthesis for solving the CSC problem.

It was investigated the Boolean satisfiability (SAT) problem defined as follows: given a Boolean formula, check whether an assignment of Boolean values to the propositional variables in the formula exists, such that the formula evaluates to true. If such an assignment exists, the formula is said to be satisfiable; otherwise, it is unsatisfiable. With using of analytical expressions of multi-valued logic 2SAT boolean satisfiability was formulated as linear programming optimization problem. The same linear programming formulation was extended to find 3SAT and kSAT boolean satisfiability for k greater than 3. So, using new analytic multi-valued logic expressions and linear programming formulation of boolean satisfiability proposition that kSAT is in P and could be solved in linear time was proved.KeywordsBoolean satisfiabilityConjunctive normal formConvex functionLinear programmingMulti-valued logicNP problemTime complexity.

Propositional Satisfiability (SAT) is a keystone in the history of computer science. SAT was the first problem shown to be NP-complete in 1971 by Stephen Cook. Having passed more than 40 years from then, SAT is now a lively research field where theory and practice have a natural intermixing. In this talk, we overview the use of SAT in practical domains, where SAT is thought in a broad sense, i.e. including SAT extensions such as Maximum Satisfiability (MaxSAT), Pseudo-Boolean Optimization (PBO) and Quantified Boolean Formulas (QBF).

Advances in methods for solving Boolean satisfiability (SAT) for large problems have motivated recent attempts to recast physical design problems as Boolean SAT problems. One persistent criticism of these approaches is their inability to supply partial solutions, i.e., to satisfy most but not all of the constraints cast in the SAT style. In this paper, we present a formulation for "subset satisfiable" Boolean SAT: we transform a "strict" SAT problem with N constraints into a new, "relaxed" SAT problem which is satisfiable just if not more than k/spl Lt/N of these constraints cannot be satisfied in the original problem. We describe a transformation based on explicit thresholding and counting for the necessary SAT relaxation. Examples from field-programmable gate-array routing show how we can determine efficiently when we can satisfy "almost all" of our geometric constraints.

Boolean Satisfiability (SAT) is a widely used modeling tool in Electronic Design Automation (EDA). It finds application in test pattern generation, delay-fault testing, equivalence checking and circuit delay computation, among many other problems. This paper starts by describing how Boolean Satisfiability algorithms can take circuit structure into account when solving instances derived from combinational circuits. Afterwards, it shows how recursive learning techniques can be incorporated into Boolean Satisfiability algorithms. The proposed algorithmic framework has several natural applications in EDA. Moreover, potential advantages include smaller run times, the utilization of circuit-specific search pruning techniques, avoiding the overspecification problem that characterizes Boolean Satisfiability testers, and reducing the time for iteratively generating instances of SAT from circuits. The experimental results obtained, on a large number of benchmark examples in different problem domains, illustrate the effectiveness of the proposed techniques.

Propositional satisfiability (SAT) is a fundamental problem of immense practical importance. While SAT is NP-complete when clauses can contain 3 literals or more, the problem can be solved in linear time when the given formula contains only binary clauses (2SAT). Many complete search algorithms for SAT solving have taken advantage of 2SAT information that occurs in the statement of the problem in order to simplify the solving process, only one that we are aware of uses 2SAT information that arises in the process of the search, as clauses are simplified. There are a number of possibilities for making use of 2SAT information to improve the SAT solving process: maintaining 2SAT satisfiability during search, detecting unit consequences of the 2SAT clauses, and Krom subsumption using 2SAT clauses. In this paper we investigate the tradeoffs of increasing complex 2SAT handling versus the search space reduction and execution time. We give experimental results illustrating that the SAT solver resulting from the best tradeoff is competitive with state of the art Davis-Putnam methods, on hard problems involving a substantial 2SAT component.

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.

A new method of DNA computing using an unusual triple-stranded DNA structure mediated by RecA protein is introduced and a minor 3-variable instance of the satisfiability (SAT) problem is solved. Separations were performed by using biotinylated oligodeoxynucleotide probes coated with RecA protein. During the computation, sequence-specific recognition of double-stranded DNA by homologous oligodeoxynucleotides was fulfilled in the presence of ATPγS. Captured by the magnetic particles coated with streptavidin, the desirable double-stranded DNA strands were retained and extracted. It is suggested that the methods employed here may help decrease the errors in DNA computation and solve some larger size NP- complete problems.

The Boolean Satisfiability Problem (SAT) is a well known NP-Complete problem. While its complexity remains a source of many interesting questions for theoretical computer scientists, the problem has found many practical applications in recent years. The emergence of efficient SAT solvers which can handle large structured SAT instances has enabled the use of SAT solvers in diverse domains such as electronic design automation and artificial intelligence. These applications continue to motivate the development of faster and more robust SAT solvers. In this paper, we describe the popular SAT solver zchaff with a focus on recent developments.

The Boolean Satisfiability Problem (SAT) is well known in computational complexity, representing the first decision problem to be proved NP-complete. SAT is also often the subject of work in proof complexity. Besides its theoretical interest, SAT finds a wide range of practical applications. Moreover, SAT solvers have been the subject of remarkable efficiency improvements since the mid-1990s, motivating their widespread use in many practical applications including Bounded and Unbounded Model Checking. The success of SAT solvers has also motivated the development of algorithms for natural extensions of SAT, including Quantified Boolean Formulas (QBF), Pseudo-Boolean constraints (PB), Maximum Satisfiability (MaxSAT) and Satisfiability Modulo Theories (SMT) which also see application in the model-checking context. This chapter first covers the organization of modern conflict-driven clause learning (CDCL) SAT solvers, which are used in the vast majority of practical applications of SAT. It then reviews the techniques shown to be effective in modern SAT solvers.