Abstract: In this paper, we present a new framework for combining complete and incomplete methods in order to deal with the Max Sat problem. The objective is to find the best assignment for a set of Boolean variables, which gives the maximum of verified clauses in a Boolean formula. Unfortunately, this problem has been shown to be NP-hard (non-deterministic polynomial-time hard) if the number of variables per clause is greater than 3. The proposed approach is based on quantum inspired genetic principles and the well known exact algorithm DPLL. The underlying idea is to harness the optimization capabilities of quantum genetic algorithm to achieve good quality solutions for Max SAT problem. A new local search based on DPLL algorithm has been embodied in the quantum genetic algorithm leading to an efficient hybrid framework which achieves better balance between diversification and intensification search. The obtained results are very encouraging and show the feasibility and effectiveness of the proposed hybrid approach.
Abstract: In this paper, we present a new framework for combining complete and incomplete methods in order to deal with the Max Sat problem. The objective is to find the best assignment for a set of Boolean variables, which gives the maximum of verified clauses in a Boolean formula. Unfortunately, this problem has been shown to be NP-hard (non-deterministic polynomial-time hard) if the number of variables per clause is greater than 3. The proposed approach is based on quantum inspired genetic principles and the well known exact algorithm DPLL. The underlying idea is to harness the optimization capabilities of quantum genetic algorithm to achieve good quality solutions for Max SAT problem. A new local search based on DPLL algorithm has been embodied in the quantum genetic algorithm leading to an efficient hybrid framework which achieves better balance between diversification and intensification search. The obtained results are very encouraging and show the feasibility and effectiveness of the proposed hybrid approach.
Abstract. In this paper we investigate the use of quantum particle swarm optimization (QPSO) principles to resolve the satisfiability problem. We describe QPSOSAT, a new iterative approach for solving the well known Maximum Satisfiability problem (MAX-SAT). This latter has been shown to be NP-hard if the number of variables per clause is greater than 3. The basic idea is to harness the optimization capabilities of QPSO algorithm to achieve good quality solutions for Max Sat problem. To enhance the efficiency of the QPSO algorithm, a local search has been used. The obtained results are very promising and show the feasibility and effectiveness of the proposed hybrid approach.
The aim of the paper is to answer a long-standing open problem on the relationship between NP and BQP. The paper shows that BQP contains NP by proposing a BQP quantum algorithm for the MAX-E3-SAT problem which is a fundamental NP-hard problem. Given an E3-CNF Boolean formula, the aim of the MAX-E3-SAT problem is to find the variable assignment that maximizes the number of satisfied clauses. The proposed algorithm runs in O(m2) for an E3-CNF Boolean formula with m clauses and in the worst case runs in O(n6) for an E3-CNF Boolean formula with n inputs. The proposed algorithm maximizes the set of satisfied clauses using a novel iterative partial negation and partial measurement technique. The algorithm is shown to achieve an arbitrary high probability of success of 1−ε for small ε>0 using a polynomial resources. In addition to solving the MAX-E3-SAT problem, the proposed algorithm can also be used to decide if an E3-CNF Boolean formula is satisfiable or not, which is an NP-complete problem, based on the maximum number of satisfied clauses.
Although some classical algorithms have been applied to solve the satisfiability problem, more effective methods are still explored because existing algorithms are constrained by inadequate computing capability of traditional computers. The parallelism of quantum computation makes quantum algorithms with promising potential to improve the computing ability, but existing quantum algorithms still require too large number of qubits to solve a simple problem effectively. In this paper, an optimized data structure was structured to solve Boolean satisfiability problem by utilizing Grover's algorithm, and then the corresponding formula was proposed to balance variables in consideration of complexity. With reasonable simplification, quantum circuits were built to decrease the number of qubits required in Grover's algorithm. The result of verification experiment further demonstrated that the proposed approach is simple, reliable and of a certain practical value.
Too often, quantum computer scientists seek to create new algorithms entirely fresh from new cloth when there are extensive and optimized classical algorithms that can be generalized wholesale. At the same time, one may seek to maintain classical advantages of performance and runtime bounds, while enabling potential quantum improvement. Hybrid quantum algorithms tap into this potential, and here we explore a class of hybrid quantum algorithms called Iterative Quantum Algorithms (IQA) that are closely related to classical greedy or local search algorithms, employing a structure where the quantum computer provides information that leads to a simplified problem for future iterations. Specifically, we extend these algorithms beyond past results that considered primarily quadratic problems to arbitrary k-local Hamiltonians, proposing a general framework that incorporates logical inference in a fundamental way. As an application we develop a hybrid quantum version of the well-known classical Davis-Putnam-Logemann-Loveland (DPLL) algorithm for satisfiability problems, which embeds IQAs within a complete backtracking based tree search framework. Our results also provide a general framework for handling problems with hard constraints in IQAs. We further show limiting cases of the algorithms where they reduce to classical algorithms, and provide evidence for regimes of quantum improvement.
The k-SAT problem is a canonical NP-complete problem for which no efficient algorithm is known. This paper proposes a hybrid quantum–classical algorithm that employs Grover's algorithm as the inner local search of a randomized outer loop. Each outer iteration measures a random assignment, prepares a problem-dependent superposition over the Hamming ball of radius r around it, and applies amplitude amplification to locate a satisfying assignment. Because the ball center is measured, the outer loop remains a classical repetition over independently sampled balls and contributes the inverse success probability rather than its square root; balancing this against the inner quadratic speedup shifts the optimal radius from n/(k+1) to n/(1+k) and yields the running time (2−2/(1+k))n, up to polynomial factors—1.268n for k=3, below the best known classical 1.307n. The contribution is not an asymptotic record—the best known quantum exponents for k-SAT, obtained by amplifying stronger classical solvers as black boxes, remain smaller—but a fully explicit, gate-level construction: the problem-dependent state over the Hamming ball and the k-SAT oracle are decomposed into elementary quantum gates, and the polynomial gate and qubit overheads are accounted for in full.
Satisfiability (SAT) problem in engineering and computer science is to find the set of assignment values of input variables for the given Boolean function that evaluate this function to TRUE or prove that such satisfying values do not exist. For POS SAT Problem, we propose a novel quantum algorithm for the maximum satisfiability (MAX-SAT) which returns the maximum number of OR terms that are satisfied for the SAT-unsatisfiable function, providing us with the information how far the given Boolean function is from the SAT satisfaction. We use Grover's algorithm with a new block called quantum counter in the oracle circuit.