Random 2-SAT with Prescribed Literal Degrees

Two classic "phase transitions" in discrete mathematics are the emergence of a giant component in a random graph as the density of edges increases, and the transition of a random 2-SAT formula from satisfiable to unsatisfiable as the density of clauses increases. The random-graph result has been extended to the case of prescribed degree sequences, where the almost-sure nonexistence or existence of a giant component is related to a simple property of the degree sequence. We similarly extend the satisfiability result, by relating the almost-sure satisfiability or unsatisfiability of a random 2-SAT formula to an analogous property of its prescribed literal-degree sequence. The extension has proved useful in analyzing literal-degree-based algorithms for (uniform) random 3-SAT.     

Approximating MIN 2-SAT and MIN 3-SAT

We obtain substantially improved approximation algorithms for the MIN k-SAT problem, for k = 2,3. More specifically, we obtain a 1.1037-approximation algorithm for the MIN 2-SAT problem, improving a previous 1.5-approximation algorithm, and a 1.2136-approximation algorithm for the MIN 3-SAT problem, improving a previous 1.75-approximation algorithm for the problem. These results are obtained by adapting techniques that were previously used to obtain approximation algorithms for the MAX k-SAT problem. We also obtain some hardness of approximation results.     

Randomized Algorithms for 3-SAT

Schoning proposed a simple yet efficient randomized algorithm for solving the k-SAT problem. In the case of 3-SAT, the algorithm has an expected running time of poly(n)·(4/3)ⁿ = O(1.334ⁿ). In this paper we present randomized algorithms and show that one of them has O(1.3302ⁿ) expected running time, improving Schoning's algorithm. (Note. At this point, the fastest randomized algorithm for 3-SAT is the one given by Iwama and Tamaki that runs in O(1.324ⁿ).)     

Quantum cooperative search algorithm for 3-SAT

Grover's search algorithm, one of the most popular quantum algorithms, provides a good solution to solve NP complexity problems, but requires a large number of quantum bits (qubits) for its functionality. In this paper, a novel algorithm called quantum cooperative search is proposed to make Grover's search algorithm work on 3-SAT problems with a small number of qubits. The proposed algorithm replaces some qubits with classical bits and finds assignments to these classical bits using the traditional 3-SAT algorithms including evolutionary algorithms and heuristic local search algorithms. In addition, the optimal configuration of the proposed algorithm is suggested by mathematical analysis. The experimental results show that the quantum cooperative search algorithm composed by Grover's search and heuristic local search performs better than other pure traditional 3-SAT algorithms in most cases. The mathematical analysis of the appropriate number of qubits is also verified by the experiments.     

Automating the DNA computer: solving n-Variable 3-SAT problems

In the decade since the first molecular computation was performed, it has been shown that DNA molecules can perform sophisticated, massively parallel computations avoiding the Von Neumann bottleneck. However, progress in the field has been slow. The largest problem solved to date is an instance of the 20-variable 3-CNF SAT problem. Performing the computation took more than two man-weeks to complete because every aspect of the computation was performed by hand. Molecular computations are extremely labor intensive and error prone–automation is necessary for further progress. The next step, (the second generation DNA computer—that of taking the laborious, laboratory bench protocols performed by hand, and automating them), has been achieved with the construction of an automated DNA computer dubbed EDNAC. It employs the same paradigm that was used to solve the labor-intensive instance of the 20-variable 3-CNF SAT problem. Using a combinatorial DNA library and complementary probes, EDNAC solves instances of the n-variable 3-CNF SAT problem. A 10 variable instance of the 3-CNF SAT problem was essayed. The computation took 28 h to perform. EDNAC correctly computed nine of the 10 variables, with a tenth variable remaining ambiguous. This result is comparable to current results in the molecular computation community. This research tested the critical properties, such as complexity, robustness, reliability, and repeatability necessary for the successful automation of a molecular computer.     

Derandomizing the HSSW Algorithm for 3-SAT

We present a (full) derandomization of HSSW algorithm for 3-SAT, proposed by Hofmeister, Schöning, Schuler, and Watanabe (in STACS 2002, pp. 192–202, 2002). Thereby, we obtain an O(1.3303 ⁿ )-time deterministic algorithm for 3-SAT, which is currently fastest.     

一种求解3-SAT问题的新方法

可满足性问题(SatisfiabilityProblem,SAT)是计算科学的典型问题之一,目前有DP算法、SAT1.3算法和遗传算法等多种求解方法。文章根据Kennedy和Eberhart提出的二进制粒子群优化算法(BinaryParticleSwarmOptimizers),基于局部随机搜索策略,给出了一种求解3-SAT问题的新方法:基于局部随机搜索的改进二进制粒子群优化算法(ModifedBinaryParticleSwarmOptimizersBasedonlocalstochasticsearch,简称MBPSO)。数值实验表明,对于随机产生的3-SAT问题测试实例,该算法是一种高效实用的新方法。     

Efficient 3-SAT algorithms in the tile assembly model

Self-assembly is a powerful process found in nature that guides simple objects assembling, on their own, into complex structures. Self-assembly is of interest to computer scientists because self-assembling systems can compute functions, assemble shapes, and guide distributed robotics systems. The tile assembly model is a formal mathematical model of self-assembly that allows the study of time and space complexities of self-assembling systems that lie at the heart of several molecular computer implementations and distributed computational software systems. These implementations and systems require efficient tile systems with small tilesets and fast execution times. The state of the art, however, requires vastly complex tile systems with large tilesets to implement fast algorithms. In this paper, I present ${\mathbb{S}}_{FS},$ a tile system that decides 3-SAT by creating $O^{\star}(1.8393^n)$ nondeterministic assemblies in parallel, improving on the previous best known solution that requires $\Uptheta(2^n)$ such assemblies. This solution directly improves the feasibility of building molecular 3-SAT solvers and efficiency of distributed software. I formally prove the correctness of the system, the number of required parallel assemblies, that the size of the system??s tileset is $147 = \Uptheta(1),$ and that the assembly time is nondeterministic linear in the size of the input.     

Is DNA computing viable for 3-SAT problems?

Adleman reported how to solve a 7-vertex instance of the Hamiltonian path problem by means of DNA manipulations. After that a major goal of subsequent research is how to use DNA manipulations to solve NP-hard problems, especially 3-SAT problems. Lipton proposed DNA experiments on test tubes to solve 3-SAT problems. Liu et al. reported how to solve a simple case of 3-SAT using DNA computing on surfaces. Lipton's model of DNA computing is simple and intuitive for 3-SAT problems. The separate (or extract) operation, which is a key manipulation of DNA computing, only extracts some of the required DNA strands and Lipton thinks that a typical percentage might be 90. But it is unknown what would happen due to imperfect extract operation. Let p be the rate, where 0<p<1. Assume that for each distinct string s in a test tube, there are 10^l (l=13 proposed by Adleman) copies of s and that extracting each of the required DNA strands is equally likely. Here, the present paper will report, no matter how large l is and no matter how close to 1 p is, there always exists a class of 3-SAT problems such that DNA computing error must occur. Therefore, DNA computing is not viable for 3-SAT.     

基于遗传算法的3-SAT问题的解决方案

本文利用遗传算法的具体求解问题无关性以及全局优化的优点,仿照自然界进化的过程,对染色体进行选择、变异、杂交,来解决3-SAT问题,并给出C 程序的实现.     

Exponential bounds for the random walk algorithm on random planted 3-SAT

Random walk algorithm （RWalkSAT） is one of the simplest and oldest heuristics for satisfiability problems. In contrast to many experimental results, relatively few rigorous analyses of RWalkSAT are available. Up to now, runtime results of small density random 3-SAT have been achieved showing RWalkSAT terminates successfully in linear time up to clause density 1.63. This paper presents a rigorous runtime analysis of RWalkSAT for 3-CNF formulas generated under the planted assignment distribution. It proves that with overwhelming probability, when starting from any initial assignment ｜x｜ 0.9999n, the expected number of steps until RWalkSAT on random planted formulas with a large constant density finds the planted assignment （1, . . . ,1） is at least Ω（1.1514n） and at most O（n21.1517n）.     

An optical solution to the 3-SAT problem using wavelength based selectors

In this paper, an optical wavelength-based method to solve a well-known NP-complete problem 3-SAT is provided. In the 3-SAT problem, a formula F in the form of conjunction of some clauses over Boolean variables is given and the question is to find if F is satisfiable or not. The provided method uses exponential number of different wavelengths in a light ray and considers each group of wavelengths as a possible value-assignment for the variables. It then uses optical devices to drop wavelengths not satisfying F from the light ray. At the end, remaining wavelengths indicate satisfiability of the formula. The method provides two ways to arrange the optical devices to select satisfying wavelengths for a given clause: simple clause selectors and combined clause selectors, both requiring exponential preprocessing time. After preprocessing phase, the provided method requires polynomial time and optical devices to solve each problem instance.     

应用OpenGL再现三维航迹

提出了一种在OpenGL中绘制三维航迹的新算法,并介绍了航迹的绘制过程。     

Uniform solutions to SAT and 3-SAT by spiking neural P systems with pre-computed resources

We consider the possibility of using spiking neural P systems for solving computationally hard problems, under the assumption that some (possibly exponentially large) pre-computed resources are given in advance. In particular, we propose two uniform families of spiking neural P systems which can be used to address the NP-complete problems sat and 3-sat, respectively. Each system in the first family is able to solve all the instances of sat which can be built using n Boolean variables and m clauses, in a time which is quadratic in n and linear in m. Similarly, each system of the second family is able to solve all the instances of 3-sat that contain n Boolean variables, in a time which is cubic in n. All the systems here considered are deterministic.     

Restarts and exponential acceleration of the Davis-Putnam-Loveland-Logemann algorithm: A large deviation analysis of the generalized unit clause heuristic for random 3-SAT

An analysis of the hardness of resolution of random 3-SAT instances using the Davis–Putnam–Loveland–Logemann (DPLL) algorithm slightly below threshold is presented. While finding a solution for such instances demands exponential effort with high probability, we show that an exponentially small fraction of resolutions require a computation scaling linearly in the size of the instance only. We compute analytically this exponentially small probability of easy resolutions from a large deviation analysis of DPLL with the Generalized Unit Clause search heuristic, and show that the corresponding exponent is smaller (in absolute value) than the growth exponent of the typical resolution time. Our study therefore gives some quantitative basis to heuristic restart solving procedures, and suggests a natural cut-off cost (the size of the instance) for the restart.     

Restarts and exponential acceleration of the Davis-Putnam-Loveland-Logemann algorithm: A large deviation analysis of the generalized unit clause heuristic for random 3-SAT

An analysis of the hardness of resolution of random 3-SAT instances using the Davis-Putnam-Loveland-Logemann (DPLL) algorithm slightly below threshold is presented. While finding a solution for such instances demands exponential effort with high probability, we show that an exponentially small fraction of resolutions require a computation scaling linearly in the size of the instance only. We compute analytically this exponentially small probability of easy resolutions from a large deviation analysis of DPLL with the Generalized Unit Clause search heuristic, and show that the corresponding exponent is smaller (in absolute value) than the growth exponent of the typical resolution time. Our study therefore gives some quantitative basis to heuristic restart solving procedures, and suggests a natural cut-off cost (the size of the instance) for the restart.     

PIC单片机应用于小区监控系统

该文谈了一种基于PIC单片机的智能小区监控系统,实现了水表、电表、煤气表的自动抄表,以及火灾、煤气泄漏、入室盗窃等安防监控。详细阐述了系统的软硬件设计以及PC机与多台PIC单片机的串行通讯方案。运用了单片机有关的软硬件技术。     

PIC单片机应用于小区监控系统

该文谈了一种基于PIC单片机的智能小区监控系统,实现了水表、电表、煤气表的自动抄表,以及火灾、煤气泄漏、入室盗窃等安防监控。详细阐述了系统的软硬件设计以及PC机与多台PIC单片机的串行通讯方案。运用了单片机有关的软硬件技术。