The satisfiability problem in regular CNF-formulas

 In this paper we deal with the propositional satisfiability (SAT) problem for a kind of multiple-valued clausal forms known as regular CNF-formulas and extend some known results from classical logic to this kind of formulas. We present a Davis–Putnam-style satisfiability checking procedure for regular CNF-formulas equipped with suitable data structures and prove its completeness. Then, we describe a series of experiments for regular random 3-SAT instances. We observe that, for the regular 3-SAT problem with this procedure, there exists a threshold of the ratio of clauses to variables such that (i) the most computationally difficult instances tend to be found near the threshold, (ii) there is a sharp transition from satisfiable to unsatisfiable instances at the threshold and (iii) the value of the threshold increases as the number of truth values considered increases. Instances in the hard part provide benchmarks for the evaluation of regular satisfiability solvers.     

两种改进的模拟退火算法求解大值域约束满足问题

随机约束满足问题的相变现象及求解算法是NP-完全问题的研究热点。RB模型（Revised B）是一个非平凡的随机约束满足问题,它具有精确的可满足性相变现象和极易产生难解实例这两个重要特征。针对RB模型这一类具有大值域的随机约束满足问题,提出了两种基于模拟退火的改进算法即RSA（Revised Simulated Annealing Algorithm）和GSA（Genetic-simulated Annealing Algorithm）。将这两种算法用于求解RB模型的随机实例,数值实验结果表明：在进入相变区域时,RSA和GSA算法依然可以有效地找到随机实例的解,并且在求解效率上明显优于随机游走算法。在接近相变阈值点时,由这两种算法得到的最优解仅使得极少数的约束无法满足。     

Spines of random constraint satisfaction problems: definition and connection with computational complexity

We study the connection between the order of phase transitions in combinatorial problems and the complexity of decision algorithms for such problems. We rigorously show that, for a class of random constraint satisfaction problems, a limited connection between the two phenomena indeed exists. Specifically, we extend the definition of the spine order parameter of Bollobás et al. [10] to random constraint satisfaction problems, rigorously showing that for such problems a discontinuity of the spine is associated with a 2^Ω(n) resolution complexity (and thus a 2^Ω(n) complexity of DPLL algorithms) on random instances. The two phenomena have a common underlying cause: the emergence of “large” (linear size) minimally unsatisfiable subformulas of a random formula at the satisfiability phase transition.We present several further results that add weight to the intuition that random constraint satisfaction problems with a sharp threshold and a continuous spine are “qualitatively similar to random 2-SAT”. Finally, we argue that it is the spine rather than the backbone parameter whose continuity has implications for the decision complexity of combinatorial problems, and we provide experimental evidence that the two parameters can behave in a different manner.AMS subject classification 68Q25, 82B27     

可满足实例的归结复杂度

GRB模型是一种随机约束满足问题模型,此模型具有精确的可满足相变现象。针对实验中出现的GRB模型在相变区域产生的可满足实例都是难解的现象,利用子句宽度和归结复杂度的关系证明了GRB模型在相变点附近产生的可满足实例对于树型归结证明具有指数下界。因此从理论上证明了在相变区域产生的可满足实例对基于归结的算法是难解的。     

Empirical investigation of stochastic local search for maximum satisfiability

The maximum satisfiability (MAX-SAT) problem is an important NP-hard problem in theory, and has a broad range of applications in practice. Stochastic local search (SLS) is becoming an increasingly popular method for solving MAX-SAT. Recently, a powerful SLS algorithm called CCLS shows efficiency on solving random and crafted MAX-SAT instances. However, the performance of CCLS on solving industrial MAX-SAT instances lags far behind. In this paper, we focus on experimentally analyzing the performance of SLS algorithms for solving industrial MAXSAT instances. First, we conduct experiments to analyze why CCLS performs poor on industrial instances. Then we propose a new strategy called additive BMS (Best from Multiple Selections) to ease the serious issue. By integrating CCLS and additive BMS, we develop a new SLS algorithm for MAXSAT called CCABMS, and related experiments indicate the efficiency of CCABMS. Also, we experimentally analyze the effectiveness of initialization methods on SLS algorithms for MAX-SAT, and combine an effective initialization method with CCABMS, resulting in an enhanced algorithm. Experimental results show that our enhanced algorithm performs better than its state-of-the-art SLS competitors on a large number of industrial MAX-SAT instances.     

Data reductions, fixed parameter tractability, and random weighted d-CNF satisfiability

Data reduction is a key technique in the study of fixed parameter algorithms. In the AI literature, pruning techniques based on simple and efficient-to-implement reduction rules also play a crucial role in the success of many industrial-strength solvers. Understanding the effectiveness and the applicability of data reduction as a technique for designing heuristics for intractable problems has been one of the main motivations in studying the phase transition of randomly-generated instances of NP-complete problems.In this paper, we take the initiative to study the power of data reductions in the context of random instances of a generic intractable parameterized problem, the weighted d-CNF satisfiability problem. We propose a non-trivial random model for the problem and study the probabilistic behavior of the random instances from the model. We design an algorithm based on data reduction and other algorithmic techniques and prove that the algorithm solves the random instances with high probability and in fixed-parameter polynomial time O(dknm) where n is the number of variables, m is the number of clauses, and k is the fixed parameter. We establish the exact threshold of the phase transition of the solution probability and show that in some region of the problem space, unsatisfiable random instances of the problem have parametric resolution proof of fixed-parameter polynomial size. Also discussed is a more general random model and the generalization of the results to the model.     

取定s的严格d-正则随机(3,2s)-SAT问题的可满足临界

3-CNF公式的随机难解实例生成对于揭示3-SAT问题的难解实质和设计满足性测试的有效算法有着重要意义.对于整数k>2和s>0,如果在一个k-CNF公式中每个变量正负出现次数均为s,则称该公式是严格正则（k,2s）-CNF公式.受严格正则（k,2s）-CNF公式的结构特征启发,提出每个变量正负出现次数之差的绝对值均为d的严格d-正则（k,2s）-CNF公式,并使用新提出的SDRRK2S模型生成严格d-正则随机（k,2s）-CNF公式.取定整数5<s<11,模拟实验显示,严格d-正则随机（3,2s）-SAT问题存在SAT-UNSAT相变现象和HARD-EASY相变现象.因此,立足于3-CNF公式的随机难解实例生成,研究了严格d-正则随机（3,2s）-SAT问题在s取定时的可满足临界.通过构造一个特殊随机实验和使用一阶矩方法,得到了严格d-正则随机（3,2s）-SAT问题在s取定时可满足临界值的一个下界.模拟实验结果验证了理论证明所得下界的正确性.     

Between 2- and 3-colorability

The recognition of 3-colorable graphs is an NP-complete problem, while 2-colorable (i.e., bipartite) graphs can be recognized in polynomial time. To make the complexity gap more precise, we study intermediate graph classes and respective problems. This note proposes a conjecture that separates difficult instances of the problem from polynomially solvable ones and proves the “polynomial” part of the conjecture.     

求解可满足性问题的改进的模拟退火算法

该文为可满足性问题的高效近似求解提出了改进的模拟退火算法。数值实验表明,对于该文随机产生的测试问题例,改进的模拟退火算法完全胜过局部搜索算法、模拟退火算法以及目前国际上流行的WSAT算法。     

An explicit semidefinite characterization of satisfiability for Tseitin instances on toroidal grid graphs

This paper is concerned with the application of semidefinite programming to the satisfiability problem, and in particular with using semidefinite liftings to efficiently obtain proofs of unsatisfiability. We focus on the Tseitin satisfiability instances which are known to be hard for many proof systems. For Tseitin instances based on toroidal grid graphs, we present an explicit semidefinite programming problem with dimension linear in the size of the Tseitin instance, and prove that it characterizes the satisfiability of these instances, thus providing an explicit certificate of satisfiability or unsatisfiability. Research partially supported by the Natural Sciences and Engineering Research Council of Canada.     

Stochastic Boolean Satisfiability

Satisfiability problems and probabilistic models are core topics of artificial intelligence and computer science. This paper looks at the rich intersection between these two areas, opening the door for the use of satisfiability approaches in probabilistic domains. The paper examines a generic stochastic satisfiability problem, SSAT, which can function for probabilistic domains as SAT does for deterministic domains. It shows the connection between SSAT and well-studied problems in belief network inference and planning under uncertainty, and defines algorithms, both systematic and stochastic, for solving SSAT instances. These algorithms are validated on random SSAT formulae generated under the fixed-clause model. In spite of the large complexity gap between SSAT (PSPACE) and SAT (NP), the paper suggests that much of what we have learned about SAT transfers to the probabilistic domain.     

Configuration landscape analysis and backbone guided local search.: Part I: Satisfiability and maximum satisfiability

Boolean satisfiability (SAT) and maximum satisfiability (Max-SAT) are difficult combinatorial problems that have many important real-world applications. In this paper, we first investigate the configuration landscapes of local minima reached by the WalkSAT local search algorithm, one of the most effective algorithms for SAT. A configuration landscape of a set of local minima is their distribution in terms of quality and structural differences relative to an optimal or a reference solution. Our experimental results show that local minima from WalkSAT form large clusters, and their configuration landscapes constitute big valleys, in that high quality local minima typically share large partial structures with optimal solutions. Inspired by this insight into WalkSAT and the previous research on phase transitions and backbones of combinatorial problems, we propose and develop a novel method that exploits the configuration landscapes of such local minima. The new method, termed as backbone-guided search, can be embedded in a local search algorithm, such as WalkSAT, to improve its performance. Our experimental results show that backbone-guided local search is effective on overconstrained random Max-SAT instances. Moreover, on large problem instances from a SAT library (SATLIB), the backbone guided WalkSAT algorithm finds satisfiable solutions more often than WalkSAT on SAT problem instances, and obtains better solutions than WalkSAT on Max-SAT problem instances, improving solution quality by 20% on average.     

Why Almost All k-Colorable Graphs Are Easy to Color

Coloring a k-colorable graph using k colors (k≥3) is a notoriously hard problem. Considering average case analysis allows for better results. In this work we consider the uniform distribution over k-colorable graphs with n vertices and exactly cn edges, c greater than some sufficiently large constant. We rigorously show that all proper k-colorings of most such graphs lie in a single “cluster”, and agree on all but a small, though constant, portion of the vertices. We also describe a polynomial time algorithm that whp finds a proper k-coloring of such a random k-colorable graph, thus asserting that most such graphs are easy to color. This should be contrasted with the setting of very sparse random graphs (which are k-colorable whp), where experimental results show some regime of edge density to be difficult for many coloring heuristics.     

随机正则(k,r)-SAT问题的可满足临界

研究k-SAT问题实例中每个变元恰好出现r=2s次,且每个变元对应的正、负文字都出现s次的严格随机正则（k,r）-SAT问题.通过构造一个特殊的独立随机实验,结合一阶矩方法,给出了严格随机正则（k,r）-SAT问题可满足临界值的上界.由于严格正则情形与正则情形的可满足临界值近似相等,因此得到了随机正则（k,r）-SAT问题可满足临界值的新上界.该上界不仅小于当前已有的随机正则（k,r）-SAT问题的可满足临界值上界,而且还小于一般的随机k-SAT问题的可满足临界值.因此,这也从理论上解释了在相变点处的随机正则（k,r）-SAT问题实例通常比在相应相变点处同规模的随机k-SAT问题实例更难满足的原因.最后,数值分析结果验证了所给上界的正确性.     

一种基于变量熵求解约束满足问题的置信传播算法

在置信传播(belief propagation,BP)算法中,提出一种基于变量熵来挑选变量从而固定变量赋值的策略,用于求解一类具有增长定义域的随机约束满足问题.RB模型是一个具有增长定义域的随机约束满足问题的典型代表,已经严格证明它不仅存在精确的可满足性相变现象,而且可以生成难解实例.在RB模型上选取两组不同的参数进行数值实验.结果表明:在接近可满足性相变点时,BP引导的消去算法仍然可以非常有效地找到随机实例的解;不断增加问题的规模,算法的运行时间呈指数级增长;并且当控制参数(约束紧度)增加时,变量的平均自由度逐渐降低.     

Disjoint essential sets of implicates of a CQ Horn function

In this paper we study a class of CQ Horn functions introduced in Boros et al. (Ann Math Artif Intell 57(3–4):249–291, 2010). We prove that given a CQ Horn function f, the maximal number of pairwise disjoint essential sets of implicates of f equals the minimum number of clauses in a CNF representing f. In other words, we prove that the maximum number of pairwise disjoint essential sets of implicates of f constitutes a tight lower bound on the size (the number of clauses) of any CNF representation of f.     

GridSAT: Design and Implementation of a Computational Grid Application

We present the latest instantiation of GridSAT [1], a distributed and complete satisfiability solver that is explicitly designed to aggregate Grid resources for application performance. GridSAT was previously shown to outperform the state-of-the-art sequential solvers. In this work, we explore the unprecedented solving power GridSAT enables through algorithmic and implementation innovations. We describe the implementation techniques that allow GridSAT to leverage a variety of high-end batch-scheduled resources, clusters, interactive workstations, and personal computing resources through autonomous scheduling, checkpoint scheduling, and work migration. These innovations have allowed GridSAT to solve a set of ‘hard’ and previously unsolved industrial and community satisfiability problems. In addition to this new solution power, GridSAT also outperforms the otherwise highest performance general solvers on the annual SAT competition [21] performance benchmarks.This work was supported by grants from the National Science Foundation, numbered CAREER-0093166, EIA-9975020, ACI-0103759, and CCR-0331654 and by the San Diego Supercomputer Center and the TeraGrid project.     

Exact satisfiability,a natural extension of set partition,and its average case behavior

The problem of determining whether a Boolean formula in conjunctive normal form is satisfiable in such a way that in each clause exactly one literal is set true, and all the other literals are set false is called the exact satisfiability problem. The exact satisfiability problem is well known to be NP-complete [5] and it contains the well known set partitioning problem as a special case. We study here the average time complexity of a simple backtracking strategy for solving the exact satisfiability problem under two probability models, the constant density model and the constant degree model. For both models we present results sharply separating classes of instances solvable in low degree polynomial time in the average from classes for which superpolynomial or exponential time is needed in the average.     

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.     

基于消息传递关系网络的布尔可满足性预测

布尔可满足性求解能够验证的问题规模通常受限, 因此, 如何高精度地预测布尔可满足性问题的可满足性是一个重要研究问题, 也是一项具有挑战性的工作.相关研究工作一般使用由文字节点和子句节点组成的图来表示布尔可满足性问题的结构, 但是这种表征方法缺少了变量、子句之间的重要关系信息.在我们的方法中, 通过将原始布尔可满足性问题实例表征为多关系异构图的方式来表达变量和句子之间的关系, 并设计使用消息传递关系网络模型来捕获实例的关系信息, 提取了更多的结构特征.结果表明, 该模型在预测精度、泛化能力和资源需求等方面均优于现有模型, 对所选数据集的平均预测精度为81%.该模型在小规模问题(变量数为100)上训练, 在大规模数据集上预测的平均预测精度达到了80.8%.同时, 该模型对随机生成的非均匀随机问题的预测精度达到99%, 这意味着它学习了预测可满足性的重要特征.此外, 模型预测所花费的时间随着问题规模的增大也只是呈线性增长. 总结而言, 本文基于关系消息传递网络提出了一个预测精度更高, 泛化能力更好的布尔可满足性预测方法.