A quantitative solution to Constraint Satisfaction Problem (CSP)

Constraint Satisfaction Problem (CSP) is an important problem in artificial intelligence and operations research. Many practical problems can be formulated as CSP, i.e., finding a consistent value assignment to variables subject to a set of constraints. In this paper, we give a quantitative approach to solve the CSPs which deals uniformly with binary constraints as well as high order,k-ary (k ≥ 2) constraints. In this quantitative approach, using variable transformation and constraint transformation, a CSP is transformed into a satisfiability (SAT) problem. The SAT problem is then solved within a continuous search space. We will evaluate the performance of this method based on randomly generated SAT problem instances and regularly generatedk-ary (k ≥ 2) CSP problem instances.     

最坏情况下X3SAT最大海明距离问题最小上界

X3SAT最大海明距离问题是指对于一个X3SAT问题实例,寻找该问题的任意两组可满足赋值之间的最大海明距离。提出了一个基于DPLL的精确算法HMX来求解X3SAT最大海明距离问题,根据公式中某个变量在两组真值赋值中的不同取值进行分支。给出了多种化简规则,这些规则很好地提高了算法的时间效率。证明了该算法可以将X3SAT最大海明距离问题的最小上界由目前最好的O(1.7107n)缩小到O(1.6760n),其中n为公式中变量的数目。     

采用二元CSP引擎求解RTL数据通路的可满足性

针对寄存器传输级(RTL)验证和测试过程中非常重要的数据通路可满足性求解问题,提出一种基于二元约束满足问题(CSP)的求解方法,包括数据通路提取、二元CSP建模和搜索求解3个步骤.数据通路提取通过对接口布尔变量和某些字变量赋值,为各个数据通路器件建立环境;二元CSP建模则根据该环境和各个数据通路器件的功能,将数据通路的可满足性问题转化为二元CSP描述;该二元CSP问题的描述被送入到二元CSP引擎,并采用冲突引导的回跳搜索策略进行求解,获得有解的例证或无解的判定.实验结果表明,即使在没有采取很多优化策略的条件下,该方法仍有较好的性能,并优于基于线性规划(LP)的求解方法.     

meSAT: multiple encodings of CSP to SAT

One approach for solving Constraint Satisfaction Problems (CSP) (and related Constraint Optimization Problems (COP)) involving integer and Boolean variables is reduction to propositional satisfiability problem (SAT). A number of encodings (e.g., direct, log, support, order) for this purpose exist as well as specific encodings for some constraints that are often encountered (e.g., cardinality constraints, global constraints). However, there is no single encoding that performs well on all classes of problems and there is a need for a system that supports multiple encodings. We present a system that translates specifications of finite linear CSP problems into SAT instances using several well-known encodings, and their combinations. We also present a methodology for selecting a suitable encoding based on simple syntactic features of the input CSP instance. Thorough evaluation has been performed on large publicly available corpora and our encoding selection method improves upon the efficiency of existing encodings and state-of-the-art tools used in comparison.     

Argumentation frameworks as constraint satisfaction problems

Argumentation is a promising approach for defeasible reasoning. It consists of justifying each plausible conclusion by arguments. Since the available information may be inconsistent, a conclusion and its negation may both be justified. The arguments are thus said to be conflicting. The main issue is how to evaluate the arguments. Several semantics were proposed for that purpose. The most important ones are: stable, preferred, complete, grounded and admissible. A semantics is a set of criteria that should be satisfied by a set of arguments, called extension, in order to be acceptable. Different decision problems related to these semantics were defined (like whether an argumentation framework has a stable extension). It was also shown that most of these problems are intractable. Consequently, developing algorithms for these problems is not trivial and thus the implementation of argumentation systems not obvious. Recently, some solutions to this problem were found. The idea is to use a reduction method where a given problem is translated in another one like SAT or ASP. This paper follows this line of research. It studies how to encode the problem of computing the extensions of an argumentation framework (under each of the previous semantics) as a constraint satisfaction problem (CSP). Such encoding is of great importance since it makes it possible to use the very efficient solvers (developed by the CSP community) for computing the extensions. Our encodings take advantage of existing reductions to SAT problems in the case of Dung’s abstract framework. Among the various ways of translating a SAT problem into a CSP one, we propose the most appropriate one in the argumentation context. We also provide encodings in case two other families of argumentation frameworks: the constrained version of Dung’s abstract framework and preference-based argumentation framework.     

On the hardness of solving edge matching puzzles as SAT or CSP problems

Edge matching puzzles have been amongst us for a long time now and traditionally they have been considered, both, a children’s game and an interesting mathematical divertimento. Their main characteristics have already been studied, and their worst-case complexity has been properly classified as a NP-complete problem. It is in recent times, specially after being used as the problem behind a money-prized contest, with a prize of 2US$ million for the first solver, that edge matching puzzles have attracted mainstream attention from wider audiences, including, of course, computer science people working on solving hard problems. We consider these competitions as an interesting opportunity to showcase SAT/CSP solving techniques when confronted to a real world problem to a broad audience, a part of the intrinsic, i.e. monetary, interest of such a contest. This article studies the NP-complete problem known as edge matching puzzle using SAT and CSP approaches for solving it. We will focus on providing, first and foremost, a theoretical framework, including a generalized definition of the problem. We will design and show algorithms for easy and fast problem instances generation, generators with easily tunable hardness. Afterwards we will provide with SAT and CSP models for the problems and we will study problem complexity, both typical case and worst-case complexity. We will also provide some specially crafted heuristics that result in a boost in solving time and study which is the effect of such heuristics.     

基于寻找可满足2SAT子问题的SAT算法*

可满足问题（SAT）是一个NP-Hard问题。提出了一种求解SAT的新算法（FFSAT）。该算法将SAT问题转换为寻找一个可满足的2-SAT子问题。SAT问题虽然是NP完全问题,但是当所有子句长度不大于2时,SAT问题可以在线性时间求解。使用2-SAT算法-BinSat求解2-SAT子问题,当它不满足时,根据赋值选择新的2-SAT子问题。实验结果表明,采用本算法的结果优于UnitWalk。     

E-CARGO模型在CSP问题中的描述

基于角色的协同RBC(Role-Based Collaboration)是一套研究角色及它们之间复杂关系的方法、理论和技术。在RBC中,群组角色分配GRA(Group Role Assignment)既是一个关键问题,也是一个难题。已有许多研究探讨了基于Q(Qualification)矩阵来处理GRA问题,但仅利用Q矩阵难以描述问题中的复杂约束关系。因此,将约束集(Constraint)引进E-CARGO模型,提出了带约束的EC-CARGO模型,研究了RBC、GRA、SAT(SATisfaction)和CSP(Constraint Satisfaction Problem)之间的联系,建立了RBC-GRA-SAT-CSP问题求解转换关系;提出应用EC-CARGO模型求解经典CSP约束满足问题的方法,进而描述了应用GRA求解CSP约束满足问题的通用框架。最后以N皇后问题为例,验证了通过GRA的约束指派求解CSP问题的有效性。     

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.     

Interactive tool support for CSP || B consistency checking

CSP || B is an integration of two well known formal notations: CSP and B. It provides a method for modelling systems with both complex state (described in B machines) and control flow (described as CSP processes). Consistency checking within this approach verifies that a controller process never calls a B operation outside its precondition. Otherwise the behaviour of the operation cannot be predicted. In previous work, this check was carried out by manually decomposing the model before preprocessing the CSP processes to perform a hand-written weakest precondition proof. In this paper, a framework is described that mechanises consistency checking in a theorem prover and removes the need for preprocessing. This work is based on an existing PVS embedding of the CSP traces model, but it is extended by introducing a notion of state so that the interaction between processes and machines can be analysed. Numerous rules have been defined (and proved) which enable consistency checking and decomposition via PVS proof. These rules also formally justify the relaxation of previous constraints on CSP || B architectures, thereby widening the scope of CSP || B modelling. The PVS embedding and rules presented in this paper are not only applicable to CSP || B specifications, but to other combined approaches which use a non-blocking semantics for the state-based operations. R. Lazic, R. Nagarajan and J. C. P. Woodcock     

Reformulation based MaxSAT robustness

The presence of uncertainty in the real world makes robustness a desirable property of solutions to constraint satisfaction problems (CSP). A solution is said to be robust if it can be easily repaired when unexpected events happen. This issue has already been addressed in the frameworks of Boolean satisfiability (SAT) and Constraint Programming (CP). Most existing works on robustness implement search algorithms to look for robust solutions instead of taking the declarative approach of reformulation, since reformulation tends to generate prohibitively large formulas, especially in the CP setting. In this paper we consider the unaddressed problem of robustness in weighted MaxSAT, by showing how robust solutions to weighted MaxSAT instances can be effectively obtained via reformulation into pseudo-Boolean formulae. Our encoding provides a reasonable balance between increase in size and performance, as shown by our experiments in the robust resource allocation framework. We also address the problem of flexible robustness, where some of the breakages may be left unrepaired if a totally robust solution does not exist. In a sense, since the use of SAT and MaxSAT encodings for solving CSP has been gaining wide acceptance in recent years, we provide an easy-to-implement new method for achieving robustness in combinatorial optimization problems.     

Survey Propagation：一种求解SAT的高效算法

Survey propagation是一种新生的SAT(CSP)算法。它基于统计物理的spin glass模型，针对具体问题进行纵览(survey)，从而极大地降低求解的复杂度。但sp算法在某些时候不收敛，或引导向错误的解。对此，G．Parisi提出一种复杂回溯(backtrack)算法，而作者在sp中加入简单回溯，也使一部分此类问题得到解决。     

Many hard examples in exact phase transitions

This paper analyzes the resolution complexity of two random constraint satisfaction problem (CSP) models (i.e. Model RB/RD) for which we can establish the existence of phase transitions and identify the threshold points exactly. By encoding CSPs into CNF formulas, it is proved that almost all instances of Model RB/RD have no tree-like resolution proofs of less than exponential size. Thus, we not only introduce new families of CSPs and CNF formulas hard to solve, which can be useful in the experimental evaluation of CSP and SAT algorithms, but also propose models with both many hard instances and exact phase transitions. Finally, conclusions are presented, as well as a detailed comparison of Model RB/RD with the Hamiltonian cycle problem and random 3-SAT, which, respectively, exhibit three different kinds of phase transition behavior in NP-complete problems.     

Earth Observation Satellite Management

The daily management of an earth observation satellite is a challenging combinatorial optimization problem. This problem can be roughly stated as follows: given (1) a set of candidate images for the next day, each one associated with a weight reflecting its importance, (2) a set of imperative constraints expressing physical limitations (no overlapping images, sufficient transition times, bounded instantaneous data flow and recording capacity), select a subset of candidates which meets all the constraints and maximizes the sum of the weights of the selected candidates. It can be easily cast in variants of the CSP, ILP or SAT frameworks. As a benchmark, we propose to the CONSTRAINTS community a set of instances, which have been produced from a simulator of the order book of the future satellite SPOT5. The fact that only some of them have been optimally solved should make them very attractive.     

Dealing with Satisfiability and n-ary CSPs in a Logical Framework

We investigate in this work a generalization of the known CNF representation that we call General Normal Form (GNF) and extend the resolution rule of Robinson to design a new sound and complete resolution proof system for the GNF. We show that by using a cardinality operator in the GNF we obtain the new representation CGNF that allows a natural and efficient Boolean encoding for n-ary finite discrete extensional CSPs. We prove that the space complexity of a CSP encoded in the CGNF is identical to its space complexity when it is expressed in the classical CSP representation. We prove that the generalized resolution rule applies for the CGNF encoding and introduce a new inference rule whose application until saturation achieves arc-consistency in a linear time complexity for n-ary CSP expressed in the CGNF encoding. Two enumerative methods for solving CSP instances encoded in this Boolean framework are studied: the first one (equivalent to MAC in CSPs) maintains full arc-consistency at each node of the search tree while the second (equivalent to FC in CSPs) performs partial arc-consistency on each node. Both methods are experimented and compared on some instances of the Ramsey problem, randomly generated 3/4-ary CSPs and promising results are obtained. We also adapted the notion of clause learning defined in SAT for the CGNF encoding. Finally, we show how the proposed inference rule can be used to achieve Path-consistency in the case of binary CSPs encoded in CGNF, and how it can be used to enforce strong path consistency when it is combined with the generalized resolution rule.     

Remarks on the cellular automaton global synchronisation problem: deterministic versus stochastic models

In the global synchronisation problem, one is asked to find a cellular automaton which has the property that every initial condition evolves into a homogeneous blinking state. We study this simple inverse problem for the case of one-dimensional systems with periodic boundary conditions. Two paradoxical observations are made: (a) despite the apparent simplicity of finding rules with good statistical results, there exist no perfect deterministic solutions to this problem, (b) if we allow the use of randomness in the local rule, constructing “perfect” stochastic solutions is easy. For the stochastic case, we give some rules for which the mean time of synchronisation varies quadratically with the number of cells and ask if this result can be improved. To explore more deeply the deterministic rules, we code our problem as a SAT problem and use SAT solvers to find rules that synchronise a large set of initial conditions (in appendix).

     

GASAT: a genetic local search algorithm for the satisfiability problem

This paper presents GASAT, a hybrid algorithm for the satisfiability problem (SAT). The main feature of GASAT is that it includes a recombination stage based on a specific crossover and a tabu search stage. We have conducted experiments to evaluate the different components of GASAT and to compare its overall performance with state-of-the-art SAT algorithms. These experiments show that GASAT provides very competitive results.     

Approximating Satisfiable Satisfiability Problems

We study the approximability of the Maximum Satisfiability Problem (MAX SAT) and of the boolean k -ary Constraint Satisfaction Problem (MAX k CSP) restricted to satisfiable instances. For both problems we improve on the performance ratios of known algorithms for the unrestricted case. Our approximation for satisfiable MAX 3CSP instances is better than any possible approximation for the unrestricted version of the problem (unless P=NP). This result implies that the requirement of perfect completeness weakens the acceptance power of non-adaptive PCP verifiers that read 3 bits. We also present the first non-trivial results about PCP classes defined in terms of free bits that collapse to P. Received August 1997; revised February 1999.     

Propositional Satisfiability Algorithm to Find Minimal Reducts for Data Mining

A fundamental problem in data mining is whether the whole information available is always necessary to represent the information system (IS). Reduct is a rough set approach in data mining that determines the set of important attributes to represent the IS. The search for minimal reduct is based on the assumption that within the dataset in an IS, there are attributes that are more important than the rest. An algorithm in finding minimal reducts based on Propositional Satisfiability (SAT) algorithm is proposed. A branch and bound algorithm is presented to solve the proposed SAT problem. The experimental result shows that the proposed algorithm has significantly reduced the number of rules generated from the obtained reducts with high percentage of classification accuracy.     

规划问题编码成SAT问题研究

智能规划问题是一个NP-hard的问题。近年来,由于在可满足问题（SAT）研究领域取得了较大进展,出现了一大批快速的能达到工业级应用的SAT solver求解器的出现,这使得运用可满足技术来求解规划问题的方法越来越得到智能规划研究者们的重视。用可满足技术求解规划问题的首要任务是必须将规划问题“翻译”成可满足问题。讨论了如何将规划问题编码成命题可满足问题的一般技术,并对“直接编码”和“基于规划图的编码”两种编码技术进行了比较,指出了两种编码技术各自的优缺点。在此基础上,深入地分析了各种不同的编码方案之间的异同点以及它们各自的优缺点。最后,指出了用SAT技术求解规划问题中存在的一些问题以及相关改进方法。