Resolution complexity of random constraint satisfaction problems: Another half of the story

AbstractLet Cn,m2,k,t be a random constraint satisfaction problem (CSP) on n binary variables, where m constraints are selected uniformly at random from all the possible k-ary constraints each of which contains exactly t tuples of the values as its restrictions. We establish an upper bound on the constraint tightness threshold for Cn,m2,k,t to have an exponential resolution complexity. The upper bound partly answers the open problem regarding the CSP resolution complexity with the tightness between the existing upper and lower bounds [D. Mitchell, Resolution complexity of random constraints, in: Proceedings Principles and Practices of Constraint Programming—CP 2002, Springer, Berlin, 2002, pp. 295–309]

A Simple Model to Generate Hard Satisfiable Instances

In this paper, we try to further demonstrate that the models of random CSP instances proposed by [Xu and Li, 2000; 2003] are of theoretical and practical interest. Indeed, these models, called RB and RD, present several nice features. First, it is quite easy to generate random instances of any arity since no particular structure has to be integrated, or property enforced, in such instances. Then, the existence of an asymptotic phase transition can be guaranteed while applying a limited restriction on domain size and on constraint tightness. In that case, a threshold point can be precisely located and all instances have the guarantee to be hard at the threshold, i.e., to have an exponential tree-resolution complexity. Next, a formal analysis shows that it is possible to generate forced satisfiable instances whose hardness is similar to unforced satisfiable ones. This analysis is supported by some representative results taken from an intensive experimentation that we have carried out, using complete and incomplete search methods.Comment: Proc. of 19th IJCAI, pp.337-342, Edinburgh, Scotland, 2005. For more information, please click http://www.nlsde.buaa.edu.cn/~kexu/papers/ijcai05-abstract.ht

Many Hard Examples in Exact Phase Transitions with Application to Generating Hard Satisfiable Instances

This paper first analyzes the resolution complexity of two random 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 CNF formulas hard for resolution, which is a central task of Proof-Complexity theory, but also propose models with both many hard instances and exact phase transitions. Then, the implications of such models are addressed. It is shown both theoretically and experimentally that an application of Model RB/RD might be in the generation of hard satisfiable instances, which is not only of practical importance but also related to some open problems in cryptography such as generating one-way functions. Subsequently, a further theoretical support for the generation method is shown by establishing exponential lower bounds on the complexity of solving random satisfiable and forced satisfiable instances of RB/RD near the threshold. 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.Comment: 19 pages, corrected mistakes in Theorems 5 and

Random Models of Very Hard 2QBF and Disjunctive Programs: An Overview

We present an overview of models of random quantified boolean formulas and their natural random disjunctive ASP program counter-parts that we have recently proposed. The models have a simple structure but also theoretical and empirical properties that make them useful for further advancement of the SAT, QBF and ASP solvers

The Satisfiability Threshold for a Seemingly Intractable Random Constraint Satisfaction Problem

We determine the exact threshold of satisfiability for random instances of a particular NP-complete constraint satisfaction problem (CSP). This is the first random CSP model for which we have determined a precise linear satisfiability threshold, and for which random instances with density near that threshold appear to be computationally difficult. More formally, it is the first random CSP model for which the satisfiability threshold is known and which shares the following characteristics with random k-SAT for k >= 3. The problem is NP-complete, the satisfiability threshold occurs when there is a linear number of clauses, and a uniformly random instance with a linear number of clauses asymptotically almost surely has exponential resolution complexity.Comment: This is the long version of a paper that will be published in the SIAM Journal on Discrete Mathematics. This long version includes an appendix and a computer program. The contents of the paper are unchanged in the latest version. The format of the arxiv submission was changed so that the computer program will appear as an ancillary file. Some comments in the computer program were update

Constraint programming for type inference in flexible model-driven engineering

Domain experts typically have detailed knowledge of the concepts that are used in their domain; however they often lack the technical skills needed to translate that knowledge into model-driven engineering (MDE) idioms and technologies. Flexible or bottom-up modelling has been introduced to assist with the involvement of domain experts by promoting the use of simple drawing tools. In traditional MDE the engineering process starts with the definition of a metamodel which is used for the instantiation of models. In bottom-up MDE example models are defined at the beginning, letting the domain experts and language engineers focus on expressing the concepts rather than spending time on technical details of the metamodelling infrastructure. The metamodel is then created manually or inferred automatically. The flexibility that bottom-up MDE offers comes with the cost of having nodes in the example models left untyped. As a result, concepts that might be important for the definition of the domain will be ignored while the example models cannot be adequately re-used in future iterations of the language definition process. In this paper, we propose a novel approach that assists in the inference of the types of untyped model elements using Constraint Programming. We evaluate the proposed approach in a number of example models to identify the performance of the prediction mechanism and the benefits it offers. The reduction in the effort needed to complete the missing types reaches up to 91.45% compared to the scenario where the language engineers had to identify and complete the types without guidance