Pseudo-Boolean Constraint Encodings for Conjunctive Normal Form and their Applications

In contrast to a single clause a pseudo-Boolean (PB) constraint is much more expressive and hence it is easier to define problems with the help of PB constraints. But while PB constraints provide us with a high-level problem description, it has been shown that solving PB constraints can be done faster with the help of a SAT solver. To apply such a solver to a PB constraint we have to encode it with clauses into conjunctive normal form (CNF). While we can find a basic encoding into CNF which is equivalent to a given PB constraint, the solving time of a SAT solver significantly depends on different properties of an encoding, e.g. the number of clauses or if generalized arc consistency (GAC) is maintained during the search for a solution. There are various PB encodings that try to optimize or balance these properties. This thesis is about such encodings. For a better understanding of the research field an overview about the state-of-the art encodings is given. The focus of the overview is a simple but complete description of each encoding, such that any reader could use, implement and extent them in his own work. In addition two novel encodings are presented: The Sequential Weight Counter (SWC) encoding and the Binary Merger Encoding. While the SWC encoding provides a very simple structure – it is listed in four lines – empirical evaluation showed its practical usefulness in various applications. The Binary Merger encoding reduces the number of clauses a PB encoding needs while having the important GAC property. To the best of our knowledge currently no other encoding has a lower upper bound for the number of clauses produced by a PB encoding with this property. This is an important improvement of the state-of-the art, since both GAC and a low number of clauses are vital for an improved solving time of the SAT solver. The thesis also contributes to the development of new applications for PB constraint encodings. The programming library PBLib provides researchers with an open source implementation of almost all PB encodings – including the encodings for the special cases at-most-one and cardinality constraints. The PBLib is also the foundation of the presented weighted MaxSAT solver optimax, the PBO solver pbsolver and the WBO, PBO and weighted MaxSAT solver npSolver

On SAT representations of XOR constraints

We study the representation of systems S of linear equations over the two-element field (aka xor- or parity-constraints) via conjunctive normal forms F (boolean clause-sets). First we consider the problem of finding an "arc-consistent" representation ("AC"), meaning that unit-clause propagation will fix all forced assignments for all possible instantiations of the xor-variables. Our main negative result is that there is no polysize AC-representation in general. On the positive side we show that finding such an AC-representation is fixed-parameter tractable (fpt) in the number of equations. Then we turn to a stronger criterion of representation, namely propagation completeness ("PC") --- while AC only covers the variables of S, now all the variables in F (the variables in S plus auxiliary variables) are considered for PC. We show that the standard translation actually yields a PC representation for one equation, but fails so for two equations (in fact arbitrarily badly). We show that with a more intelligent translation we can also easily compute a translation to PC for two equations. We conjecture that computing a representation in PC is fpt in the number of equations.Comment: 39 pages; 2nd v. improved handling of acyclic systems, free-standing proof of the transformation from AC-representations to monotone circuits, improved wording and literature review; 3rd v. updated literature, strengthened treatment of monotonisation, improved discussions; 4th v. update of literature, discussions and formulations, more details and examples; conference v. to appear LATA 201

Certifying Correctness for Combinatorial Algorithms : by Using Pseudo-Boolean Reasoning

Over the last decades, dramatic improvements in combinatorialoptimisation algorithms have significantly impacted artificialintelligence, operations research, and other areas. These advances,however, are achieved through highly sophisticated algorithms that aredifficult to verify and prone to implementation errors that can causeincorrect results. A promising approach to detect wrong results is touse certifying algorithms that produce not only the desired output butalso a certificate or proof of correctness of the output. An externaltool can then verify the proof to determine that the given answer isvalid. In the Boolean satisfiability (SAT) community, this concept iswell established in the form of proof logging, which has become thestandard solution for generating trustworthy outputs. The problem isthat there are still some SAT solving techniques for which prooflogging is challenging and not yet used in practice. Additionally,there are many formalisms more expressive than SAT, such as constraintprogramming, various graph problems and maximum satisfiability(MaxSAT), for which efficient proof logging is out of reach forstate-of-the-art techniques.This work develops a new proof system building on the cutting planesproof system and operating on pseudo-Boolean constraints (0-1 linearinequalities). We explain how such machine-verifiable proofs can becreated for various problems, including parity reasoning, symmetry anddominance breaking, constraint programming, subgraph isomorphism andmaximum common subgraph problems, and pseudo-Boolean problems. Weimplement and evaluate the resulting algorithms and a verifier for theproof format, demonstrating that the approach is practical for a widerange of problems. We are optimistic that the proposed proof system issuitable for designing certifying variants of algorithms inpseudo-Boolean optimisation, MaxSAT and beyond

SAT-based approaches for constraint optimization

La optimització amb restriccions ha estat utilitzada amb èxit par a resoldre problemes en molts dominis reals (industrials). Aquesta tesi es centra en les aproximacions lògiques, concretament en Màxima Satisfactibilitat (MaxSAT) que és la versió d’optimització del problema de Satisfactibilitat booleana (SAT). A través de MaxSAT, s’han resolt molts problemes de forma eficient. Famílies d’instàncies de la majoria d’aquests problemes han estat sotmeses a la MaxSAT Evaluation (MSE), creant així una col•lecció pública i accessible d’instàncies de referència. En les edicions recents de la MSE, els algorismes SAT-based han estat les aproximacions que han tingut un millor comportament per a les instàncies industrials. Aquesta tesi està centrada en millorar els algorismes SAT-based . El nostre treball ha contribuït a tancar varies instàncies obertes i a reduir dramàticament el temps de resolució en moltes altres. A més, hem trobat sorprenentment que reformular y resoldre el problema MaxSAT a través de programació lineal sencera era especialment adequat per algunes famílies. Finalment, hem desenvolupat el primer portfoli altament eficient par a MaxSAT que ha dominat en totes las categories de la MSE des de 2013.La optimización con restricciones ha sido utilizada con éxito para resolver problemas en muchos dominios reales (industriales). Esta tesis se centra en las aproximaciones lógicas, concretamente en Máxima Satisfacibilidad (MaxSAT) que es la versión de optimización del problema de Satisfacibilidad booleana (SAT). A través de MaxSAT, se han resuelto muchos problemas de forma eficiente. Familias de instancias de la mayoría de ellos han sido sometidas a la MaxSAT Evaluation (MSE), creando así una colección pública y accesible de instancias de referencia. En las ediciones recientes de la MSE, los algoritmos SAT-based han sido las aproximaciones que han tenido un mejor comportamiento para las instancias industriales. Esta tesis está centrada en mejorar los algoritmos SAT-based. Nuestro trabajo ha contribuido a cerrar varias instancias abiertas y a reducir dramáticamente el tiempo de resolución en muchas otras. Además, hemos encontrado sorprendentemente que reformular y resolver el problema MaxSAT a través de programación lineal entera era especialmente adecuado para algunas familias. Finalmente, hemos desarrollado el primer portfolio altamente eficiente para MaxSAT que ha dominado en todas las categorías de la MSE desde 2013.Constraint optimization has been successfully used to solve problems in many real world (industrial) domains. This PhD thesis is focused on logic-based approaches, in particular, on Maximum Satisfiability (MaxSAT) which is the optimization version of Satisfiability (SAT). There have been many problems efficiency solved through MaxSAT. Instance families on the majority of them have been submitted to the international MaxSAT Evaluation (MSE), creating a collection of publicly available benchmark instances. At recent editions of MSE, SAT-based algorithms were the best performing single algorithm approaches for industrial problems. This PhD thesis is focused on the improvement of SAT-based algorithms. All this work has contributed to close up some open instances and to reduce dramatically the solving time in many others. In addition, we have surprisingly found that reformulating and solving the MaxSAT problem through Integer Linear Programming (ILP) was extremely well suited for some families. Finally, we have developed the first highly efficient MaxSAT portfolio that dominated all categories of MSE since 2013

Towards Next Generation Sequential and Parallel SAT Solvers

This thesis focuses on improving the SAT solving technology. The improvements focus on two major subjects: sequential SAT solving and parallel SAT solving. To better understand sequential SAT algorithms, the abstract reduction system Generic CDCL is introduced. With Generic CDCL, the soundness of solving techniques can be modeled. Next, the conflict driven clause learning algorithm is extended with the three techniques local look-ahead, local probing and all UIP learning that allow more global reasoning during search. These techniques improve the performance of the sequential SAT solver Riss. Then, the formula simplification techniques bounded variable addition, covered literal elimination and an advanced cardinality constraint extraction are introduced. By using these techniques, the reasoning of the overall SAT solving tool chain becomes stronger than plain resolution. When using these three techniques in the formula simplification tool Coprocessor before using Riss to solve a formula, the performance can be improved further. Due to the increasing number of cores in CPUs, the scalable parallel SAT solving approach iterative partitioning has been implemented in Pcasso for the multi-core architecture. Related work on parallel SAT solving has been studied to extract main ideas that can improve Pcasso. Besides parallel formula simplification with bounded variable elimination, the major extension is the extended clause sharing level based clause tagging, which builds the basis for conflict driven node killing. The latter allows to better identify unsatisfiable search space partitions. Another improvement is to combine scattering and look-ahead as a superior search space partitioning function. In combination with Coprocessor, the introduced extensions increase the performance of the parallel solver Pcasso. The implemented system turns out to be scalable for the multi-core architecture. Hence iterative partitioning is interesting for future parallel SAT solvers. The implemented solvers participated in international SAT competitions. In 2013 and 2014 Pcasso showed a good performance. Riss in combination with Copro- cessor won several first, second and third prices, including two Kurt-Gödel-Medals. Hence, the introduced algorithms improved modern SAT solving technology