Boolean Satisfiability (SAT) has seen many successful applications in various fields such as Electronic Design Automation and Artificial Intelligence. However, in some cases, it may be required/preferable to use variations of the general SAT problem. In this paper, we consider one important variation, the Partial MAX-SAT problem. Unlike SAT, Partial MAX-SAT has certain constraints (clauses) that are marked as relaxable and the rest are hard, i.e. non-relaxable. The objective is to find a variable assignment that satisfies all non-relaxable clauses together with the maximum number of relaxable ones. We have implemented two solvers for the Partial MAX-SAT problem using a contemporary SAT solver, zChaff. The first approach is a novel diagnosis based algorithm; it iteratively analyzes the UNSAT core of the current SAT instance and eliminates the core through a modification of the problem instance by adding relaxation variables. The second approach is encoding based; it constructs an efficient auxiliary counter that constrains the number of relaxed clauses and supports binary search or linear scan for the optimal solution. Both solvers are complete as they guarantee the optimality of the solution. We discuss the relative strengths and thus applicability of the two solvers for different solution scenarios. Further, we show how both techniques benefit from the persistent learning techniques of incremental SAT. Experiments using practical instances of this problem show significant improvements over the best known solvers.

On solving the partial MAX-SAT problem

The crossing number is a popular tool in graph drawing and visualization, but there is not really just one crossing number; there is a large family of crossing number notions of which the crossing number is the best known. We survey the rich variety of crossing number variants that have been introduced in the literature for purposes that range from studying the theoretical underpinnings of the crossing number to crossing minimization for visualization problems.

The Graph Crossing Number and its Variants: A Survey

Combinatorial Mathematics: THE PRINCIPLE OF INCLUSION AND EXCLUSION

Learnt clauses in CDCL SAT solvers often contain redundant literals. This may have a negative impact on performance because redundant literals may deteriorate both the effectiveness of Boolean constraint propagation and the quality of subsequent learnt clauses. To overcome this drawback, we define a new inprocessing SAT approach which eliminates redundant literals from learnt clauses by applying Boolean constraint propagation. Learnt clause minimization is activated before the SAT solver triggers some selected restarts, and affects only some learnt clauses during the search process. Moreover, we conducted an empirical evaluation on instances coming from the hard combinatorial and application categories of recent SAT competitions. The results show that a remarkable number of additional instances are solved when the approach is incorporated into five of the best performing CDCL SAT solvers (Glucose, TC_Glucose, COMiniSatPS, MapleCOMSPS and MapleCOMSPS_LRB).

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An effective learnt clause minimization approach for CDCL SAT solvers

We study Hanani-Tutte style theorems for various notions of planarity, including partially embedded planarity, and simultaneous planarity. This approach brings together the combinatorial, computational and algebraic aspects of planarity notions and may serve as a uniform foundation for planarity, as suggested in the writings of Tutte and Wu.

/pdf/toward-a-theory-of-planarity-hanani-tutte-and-planarity-34pv9pl9j2.pdf

Toward a theory of planarity: hanani-tutte and planarity variants

XSAT and NAE-SAT of linear CNF classes

Generalized k-ary tanglegrams on level graphs: A satisfiability-based approach and its evaluation

Applying pre- and inprocessing techniques to simplify CNF formulas both before and during search can considerably improve the performance of modern SAT solvers. These algorithms mostly aim at reducing the number of clauses, literals, and variables in the formula. However, to be worthwhile, it is necessary that their additional runtime does not exceed the runtime saved during the subsequent SAT solver execution. In this paper we investigate the efficiency and the practicability of selected simplification algorithms for CDCL-based SAT solving. We first analyze them by means of their expected impact on the CNF formula and SAT solving at all. While testing them on real-world and combinatorial SAT instances, we show which techniques and combinations of them yield a desirable speedup and which ones should be avoided.

Effectiveness of pre- and inprocessing for CDCL-based SAT solving.

Given a simple undirected graph $G$, the maximum $k$-club problem is to find a maximum-cardinality subset of nodes inducing a subgraph of diameter at most $k$ in $G$. This NP-hard generalization of clique, originally introduced to model low diameter clusters in social networks, is of interest in network-based data mining and clustering applications. We give two MAX-SAT formulations of the problem and show that two exact methods resulting from our encodings outperform significantly the state-of-the-art exact methods when evaluated both on sparse and dense random graphs as well as on diverse real-life graphs from the literature.

On Solving the Maximum $k$-club Problem.

In this paper, the architecture and evaluation of a new system for recognizing textual entailment (RTE) is presented. It is conceived as an adaptable and modular environ- ment allowing for a high-coverage syntactic and semantic text analysis combined with logical inference. For the syntactic and semantic analysis it combines an HPSG-based deep semantic analysis with a shallow one supported by statistical models in order to increase the quality and accuracy of results. For recognizing textual entailment we use logical inference of first- order employing model-theoretic techniques and automated reasoning tools. The inference is supported with problem- relevant background knowledge extracted automatically and on demand from external sources like, e.g., WordNet, YAGO, and OpenCyc, or other, experimental sources with, e.g., manually defined presupposition resolutions, or with general and com- mon sense knowledge. The system comes with a graphical user interface for control and presentation purposes. The evaluation shows that the success rate of the presented RTE system is comparable with that of the best logic-based approaches.

Andreas Wotzlaw

Papers

XSAT and NAE-SAT of linear CNF classes

Generalized k-ary tanglegrams on level graphs: A satisfiability-based approach and its evaluation

Effectiveness of pre- and inprocessing for CDCL-based SAT solving.

On Solving the Maximum $k$-club Problem.

Recognizing Textual Entailment with Deep-Shallow Semantic Analysis and Logical Inference