Many popular algorithms that work with Boolean functions are dramatically dependent on the order of variables in input representations of Boolean functions. Such algorithms include satisfiability (SAT) solvers that are critical in formal verification and Binary Decision Diagrams (BDDs) manipulation algorithms that are increasingly popular in synthesis and verification. Finding better variable-orderings is a well-recognized problem in each of those contexts. Currently, all leading-edge variable-ordering algorithms are dynamic in the sense that they are invoked many times in the course of the "host" algorithm that solves SAT or manipulates BDDs. Examples include the DLIS ordering for SAT solvers and variable sifting during BDD manipulations. In this work we propose a universal variable ordering MINCE (MIN Cut Etc.) that pre-processes a given Boolean formula in CNF. MINCE is completely independent from target algorithms and outperforms both DLIS for SAT and variable sifting for BDDs. We argue that MINCE tends to capture structural properties of Boolean functions arising from real-world applications.

Abstract: The increasing popularity of SAT and BDD techniques in formal hardware verification and automated synthesis of logic circuits encourages the search for additional speedups. Since typical SAT and BDD algorithms are exponential in the worst-case, the structure of realworld instances is a natural source of improvements. While SAT and BDD techniques are often presented as mutually exclusive alternatives, our work points out that both can be improved via the use of the same structural properties of instances. Our proposed methods are based on efficient problem partitioning and can be easily applied as pre-processing with arbitrary SAT solvers and BDD packages without modifying the source code of SAT/BDD tools. Finding a better variable ordering is a well recognized problem for both SAT solvers and BDD packages. Currently, the best variable-ordering algorithms are dynamic, in the sense that they are invoked many times in the course of the host algorithm that solves SAT or manipulates BDDs. Examples include the DLCS ordering for SAT solvers and variable sifting during BDD manipulations. In this work we propose a universal variable-ordering algorithm MINCE (MIN Cut Etc.) that pre-processes a given Boolean formula in CNF. MINCE is completely independent from target SAT algorithms and in some cases outperforms both the variable state independent

The Pseudo-Boolean Optimization (PBO) and Maximum Satisfiability (MaxSAT) problems are natural optimization extensions of Boolean Satisfiability (SAT). In the recent past, different algorithms have been proposed for PBO and for MaxSAT, despite the existence of straightforward mappings from PBO to MaxSAT and viceversa. This papers proposes Weighted Boolean Optimization (WBO), a new unified framework that aggregates and extends PBO and MaxSAT. In addition, the paper proposes a new unsatisfiability-based algorithm for WBO, based on recent unsatisfiability-based algorithms for MaxSAT. Besides standard MaxSAT, the new algorithm can also be used to solve weighted MaxSAT and PBO, handling pseudo-Boolean constraints either natively or by translation to clausal form. Experimental results illustrate that unsatisfiability-based algorithms for MaxSAT can be orders of magnitude more efficient than existing dedicated algorithms. Finally, the paper illustrates how other algorithms for either PBO or MaxSAT can be extended to WBO. 1

Abstract — For complex optimization problems, several population-based heuristics like Multi-Objective Evolutionary Algorithms have been developed. These algorithms are aiming to deliver sufficiently good solutions in an acceptable time. However, for discrete problems that are restricted by several constraints it is mostly a hard problem to even find a single feasible solution. In these cases, the optimization heuristics typically perform poorly as they mainly focus on searching feasible solutions rather than optimizing the objectives. In this paper, we propose a novel methodology to obtain feasible solutions from constrained discrete problems in populationbased optimization heuristics. At this juncture, the constraints have to be converted into the Propositional Satisfiability Problem (SAT). Obtaining a feasible solution is done by the DPLL algorithm which is the core of most modern SAT solvers. It is shown in detail how this methodology is implemented in Multiobjective Evolutionary Algorithms. The SAT solver is used to obtain feasible solutions from the genetic encoded information on arbitrarily hard solvable problems where common methods like penalty functions or repair strategies are failing. Handmade test cases are used to compare various configurations of the SAT solver. On an industrial example, the proposed methodology is compared to common strategies which are used to obtain feasible solutions. I.

We investigate the problem of generalizing acceleration techniques as found in recent satisfiability engines for conjunctive normal forms (CNFs) to linear constraint systems over the Booleans. The rationale behind this research is that rewriting the propositional formulae occurring in e.g. bounded model checking (BMC) [5] to CNF requires a blowup in either the formula size (worst-case exponential) or in the number of propositional variables (linear, thus yielding a worst-case exponential blow-up of the search space). We demonstrate that acceleration techniques like observation lists and lazy clause evaluation [14] as well as the more traditional non-chronological backtracking and learning techniques generalize smoothly to Davis-Putnam-like resolution procedures for the very concise propositional logic of linear constraint systems over the Booleans. Despite the more expressive input language, the performance of our prototype implementation comes surprisingly close to that of state-of-the-art CNF-SAT engines like ZCha [14]. First experiments with bounded model-construction problems show that the overhead in the satisfiability engine that can be attributed to the richer input language is often amortized by the conciseness gained in the propositional encoding of the BMC problem.

Recent algorithmic advances in Boolean satisfiability (SAT), along with highly efficient solver implementations, have enabled the successful deployment of SAT technology in a wide range of applications domains, and particularly in electronic design automation (EDA). SAT is increasingly being used as the underlying model for a number of applications in EDA. This paper describes how to formulate two problems in power estimation of CMOS combinational circuits as SAT problems or 0–1 integer linear programming (ILP). In these circuits, it was proven that maximizing dissipation is equivalent to maximizing gate output activity, appropriately weighted to account for differing load capacitances. The first problem in this work deals with identifying an input vector pair that maximizes the weighted circuit activity. In the second application we attempt to find an estimate for the maximum power-up current in circuits where power cut-off or gating techniques are used to reduce leakage current. Both problems were successfully formulated as SAT problems. SAT-Based and generic Integer Linear Programming (ILP) solvers are then used to find a solution. The experimental results obtained on a large number of benchmark circuits provide promising evidence that the proposed complete approach is both viable and useful and outperforms the random approach.

We present and evaluate new techniques for designing algorithm portfolios. In our view, the problem has both a scheduling aspect and a machine learning aspect. Prior work has largely addressed one of the two aspects in isolation. Building on recent work on the scheduling aspect of the problem, we present a technique that addresses both aspects simultaneously and has attractive theoretical guarantees. Experimentally, we show that this technique can be used to improve the performance of state-of-the-art algorithms for Boolean satisfiability, zero-one integer programming, and A.I. planning. 1

Abstract. Many problems are naturally expressed using CNF clauses and boolean cardinality constraints. It is generally believed that solving such problems through pure CNF encoding is inefficient, so many authors has proposed specialized algorithms: the pseudo-boolean solvers. In this paper we show that an appropriate pure CNF encoding can be competitive with these specialized methods. In conjunction with our encoding, we propose a slight modification of the DLL procedure that allows any DLL-based SAT solver to solve boolean cardinality optimization problems. We show experimentally that our encoding allows zchaff to be competitive with pseudo-boolean solvers on some decision and optimization problems. 1

We present a concept to significantly advance the state of the art for bounded model checking (BMC) and inductive verification (IV) of hybrid discrete-continuous systems. Our approach combines the expertise of partners coming from different domains, like hybrid systems modeling and digital circuit verification, bounded planning and heuristic search, combinatorial optimization and integer programming. After sketching the overall verification flow we present first results indicating that the combination and tight integration of di#erent verification engines is a first step to pave the way to fully automated BMC and IV of medium to large-scale networks of hybrid automata.

Abstract. The utilization of cutting planes is a key technique in Integer Linear Programming (ILP). However, cutting planes have seldom been applied in Pseudo-Boolean Optimization (PBO) algorithms derived from the Davis-Logemann-Loveland (DLL) procedure for Propositional Satisfiability (SAT). This paper proposes the utilization of cutting planes in a DLL-style PBO algorithm, which incorporates the most effective techniques for PBO. We propose the utilization of cutting planes both during preprocessing and during the search process. Moreover, we also establish conditions that enable clause learning and non-chronological backtracking in the presence of conflicts involving constraints generated by cutting plane techniques. The experimental results, obtained on a large number of classes of instances, indicate that the integration of cutting planes with backtrack search is an extremely effective technique for PBO. 1