Runtime Distributions and Criteria for Restarts

Randomized algorithms sometimes employ a restart strategy. After a certain number of steps, the current computation is aborted and restarted with a new, independent random seed. In some cases, this results in an improved overall expected runtime. This work introduces properties of the underlying runtime distribution which determine whether restarts are advantageous. The most commonly used probability distributions admit the use of a scale and a location parameter. Location parameters shift the density function to the right, while scale parameters affect the spread of the distribution. It is shown that for all distributions scale parameters do not influence the usefulness of restarts and that location parameters only have a limited influence. This result simplifies the analysis of the usefulness of restarts. The most important runtime probability distributions are the log-normal, the Weibull, and the Pareto distribution. In this work, these distributions are analyzed for the usefulness of restarts. Secondly, a condition for the optimal restart time (if it exists) is provided. The log-normal, the Weibull, and the generalized Pareto distribution are analyzed in this respect. Moreover, it is shown that the optimal restart time is also not influenced by scale parameters and that the influence of location parameters is only linear

The Potential of Restarts for ProbSAT

This work analyses the potential of restarts for probSAT, a quite successful algorithm for k-SAT, by estimating its runtime distributions on random 3-SAT instances that are close to the phase transition. We estimate an optimal restart time from empirical data, reaching a potential speedup factor of 1.39. Calculating restart times from fitted probability distributions reduces this factor to a maximum of 1.30. A spin-off result is that the Weibull distribution approximates the runtime distribution for over 93% of the used instances well. A machine learning pipeline is presented to compute a restart time for a fixed-cutoff strategy to exploit this potential. The main components of the pipeline are a random forest for determining the distribution type and a neural network for the distribution's parameters. ProbSAT performs statistically significantly better than Luby's restart strategy and the policy without restarts when using the presented approach. The structure is particularly advantageous on hard problems.Comment: Eurocast 201

Efficient satisfiability solver

The past few decades saw great improvements in the performance of satisfiability (SAT) solvers. In this thesis, we discuss the state-of-the-art techniques used in building an efficient SAT solver. Modern SAT solvers are mainly constituted by the following components: decision heuristics, Boolean constraint propagation, conflict analysis, restart, clause deletion and preprocessing. Various algorithms and implementations in each component will be discussed and analyzed. Then we propose a new backtracking strategy, partial backtracking, which can be easily implemented in SAT solvers. It is essentially an extension of the backtracking strategy used in most SAT solvers. With partial backtracking, the solver consecutively amends the variable assignments instead of discarding them completely so that it does not backtrack as many levels as the classic strategy does after analyzing a conflict. We implemented this strategy in our solver Nigma and the experiments show that the solver benefits from this adjustment

Using Restarts in Constraint Programming over Finite Domains - An Experimental Evaluation

The use of restart techniques in complete Satisfiability (SAT) algorithms has made solving hard real world instances possible. Without restarts such algorithms could not solve those instances, in practice. State of the art algorithms for SAT use restart techniques, conflict clause recording (nogoods), heuristics based on activity variable in conflict clauses, among others. Algorithms for SAT and Constraint problems share many techniques; however, the use of restart techniques in constraint programming with finite domains (CP(FD)) is not widely used as it is in SAT. We believe that the use of restarts in CP(FD) algorithms could also be the key to efficiently solve hard combinatorial problems. In this PhD thesis we study restarts and associated techniques in CP(FD) solvers. In particular, we propose to including in a CP(FD) solver restarts, nogoods and heuristics based in nogoods as this should improve search algorithms, and, consequently, efficiently solve hard combinatorial problems. We thus intend to: a) implement restart techniques (successfully used in SAT) to solve constraint problems with finite domains; b) implement nogoods (learning) and heuristics based on nogoods, already in use in SAT and associated with restarts; and c) evaluate the use of restarts and the interplay with the other implemented techniques. We have conducted the study in the context of domain splitting backtrack search algorithms with restarts. We have defined domain splitting nogoods that are extracted from the last branch of the search algorithm before the restart. And, inspired by SAT solvers, we were able to use information within those nogoods to successfully help the variable selection heuristics. A frequent restart strategy is also necessary, since our approach learns from restarts

Scavenger 0.1: A Theorem Prover Based on Conflict Resolution

This paper introduces Scavenger, the first theorem prover for pure first-order logic without equality based on the new conflict resolution calculus. Conflict resolution has a restricted resolution inference rule that resembles (a first-order generalization of) unit propagation as well as a rule for assuming decision literals and a rule for deriving new clauses by (a first-order generalization of) conflict-driven clause learning.Comment: Published at CADE 201

A Duality-Aware Calculus for Quantified Boolean Formulas

Wir präsentieren ein formales Rahmenwerk, das es ermöglicht das Verhalten von QBF-Beweisen zu beschreiben.Learning and backjumping are essential features in search-based decision procedures for Quantified Boolean Formulas (QBF). To obtain a better understanding of such procedures, we present a formal framework, which allows to simultaneously reason on prenex conjunctive and disjunctive normal form. It captures both satisfying and falsifying search states in a symmetric way. This symmetry simplifies the framework and offers potential for further variants.W1255-N23S11408-N23(VLID)193237