It has been widely observed that there is no single “dominant ” SAT solver; instead, different solvers perform best on different instances. Rather than following the traditional approach of choosing the best solver for a given class of instances, we advocate making this decision online on a per-instance basis. Building on previous work, we describe SATzilla, an automated approach for constructing per-instance algorithm portfolios for SAT that use so-called empirical hardness models to choose among their constituent solvers. This approach takes as input a distribution of problem instances and a set of component solvers, and constructs a portfolio optimizing a given objective function (such as mean runtime, percent of instances solved, or score in a competition). The excellent performance of our SATzilla portfolios has been independently verified in the 2007 SAT Competition, where our SATzilla-07 solvers won three gold, one silver and one bronze medal. In this article, we go well beyond SATzilla-07 by making the portfolio construction scalable and completely automated, and improving it by integrating local search solvers as candidate solvers, by predicting performance score instead of runtime, and by using hierarchical hardness models that take into account different types of SAT instances. We demonstrate the effectiveness of these new techniques in extensive experimental results on data sets including instances from the most recent SAT competition. 1.

Abstract. Machine learning can be used to build models that predict the runtime of search algorithms for hard combinatorial problems. Such empirical hardness models have previously been studied for complete, deterministic search algorithms. In this work, we demonstrate that such models can also make surprisingly accurate predictions of the run-time distributions of incomplete and randomized search methods, such as stochastic local search algorithms. We also show for the first time how information about an algorithm’s parameter settings can be incorporated into a model, and how such models can be used to automatically adjust the algorithm’s parameters on a per-instance basis in order to optimize its performance. Empirical results for Novelty + and SAPS on structured and unstructured SAT instances show very good predictive performance and significant speedups of our automatically determined parameter settings when compared to the default and best fixed distribution-specific parameter settings. 1

The determination of appropriate values for free algorithm parameters is a challenging and tedious task in the design of effective algorithms for hard problems. Such parameters include categorical choices (e.g., neighborhood structure in local search or variable/value ordering heuristics in tree search), as well as numerical parameters (e.g., noise or restart timing). In practice, tuning of these parameters is largely carried out manually by applying rules of thumb and crude heuristics, while more principled approaches are only rarely used. In this paper, we present a local search approach for algorithm configuration and prove its convergence to the globally optimal parameter configuration. Our approach is very versatile: it can, e.g., be used for minimising run-time in decision problems or for maximising solution quality in optimisation problems. It further applies to arbitrary algorithms, including heuristic tree search and local search algorithms, with no limitation on the number of parameters. Experiments in four algorithm configuration scenarios demonstrate that our automatically determined parameter settings always outperform the algorithm defaults, sometimes by several orders of magnitude. Our approach also shows better performance and greater flexibility than the recent CALIBRA system. Our ParamILS code, along with instructions on how to use it for tuning your own algorithms, is available on-line at

Abstract. In this paper we introduce UBCSAT, a new implementation and experimentation environment for Stochastic Local Search (SLS) algorithms for SAT and MAX-SAT. Based on a novel triggered procedure architecture, UBCSAT provides implementations of numerous well-known and widely used SLS algorithms for SAT and MAX-SAT, including GSAT, WalkSAT, and SAPS; these implementations generally match or exceed the efficiency of the respective original reference implementations. Through numerous reporting and statistical features, including the measurement of run-time distributions, UBCSAT facilitates the advanced empirical analysis of these algorithms. New algorithm variants, SLS algorithms, and reporting features can be added to UBCSAT in a straightforward and efficient way. UBCSAT is implemented in C and runs on numerous platforms and operating systems; it is publicly and freely available at

Propositional Satisfiability (SAT) and Constraint Programming (CP) have developed as two relatively independent threads of research, cross-fertilising occasionally. These two approaches to problem solving have a lot in common, as evidenced by similar ideas underlying the branch and prune algorithms that are most successful at solving both kinds of problems. They also exhibit differences in the way they are used to state and solve problems, since SAT’s approach is in general a black-box approach, while CP aims at being tunable and programmable. This survey overviews the two areas in a comparative way, emphasising the similarities and differences between the two and the points where we feel that one technology can benefit from ideas or experience acquired

Designing high-performance algorithms for computationally hard problems is a difficult and often time-consuming task. In this work, we demonstrate that this task can be automated in the context of stochastic local search (SLS) solvers for the propositional satisfiability problem (SAT). We first introduce a generalised, highly parameterised solver framework, dubbed SATenstein, that includes components gleaned from or inspired by existing high-performance SLS algorithms for SAT. The parameters of SATenstein control the selection of components used in any specific instantiation and the behaviour of these components. SATenstein can be configured to instantiate a broad range of existing high-performance SLSbased SAT solvers, and also billions of novel algorithms. We used an automated algorithm configuration procedure to find instantiations of SATenstein that perform well on several well-known, challenging distributions of SAT instances. Overall, we consistently obtained significant improvements over the previously best-performing SLS algorithms, despite expending minimal manual effort. 1 1

Algorithms based on following local gradient information are surprisingly effective for certain classes of constraint satisfaction problems. Unfortunately, previous local search algorithms are notoriously incomplete: They are not guaranteed to find a feasible solution if one exists and they cannot be used to determine unsatisfiability. We present an algorithmic framework for complete local search and discuss in detail an instantiation for the propositional satisfiability problem (SAT). The fundamental idea is to use constraint learning in combination with a novel objective function that converges during search to a surface without local minima. Although the algorithm has worst-case exponential space complexity, we present empirical results on challenging SAT competition benchmarks that suggest that our implementation can perform as well as state-of-the-art solvers based on more mature techniques. Our framework suggests a range of possible algorithms lying between tree-based search and local search.

This paper examines the relative performance of additive and multiplicative clause weighting schemes for propositional satisfiability testing. Starting with one of the most recently developed multiplicative algorithms (SAPS), an experimental study was constructed to isolate the effects of multiplicative in comparison to additive weighting, while controlling other key features of the two approaches, namely the use of random versus flat moves, deterministic versus probabilistic weight smoothing and multiple versus single inclusion of literals in the local search neighborhood. As a result

Recent dynamic local search (DLS) algorithms such as SAPS are amongst the state-of-the-art methods for solving the propositional satisfiability problem (SAT). DLS algorithms modify the search landscape during the search process by means of dynamically changing clause penalties. In this work, we study whether the resulting, ‘warped ’ landscapes are easier to search than the landscapes that correspond to the original problem instances. We present empirical evidence indicating that (somewhat contrary to common belief) this is not the case, and that the main benefit of the dynamic penalty update mechanism in SAPS is an effective diversification of the search process. In most other high-performance stochastic local search algorithms, the same effect is achieved by the strong use of randomised decisions throughout the search. We demonstrate that in SAPS, random decisions are only required in the (standard) search initialisation procedure, and can be completely eliminated from the remainder of the subsequent search process without any significant change in the behaviour or performance of the resulting algorithms compared to the original, fully randomised SAPS algorithm. We conjecture that the reason for this unexpected result lies in the ability of the deterministic variants of the scaling and smoothing mechanism and the subsidiary iterative best improvement mechanism underlying SAPS to effectively propagate the effects of initial randomisation throughout a search process that shows the sensitive dependence on inditial conditions that is characteristic for chaotic processes. 1

1 Introduction: the role of the user in heuristics Most state-of-the-art heuristics are characterized by a certain number of choices and free parameters, whose appropriate setting is a subject that raises issues of research methodology [5, 41, 51]. In some cases, these parameters are tuned through a feedback loop that includes the user as a crucial learning component: depending on preliminary algorithm tests some parameter values are changed by the