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

In this paper, we introduce DLS-MC, a new stochastic local search algorithm for the maximum clique problem. DLS-MC alternates between phases of iterative improvement, during which suitable vertices are added to the current clique, and plateau search, during which vertices of the current clique are swapped with vertices not contained in the current clique. The selection of vertices is solely based on vertex penalties that are dynamically adjusted during the search, and a perturbation mechanism is used to overcome search stagnation. The behaviour of DLS-MC is controlled by a single parameter, penalty delay, which controls the frequency at which vertex penalties are reduced. We show empirically that DLS-MC achieves substantial performance improvements over state-of-the-art algorithms for the maximum clique problem over a large range of the commonly used DIMACS benchmark instances. 1.

Abstract. In recent years, dynamic local search (DLS) clause weighting algorithms have emerged as the local search state-of-the-art for solving propositional satisfiability problems. However, most DLS algorithms require the tuning of domain dependent parameters before their performance becomes competitive. If manual parameter tuning is impractical then various mechanisms have been developed that can automatically adjust a parameter value during the search. To date, the most effective adaptive clause weighting algorithm is RSAPS. However, RSAPS is unable to convincingly outperform the best non-weighting adaptive algorithm AdaptNovelty +, even though manually tuned clause weighting algorithms can routinely outperform the Novelty + heuristic on which AdaptNovelty + is based. In this study we introduce R+DDFW +, an enhanced version of the DDFW clause weighting algorithm developed in 2005, that not only adapts the total amount of weight according to the degree of stagnation in the search, but also incorporates the latest resolution-based preprocessing approach used by the winner of the 2005 SAT competition (R+AdaptNovelty +). In an empirical study we show R+DDFW + improves on DDFW and outperforms the other leading adaptive (R+Adapt-Novelty +, R+RSAPS) and non-adaptive (R+G 2 WSAT) local search solvers over a range of random and structured benchmark problems. 1

Abstract. Recent research has shown that it is often preferable to encode realworld problems as propositional satisfiability (SAT) problems, and then solve using general purpose solvers. In this way the efficiencies of SAT technology can be exploited, and the development of specialised solution techniques can be avoided. However, in the interval algebra (IA) domain of temporal reasoning, the state-of-the-art still involves the use of specialised techniques that exploit the particular structure of interval relations. In this paper we investigate the feasibility of representing and solving IA problems as SAT problems. We firstly introduce two methods of representing IA as a constraint satisfaction problem (CSP), and then use three SAT-encoding schemes to produce six different IA to SAT representations. In an empirical study, we examine the performance of existing SAT local and complete search solvers on these SAT representations, and perform a comparison with solvers that operate on native IA representations. Our results show that the best performance over a range of algorithms is produced using a support SAT encoding of a point algebra-based CSP. The results also show that a state-of-the-art complete SAT solver (zChaff) can solve these instances significantly faster than existing IA solvers working on equivalent native IA representations. 1

Abstract. In recent years, dynamic local search (DLS) clause weighting algorithms have emerged as the local search state-of-the-art for solving propositional satisfiability problems. This paper introduces a new approach to clause weighting, known as Divide and Distribute Fixed Weights (DDFW), that transfers weights from neighbouring satisfied clauses to unsatisfied clauses in order to break out from local minima. Unlike earlier approaches, DDFW continuously redistributes a fixed quantity of weight between clauses, and so does not require a weight smoothing heuristic to control weight growth. It also exploits inherent problem structure by redistributing weights between neighbouring clauses. To evaluate our ideas, we compared DDFW with two of the best reactive local search algorithms, AdaptNovelty+ and RSAPS. In both these algorithms, a problem sensitive parameter is automatically adjusted during the search, whereas DDFW uses a fixed default parameter. Our empirical results show that DDFW has consistently better performance over a range of SAT benchmark problems. This gives a strong indication that neighbourhood weight redistribution strategies could be the key to a next generation of structure exploiting, parameter-free local search SAT solvers. 1

Recent research has focused on bridging the gap between the satisfiability (SAT) and constraint satisfaction problem (CSP) formalisms. One approach has been to develop a many-valued SAT formula (MV-SAT) as an intermediate paradigm between SAT and CSP, and then to translate existing highly efficient SAT solvers to the MV-SAT domain. Experimental results have shown this approach can achieve significant improvements in performance compared with the traditional SAT and CSP approaches. In this paper, we follow a different route, developing SAT solvers that can automatically recognise CSP structure hidden in SAT encodings. This allows us to look more closely at how constraint weighting can be implemented in the SAT and CSP domains. Our experimental results show that a SAT-based approach to handle weights, together with CSP-based approach to variable instantiation, is superior to other combinations of SAT and CSP-based approaches. A further experiment on the round robin scheduling problem indicates that this many-valued constraint weighting approach outperforms other state-of-the-art solvers.

Abstract. We introduce a novel stochastic local search algorithm for the vertex cover problem. Compared to current exhaustive search techniques, our algorithm achieves excellent performance on a suite of problems drawn from the field of biology. We also evaluate our performance on the commonly used DIMACS benchmarks for the related clique problem, finding that our approach is competitive with the current best stochastic local search algorithm for finding cliques. On three very large problem instances, our algorithm establishes new records in solution quality. 1

An incomplete method for solving the propositional satisfiability problem (or a general constraint satisfaction problem) is one that does not provide the guarantee that it will eventually either report a satisfying assignment or declare that the given formula is unsatisfiable. In practice, most such methods are biased towards the satisfiable side: they are typically run with a pre-set resource limit, after which they either produce a valid solution or report failure; they never declare the formula to be unsatisfiable. These are the kind of algorithms we will discuss in this chapter. In complexity theory terms, such algorithms are referred to as having one-sided error. In principle, an incomplete algorithm could instead be biased towards the unsatisfiable side, always providing proofs of unsatisfiability but failing to find solutions to some satisfiable instances, or be incomplete with respect to both satisfiable and unsatisfiable instances (and thus have two-sided error). Unlike systematic solvers often based on an exhaustive branching and backtracking search, incomplete methods are generally based on stochastic local search,

Abstract. Clause weighting local search methods are widely used for satisfiability testing. A feature of particular importance for such methods is the scheme used to maintain the clause weight distribution relevant to different areas of the search landscape. Existing methods periodically adjust clause weights either multiplicatively or additively. Tie breaking strategies are used whenever a method’s evaluation function encounters more than one optimal candidate flip, with the dominant approach being to break such ties randomly. Although this is acceptable for multiplicative methods as they rarely encounter such situations, additive methods encounter significantly more tie breaking scenarios in their landscapes, and therefore a more refined tie breaking strategy is of much greater relevance. This paper proposes a new way of handling the tie breaking situations frequently encountered in the landscapes of additive constraint weighting local search methods. We demonstrate through an empirical study that when this idea is used to modify the purely random tie breaking strategy of a state-of-the-art solver, the modified method significantly outperforms the existing one on a range of benchmarks, especially when we consider the encodings of large and structured problems. Content Areas: Search, Constraint Satisfaction 1

Abstract. Although clause weighting local search algorithms have produced some of the best results on a range of challenging satisfiability (SAT) benchmarks, this performance is dependent on the careful handtuning of sensitive parameters. When such hand-tuning is not possible, clause weighting algorithms are generally outperformed by self-tuning WalkSAT-based algorithms such as AdaptNovelty + and AdaptG 2 WSAT. In this paper we investigate tuning the weight decay parameter of two clause weighting algorithms using the statistical properties of cost distributions that are dynamically accumulated as the search progresses. This method selects a parameter setting both according to the speed of descent in the cost space and according to the shape of the accumulated cost distribution, where we take the shape to be a predictor of future performance. In a wide ranging empirical study we show that this automated approach to parameter tuning can outperform the default settings for two state-of-the-art algorithms that employ clause weighting (PAWS and gNovelty +). We also show that these self-tuning algorithms are competitive with three of the best-known self-tuning SAT local search techniques: RSAPS, AdaptNovelty + and AdaptG 2 WSAT. Key words: Local search, clause weighting, automated parameter tuning, satisfiability.