Horst Samulowitz

Algorithm selection and scheduling

Algorithm portfolios aim to increase the robustness of our ability to solve problems efficiently. While recently proposed algorithm selection methods come ever closer to identifying the most appropriate solver given an input instance, they are bound to make wrong and, at times, costly decisions. Solver scheduling has been proposed to boost the performance of algorithm selection. Scheduling tries to allocate time slots to the given solvers in a portfolio so as to maximize, say, the number of solved instances within a given time limit. We show how to solve the corresponding optimization problem at a low computational cost using column generation, resulting in fast and high quality solutions. We integrate this approach with a recently introduced algorithm selector, which we also extend using other techniques. We propose various static as well as dynamic scheduling strategies, and demonstrate that in comparison to pure algorithm selection, our novel combination of scheduling and solver selection can significantly boost performance.

Preprocessing QBF

In this paper we investigate the use of preprocessing when solving Quantified Boolean Formulas (QBF). Many different problems can be efficiently encoded as QBF instances, and there has been a great deal of recent interest and progress in solving such instances efficiently. Ideas from QBF have also started to migrate to CSP with the exploration of Quantified CSPs which offer an intriguing increase in representational power over traditional CSPs. Here we show that QBF instances can be simplified using techniques related to those used for preprocessing SAT. These simplifications can be performed in polynomial time, and are used to preprocess the instance prior to invoking a worst case exponential algorithm to solve it. We develop a method for preprocessing QBF instances that is empirically very effective. That is, the preprocessed formulas can be solved significantly faster, even when we account for the time required to perform the preprocessing. Our method significantly improves the efficiency of a range of state-of-the-art QBF solvers. Furthermore, our method is able to completely solve some instances just by preprocessing, including some instances that to our knowledge have never been solved before by any QBF solver.

Search combinators

The ability to model search in a constraint solver can be an essential asset for solving combinatorial problems. However, existing infrastructure for defining search heuristics is often inadequate. Either modeling capabilities are extremely limited or users are faced with a general-purpose programming language whose features are not tailored towards writing search heuristics. As a result, major improvements in performance may remain unexplored. This article introduces search combinators, a lightweight and solver-independent method that bridges the gap between a conceptually simple modeling language for search (high-level, functional and naturally compositional) and an efficient implementation (low-level, imperative and highly non-modular). By allowing the user to define application-tailored search strategies from a small set of primitives, search combinators effectively provide a rich domain-specific language (DSL) for modeling search to the user. Remarkably, this DSL comes at a low implementation cost to the developer of a constraint solver. The article discusses two modular implementation approaches and shows, by empirical evaluation, that search combinators can be implemented without overhead compared to a native, direct implementation in a constraint solver.

Parallel SAT solver selection and scheduling

Combining differing solution approaches by means of solver portfolios has proven as a highly effective technique for boosting solver performance. We consider the problem of generating parallel SAT solver portfolios. Our approach is based on a recently introduced sequential SAT solver portfolio that excelled at the last SAT competition. We show how the approach can be generalized for the parallel case, and how obstacles like parallel SAT solvers and symmetries induced by identical processors can be overcome. We compare different ways of computing parallel solver portfolios with the best performing parallel SAT approaches to date. Extensive experimental results show that the developed methodology very significantly improves our current parallel SAT solving capabilities.

Guiding combinatorial optimization with UCT

We propose a new approach for search tree exploration in the context of combinatorial optimization, specifically Mixed Integer Programming (MIP), that is based on UCT, an algorithm for the multi-armed bandit problem designed for balancing exploration and exploitation in an online fashion. UCT has recently been highly successful in game tree search. We discuss the differences that arise when UCT is applied to search trees as opposed to bandits or game trees, and provide initial results demonstrating that the performance of even a highly optimized state-of-the-art MIP solver such as CPLEX can be boosted using UCTs guidance on a range of problem instances.

Non-model-based algorithm portfolios for SAT

When tackling a computationally challenging combinatorial problem, one often observes that some solution approaches work well on some instances, while other approaches work better on other instances. This observation has given rise to the idea of building algorithm portfolios [5]. Leyton-Brown et al. [1], for instance, proposed to select one of the algorithms in the portfolio based on some features of the instance to be solved. This approach has been blessed with tremendous success in the past. Especially in SAT, the SATzilla portfolios [7] have performed extremely well in past SAT Competitions [6].

Resolution and parallelizability: barriers to the efficient parallelization of SAT solvers

Recent attempts to create versions of Satisfiability (SAT) solvers that exploit parallel hardware and information sharing have met with limited success. In fact, the most successful parallel solvers in recent competitions were based on portfolio approaches with little to no exchange of information between processors. This experience contradicts the apparent parallelizability of exploring a combinatorial search space. We present evidence that this discrepancy can be explained by studying SAT solvers through a proof complexity lens, as resolution refutation engines. Starting with the observation that a recently studied measure of resolution proofs, namely depth, provides a (weak) upper bound to the best possible speedup achievable by such solvers, we empirically show the existence of bottlenecks to parallelizability that resolution proofs typically generated by SAT solvers exhibit. Further, we propose a new measure of parallelizability based on the best-case makespan of an offline resource constrained scheduling problem. This measure explicitly accounts for a bounded number of parallel processors and appears to empirically correlate with parallel speedups observed in practice. Our findings suggest that efficient parallelization of SAT solvers is not simply a matter of designing the right clause sharing heuristics; even in the best case, it can be -- and indeed is -- hindered by the structure of the resolution proofs current SAT solvers typically produce.

On the stochastic constraint satisfaction framework

Stochastic constraint satisfaction is a framework that allows to make decisions taking into account possible futures. We study two challenging aspects of this framework: (1) variables in stochastic CSP are ordered sequentially, which is adequate for the representation of a number of problems, but is not a natural choice for the modeling of problems in which the future can follow different branches (2) the framework was designed to allow multi-objective decision-making, yet this issue has been treated only superficially in the literature. We bring a number of clarifications to these two aspects. In particular, we show how minor modifications allow the framework to deal with non-sequential forms, we identify a number of technicalities related to the use of the sequential ordering of variables and of the use of multiple objectives, and in addition we propose the first search algorithm that solves multi-objective stochastic problems in polynomial space.

Boosting Sequential Solver Portfolios: Knowledge Sharing and Accuracy Prediction

Sequential algorithm portfolios for satisfiability testing SAT, such as SATzilla and 3S , have enjoyed much success in the last decade. By leveraging the differing strengths of individual SAT solvers, portfolios employing older solvers have often fared as well or better than newly designed ones, in several categories of the annual SAT Competitions and Races. We propose two simple yet powerful techniques to further boost the performance of sequential portfolios, namely, a generic way of knowledge sharing suitable for sequential SAT solver schedules which is commonly employed in parallel SAT solvers, and a meta-level guardian classifier for judging whether to switch the main solver suggested by the portfolio with a recourse action solver. With these additions, we show that the performance of the sequential portfolio solver 3S , which dominated other sequential categories but was ranked 10th in the application category of the 2011 SAT Competition, can be boosted significantly, bringing it just one instance short of matching the performance of the winning application track solver, while still outperforming all other solvers submitted to the crafted and random categories.

Parallel Combinatorial Optimization with Decision Diagrams

We propose a new approach for parallelizing search for combinatorial optimization that is based on a recursive application of approximate Decision Diagrams. This generic scheme can, in principle, be applied to any combinatorial optimization problem for which a decision diagram representation is available. We consider the maximum independent set problem as a specific case study, and show how a recently proposed sequential branch-and-bound scheme based on approximate decision diagrams can be parallelized efficiently using the X10 parallel programming and execution framework. Experimental results using our parallel solver, DDX10, running on up to 256 compute cores spread across a cluster of machines indicate that parallel decision diagrams scale effectively and consistently. Moreover, on graphs of relatively high density, parallel decision diagrams often outperform state-of-the-art parallel integer programming when both use a single 32-core machine.