This paper reports on the Second International Nurse Rostering Competition (INRC-II). Its contributions are (1) a new problem formulation which, differently from INRC-I, is a multi-stage procedure, (2) a competition environment that, as in INRC-I, will continue to serve as a growing testbed for search approaches to the INRC-II problem, and (3) final results of the competition. We discuss also the competition environment, which is an infrastructure including problem and instance definitions, testbeds, validation/simulation tools and rules. The hardness of the competition instances has been evaluated through the behaviour of our own solvers, and confirmed by the solvers of the participants. Finally, we discuss general issues about both nurse rostering problems and optimisation competitions in general.

Pseudo-Boolean (PB) constraints often have a critical role in constraint satisfaction and optimisation problems. Encoding PB constraints to SAT has proven to be an efficient approach in many applications, however care must be taken to encode them compactly and with good propagation properties. It has been shown that at-most-one (AMO) and exactly-one (EO) relations over subsets of the variables can be exploited in various encodings of PB constraints, improving their compactness and solving performance. In this paper we detect AMO and EO relations completely automatically and exploit them to improve SAT encodings that are based on Multi-Valued Decision Diagrams (MDDs). Our experiments show substantial reductions in encoding size and dramatic improvements in solving time thanks to automatic AMO and EO detection.

Access to good benchmark instances is always desirable when developing new algorithms, new constraint models, or when comparing existing ones. Handwritten instances are of limited utility and are timeconsuming to produce. A common method for generating instances is constructing special purpose programs for each class of problems. This can be better than manually producing instances, but developing such instance generators also has drawbacks. In this paper, we present a method for generating graded instances completely automatically starting from a class-level problem specification. A graded instance in our present setting is one which is neither too easy nor too difficult for a given solver. We start from an abstract problem specification written in the Essence language and provide a system to transform the problem specification, via automated type-specific rewriting rules, into a new abstract specification which we call a generator specification. The generator specification is itself parameterised by a number of integer parameters; these are used to characterise a certain region of the parameter space. The solutions of each such generator instance form valid problem instances. We use the parameter tuner irace to explore the space of possible generator parameters, aiming to find parameter values that yield graded instances. We perform an empirical evaluation of our system for five problem classes from CSPlib, demonstrating promising results.

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