Zero-One Designs Produce Small Hard SAT Instances

Some basics of combinatorial block design are combined with certain constraint satisfaction problems of interest to the satisfiability community. The paper shows how such combinations lead to satisfiability problems, and shows empirically that these are some of the smallest very hard satisfiability problems ever constructed. Partially balanced (0,1) designs (PB01Ds) are introduced as an extension of balanced incomplete block designs (BIBDs) and (0,1) designs. Also, (0,1) difference sets are introduced as an extension of certain cyclical difference sets. Constructions based on (0,1) difference sets enable generation of PB01Ds over a much wider range of parameters than is possible for BIBDs. Building upon previous work of Spence, it is shown how PB01Ds lead to small, very hard, unsatisfiable formulas. A new three-dimensional form of combinatorial block design is introduced, and leads to small, very hard, satisfiable formulas. The methods are validated on solvers that performed well in the SAT 2009 and earlier competitions.