The generation of SAT instances is an important issue in computer science, and it is useful for researchers to verify the effectiveness of SAT solvers. Addressing this issue could inspire researchers to propose new search strategies. SAT problems exist in various real-world applications, some of which have more than one solution. However, although several algorithms for generating random SAT instances have been proposed, few can be used to generate hard instances that have multiple predefined solutions. In this paper, we propose the KHidden-M algorithm to generate SAT instances with multiple predefined solutions that could be hard to solve by the local search strategy when the number of predefined solutions is small enough and the Hamming distance between them is not less than half of the solution length. Specifically, first, we generate an SAT instance that is satisfied by all of the predefined solutions. Next, if the generated SAT instance does not satisfy the hardness condition, then a strategy will be conducted to adjust clauses through multiple iterations to improve the hardness of the whole instance. We propose three strategies to generate the SAT instance in the first part. The first strategy is called the random strategy, which randomly generates clauses that are satisfied by all of the predefined solutions. The other two strategies are called the estimating strategy and greedy strategy, and using them, we attempt to generate an instance that directly satisfies or is closer to the hardness condition for the local search strategy. We employ two SAT solvers (i.e., WalkSAT and Kissat) to investigate the hardness of the SAT instances generated by our algorithm in the experiments. The experimental results show the effectiveness of the random, estimating and greedy strategies. Compared to the state-of-the-art algorithm for generating SAT instances with predefined solutions, namely, M-hidden, our algorithm could be more effective in generating hard SAT instances.

It is crucial to generate crafted SAT formulas with predefined solutions for the testing and development of SAT solvers since many SAT formulas from real-world applications have solutions. Although some generating algorithms have been proposed to generate SAT formulas with predefined solutions, community structures of SAT formulas are not considered. We propose a 3-SAT formula generating algorithm that not only guarantees the existence of a predefined solution, but also simultaneously considers community structures and clause distributions. The proposed 3-SAT formula generating algorithm controls the quality of community structures through controlling (1) the number of clauses whose variables have a common community, which we call intra-community clauses, and (2) the number of variables that only belong to one community, which we call intra-community variables. To study the combined effect of community structures and clause distributions on the hardness of SAT formulas, we measure solving runtimes of two solvers, gluHack (a leading CDCL solver) and CPSparrow (a leading SLS solver), on the generated SAT formulas under different groups of parameter settings. Through extensive experiments, we obtain some noteworthy observations on the SAT formulas generated by the proposed algorithm: (1) The community structure has little or no effects on the hardness of SAT formulas with regard to CPSparrow but a strong effect with regard to gluHack. (2) Only when the proportion of true literals in a SAT formula in terms of the predefined solution is 0.5, SAT formulas are hard-to-solve with regard to gluHack; when this proportion is below 0.5, SAT formulas are hard-to-solve with regard to CPSparrow. (3) When the ratio of the number of clauses to that of variables is around 4.25, the SAT formulas are hard-to-solve with regard to both gluHack and CPSparrow.