A multiparametric view on answer set programming

Disjunctive answer set programming (ASP) is an important framework for declarative modeling and problem solving, where the computational complexity of basic decision problems like consistency (deciding whether a program has an answer set) is located on the second level of the polynomial hierarchy. During the last decades different approaches have been applied to find tractable fragments of programs, in particular, also using parameterized complexity. However, the full potential of parameterized complexity has not been unlocked since only one or very few parameters have been considered at once. In this paper, we consider several natural parameters for the consistency problem of disjunctive ASP. In addition, we also take the sizes of the answer sets into account; a restriction that is particularly interesting for applications requiring small solutions as encoding subset minimization problems in ASP can be done directly due to inherent minimization in its semantics. Previous work on parameterizing the consistency problem by the size of answer sets yielded mostly negative results. In contrast, we start from recent findings for the problem WMMSAT and show several novel fixed-parameter tractability (fpt) results based on combinations of parameters. Moreover, we establish a variety of hardness results (paraNP, W[2], and W[1]-hardness) to assess tightness of our parameter combinations.