Parameterized complexity of completeness reasoning for conjunctive queries

Abstract

Incompleteness management has become a popular research topic and been viewed in many applications in the area of data quality and data management. Traditional methods for handling incompleteness assume data is totally complete or incomplete. However, in practical applications, data is often partial complete, which means that data is not totally complete but some special parts of the data satisfying given semantic specifications are complete. Intuitively, partial complete data can still give complete answers for queries consistent with the semantic specifications. Therefore, it is highly needed to study the fundamental problems for managing partial complete data. However, as far as known by us, there are only few works focusing on this area. The most important and fundamental problem, completeness reasoning, is studied from the aspect of parameterized complexity by this paper. The completeness reasoning problem, TC-QC (Table Completeness to Query Completeness), is first formally defined and studied by Razniewski et al. [1]. Given completeness statements of data, the goal of the TC-QC problem is to determine whether the result of a special query Q is complete, that is to reason query completeness based on given data completeness. Razniewski et al. have shown that the TC-QC problem is NP-hard even for conjunctive queries, and a natural and interesting question is whether or not TC-QC can be solved efficiently by parameterized algorithms. To answer that, the parameterized complexities of completeness reasoning for conjunctive queries are studied by the paper. First, it is shown that, considering the parameterizations defined by the size of query completeness or table completeness, the parameterized TC-QC problem for conjunctive queries is para-NP-complete, which strongly indicate that the TC-QC problems parameterized by the above two parameters do not admit fixed-parameter tractable algorithms. Then, for more special cases parameterized by different constraints on query structures like degree, tree-width and number of variables, the TC-QC problems are still not fixed parameter tractable. Finally, on the positive side, if each data completeness statement has a constant size bound, the parameterized TC-QC problem defined by the query completeness size can be solved by a fixed-parameter tractable algorithm.