How Hard Is Completeness Reasoning for Conjunctive Queries?

Abstract

Incomplete data has been wildly viewed in many real applications. In practical, data is often partial complete, it means that the whole part of data is incomplete but there exist special complete parts of data which can still support answering related queries. However, as far as we know, there are only few works focusing on managing partial complete data. Therefore, efficient methods for representing partial complete data and deciding which queries can be answered over the complete parts are seriously needed. The completeness reasoning problem, TC-QC (Table Completeness to Query Completeness), has been recognized as an important fundamental problem in managing partial complete data. 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. Previous works show 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. The paper investigates the parameterized complexity of completeness reasoning for conjunctive queries. We show that, if the query completeness size or the table completeness size is considered as a parameter, then the parameterized TC-QC problem for conjunctive queries is para-NP-complete. These results strongly indicate that no fixed-parameter tractable algorithms exist for the TC-QC problems parameterized by the above two parameters. Thus, more special cases of TC-QC defined by different constraints are studied. It is shown that, when the constraints about query structures like degree, tree-width and number of variables are considered, the parameterized TC-QC problems are still intractable. On the positive side, we show that, 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.