Towards a Dichotomy for Minimally Factorizing the Provenance of Self-Join Free Conjunctive Queries

We consider the problem of finding the minimal-size factorization of the provenance of self-join-free conjunctive queries, i.e., we want to find a formula that \emph{minimizes the number of variable repetitions}. This problem is equivalent to solving the fundamental Boolean formula factorization problem for the restricted setting of the provenance formulas of self-join free queries. While general Boolean formula minimization is $\Sigma^p_2$-complete, we show that the problem is NP-C in our case. Additionally, we identify a large category of queries that can be solved in PTIME, expanding beyond the previously known tractable cases of read-once formulas and hierarchical queries. We describe connections between factorizations, variable elimination orders and minimal query plans. We leverage these insights to create an Integer Linear Program (ILP) that can solve the minimal factorization problem exactly. We also propose a Max-Flow Min-Cut (MFMC) based algorithm that gives an efficient approximate solution. Importantly, we show that both the Linear Programming (LP) relaxation of our proposed ILP, and our MFMC based algorithm are \emph{always correct for all known PTIME cases}. Thus, we present two unified algorithms that can recover all known PTIME cases in PTIME, yet also solve NP-C cases exactly or approximately as desired.