Jaguar: A Primal Algorithm for Conjunctive Query Evaluation in Submodular-Width Time
The submodular width is a complexity measure of conjunctive queries (CQs), which assigns a nonnegative real number, subw(Q), to each CQ Q. An existing algorithm, called PAND, performs CQ evaluation in polynomial time where the exponent is essentially subw(Q). Formally, for every Boolean CQ Q, PANDA evaluates Q in time $O(N^{\mathsf{subw}(Q)} \cdot \mathsf{polylog}(N))$, where N denotes the input size; moreover, there is complexity-theoretic evidence that, for a number of Boolean CQs, no exponent strictly below subw(Q) can be achieved by combinatorial algorithms. On a high level, the submodular width of a CQ Q can be described as the maximum over all polymatroids, which are set functions on the variables of Q that satisfy Shannon inequalities. The PANDA algorithm in a sense works in the dual space of this maximization problem, makes use of information theory, and transforms a CQ into a set of disjunctive datalog programs which are individually solved. In this article, we introduce a new algorithm for CQ evaluation which achieves, for each Boolean CQ Q and for all epsilon > 0, a running time of $O(N^{\mathsf{subw}(Q)+\epsilon})$. This new algorithm's description and analysis are, in our view, significantly simpler than those of PANDA. We refer to it as a "primal" algorithm as it operates in the primal space of the described maximization problem, by maintaining a feasible primal solution, namely, a polymatroid. Indeed, this algorithm deals directly with the input CQ and adaptively computes a sequence of joins, in a guided fashion, so that the cost of these join computations is bounded. Additionally, this algorithm can achieve the stated runtime for the generalization of the submodular width incorporating degree constraints. We dub our algorithm Jaguar, as it is a join-adaptive guided algorithm.