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arXiv · 2304.14557

The Fine-Grained Complexity of Boolean Conjunctive Queries and Sum-Product Problems

Austen Z. Fan·Paraschos Koutris·Hangdong Zhao
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Abstract

We study the fine-grained complexity of evaluating Boolean Conjunctive Queries and their generalization to sum-of-product problems over an arbitrary semiring. For these problems, we present a general semiring-oblivious reduction from the k-clique problem to any query structure (hypergraph). Our reduction uses the notion of embedding a graph to a hypergraph, first introduced by Marx. As a consequence of our reduction, we can show tight conditional lower bounds for many classes of hypergraphs, including cycles, Loomis-Whitney joins, some bipartite graphs, and chordal graphs. These lower bounds have a dependence on what we call the clique embedding power of a hypergraph H, which we believe is a quantity of independent interest. We show that the clique embedding power is always less than the submodular width of the hypergraph, and present a decidable algorithm for computing it. We conclude with many open problems for future research.

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Austen Z. Fan, Paraschos Koutris, Hangdong Zhao. 2023-04-27. The Fine-Grained Complexity of Boolean Conjunctive Queries and Sum-Product Problems. https://arxiv.org/abs/2304.14557

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