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.

翻译：我们研究布尔合取查询及其在半环上的推广——求和-乘积问题的细粒度复杂度。针对这些问题，我们提出了一种通用的半环无关归约方法，可将k-团问题归约至任意查询结构（超图）。该归约使用了马克思首次引入的图嵌入超图概念。由此归约可推出多类超图的严格条件下界，包括循环图、Loomis-Whitney连接图、某些二部图以及弦图。这些下界依赖于我们称为超图H的团嵌入幂次的概念，该量本身具有独立研究价值。我们证明团嵌入幂次始终小于超图的子模宽度，并给出其可判定计算算法。文末提出若干开放问题供后续研究。