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~\cite{Marx13}. 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：我们研究了评估布尔合取查询及其在半环上的和积问题推广形式的细粒度复杂性。针对这些问题，我们提出了一种通用的半环无关归约方法，将k-团问题归约到任意查询结构（超图）。该归约使用了Marx~\cite{Marx13}首次提出的图到超图的嵌入概念。作为归约的结果，我们为多类超图（包括循环、Loomis-Whitney连接、某些二分图和弦图）建立了严格的条件下限。这些下界依赖于我们称之为超图H的团嵌入能力这一度量——我们相信该度量本身具有独立的研究价值。我们证明了团嵌入能力总是小于超图的子模宽度，并给出了计算它的可判定算法。最后，我们提出了许多尚待未来研究的开放性问题。