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

Authors Austen Z. Fan , Paraschos Koutris , Hangdong Zhao

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Austen Z. Fan
  • Department of Computer Sciences, University of Wisconsin-Madison, WI, USA
Paraschos Koutris
  • Department of Computer Sciences, University of Wisconsin-Madison, WI, USA
Hangdong Zhao
  • Department of Computer Sciences, University of Wisconsin-Madison, WI, USA

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Austen Z. Fan, Paraschos Koutris, and Hangdong Zhao. The Fine-Grained Complexity of Boolean Conjunctive Queries and Sum-Product Problems. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 127:1-127:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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 [Dániel Marx, 2013]. 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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Database theory
  • Theory of computation → Parameterized complexity and exact algorithms
  • Fine-grained complexity
  • conjunctive queries
  • semiring-oblivious reduction


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