Counting Answers to Existential Questions (Track B: Automata, Logic, Semantics, and Theory of Programming)

Authors Holger Dell , Marc Roth , Philip Wellnitz

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Author Details

Holger Dell
  • Cluster of Excellence (MMCI), Saarland Informatics Campus (SIC), Saarbrücken, Germany
Marc Roth
  • Cluster of Excellence (MMCI), Saarland Informatics Campus (SIC), Saarbrücken, Germany
Philip Wellnitz
  • Max Planck Institute for Informatics, Saarland Informatics Campus (SIC), Saarbrücken, Germany


We thank Cornelius Brand, Karl Bringmann, Radu Curticapean, Reinhard Diestel, Joshua Erde, Stephan Kreutzer, Stefan Mengel, Daniel Weißauer, and an anonymous reviewer for discussions and advice.

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Holger Dell, Marc Roth, and Philip Wellnitz. Counting Answers to Existential Questions (Track B: Automata, Logic, Semantics, and Theory of Programming). In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 113:1-113:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


Conjunctive queries select and are expected to return certain tuples from a relational database. We study the potentially easier problem of counting all selected tuples, rather than enumerating them. In particular, we are interested in the problem’s parameterized and data complexity, where the query is considered to be small or even fixed, and the database is considered to be large. We identify two structural parameters for conjunctive queries that capture their inherent complexity: The dominating star size and the linked matching number. If the dominating star size of a conjunctive query is large, then we show that counting solution tuples to the query is at least as hard as counting dominating sets, which yields a fine-grained complexity lower bound under the Strong Exponential Time Hypothesis (SETH) as well as a #W[2]-hardness result in parameterized complexity. Moreover, if the linked matching number of a conjunctive query is large, then we show that the structure of the query is so rich that arbitrary queries up to a certain size can be encoded into it; in the language of parameterized complexity, this essentially establishes a #A[2]-completeness result. Using ideas stemming from Lovász (1967), we lift complexity results from the class of conjunctive queries to arbitrary existential or universal formulas that might contain inequalities and negations on constraints over the free variables. As a consequence, we obtain a complexity classification that refines and generalizes previous results of Chen, Durand, and Mengel (ToCS 2015; ICDT 2015; PODS 2016) for conjunctive queries and of Curticapean and Marx (FOCS 2014) for the subgraph counting problem. Our proof also relies on graph minors, and we show a strengthening of the Excluded-Grid-Theorem which might be of independent interest: If the linked matching number (and thus the treewidth) is large, then not only can we find a large grid somewhere in the graph, but we can find a large grid whose diagonal has disjoint paths leading into an assumed node-well-linked set.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Theory of computation → Problems, reductions and completeness
  • Conjunctive queries
  • graph homomorphisms
  • counting complexity
  • parameterized complexity
  • fine-grained complexity


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