Existential Second-Order Logic over Graphs: Parameterized Complexity

Authors Max Bannach , Florian Chudigiewitsch , Till Tantau

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Author Details

Max Bannach
  • European Space Agency, Advanced Concepts Team, Noordwijk, The Netherlands
Florian Chudigiewitsch
  • Universität zu Lübeck, Germany
Till Tantau
  • Universität zu Lübeck, Germany


We thank Marcel Wienöbst for fruitful discussions and helpful comments on an earlier draft.

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Max Bannach, Florian Chudigiewitsch, and Till Tantau. Existential Second-Order Logic over Graphs: Parameterized Complexity. In 18th International Symposium on Parameterized and Exact Computation (IPEC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 285, pp. 3:1-3:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


By Fagin’s Theorem, NP contains precisely those problems that can be described by formulas starting with an existential second-order quantifier, followed by only first-order quantifiers (eso formulas). Subsequent research refined this result, culminating in powerful theorems that characterize for each possible sequence of first-order quantifiers how difficult the described problem can be. We transfer this line of inquiry to the parameterized setting, where the size of the set quantified by the second-order quantifier is the parameter. Many natural parameterized problems can be described in this way using simple sequences of first-order quantifiers: For the clique or vertex cover problems, two universal first-order quantifiers suffice ("for all pairs of vertices ... must hold"); for the dominating set problem, a universal followed by an existential quantifier suffice ("for all vertices, there is a vertex such that ..."); and so on. We present a complete characterization that states for each possible sequence of first-order quantifiers how high the parameterized complexity of the described problems can be. The uncovered dividing line between quantifier sequences that lead to tractable versus intractable problems is distinct from that known from the classical setting, and it depends on whether the parameter is a lower bound on, an upper bound on, or equal to the size of the quantified set.

Subject Classification

ACM Subject Classification
  • Theory of computation → Finite Model Theory
  • Theory of computation → Complexity theory and logic
  • Theory of computation → Fixed parameter tractability
  • Theory of computation → W hierarchy
  • existential second-order logic
  • graph problems
  • parallel algorithms
  • fixed-parameter tractability
  • descriptive complexity


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