Detecting and Counting Small Subgraphs, and Evaluating a Parameterized Tutte Polynomial: Lower Bounds via Toroidal Grids and Cayley Graph Expanders

Authors Marc Roth , Johannes Schmitt , Philip Wellnitz

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Marc Roth
  • Merton College, University of Oxford, UK
Johannes Schmitt
  • Mathematical Institute, University of Bonn, Germany
Philip Wellnitz
  • Max Planck Institute for Informatics, Saarland Informatics Campus (SIC), Saarbrücken, Germany


We thank Alina Vdovina and Norbert Peyerimhoff for explaining their construction of 2-group Cayley graph expanders [Peyerimhoff and Vdovina, 2011]. We thank Johannes Lengler and Holger Dell for helpful discussions on early drafts of this work. The second author was supported by the SNF early postdoc mobility grant 184245 and thanks the Max Planck Institute for Mathematics in Bonn for its hospitality.

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Marc Roth, Johannes Schmitt, and Philip Wellnitz. Detecting and Counting Small Subgraphs, and Evaluating a Parameterized Tutte Polynomial: Lower Bounds via Toroidal Grids and Cayley Graph Expanders. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 108:1-108:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


Given a graph property Φ, we consider the problem EdgeSub(Φ), where the input is a pair of a graph G and a positive integer k, and the task is to decide whether G contains a k-edge subgraph that satisfies Φ. Specifically, we study the parameterized complexity of EdgeSub(Φ) and of its counting problem #EdgeSub(Φ) with respect to both approximate and exact counting. We obtain a complete picture for minor-closed properties Φ: the decision problem EdgeSub(Φ) always admits an FPT ("fixed-parameter tractable") algorithm and the counting problem #EdgeSub(Φ) always admits an FPTRAS ("fixed-parameter tractable randomized approximation scheme"). For exact counting, we present an exhaustive and explicit criterion on the property Φ which, if satisfied, yields fixed-parameter tractability and otherwise #W[1]-hardness. Additionally, most of our hardness results come with an almost tight conditional lower bound under the so-called Exponential Time Hypothesis, ruling out algorithms for #EdgeSub(Φ) that run in time f(k)⋅ |G|^{o(k/log k)} for any computable function f. As a main technical result, we gain a complete understanding of the coefficients of toroidal grids and selected Cayley graph expanders in the homomorphism basis of #EdgeSub(Φ). This allows us to establish hardness of exact counting using the Complexity Monotonicity framework due to Curticapean, Dell and Marx (STOC'17). This approach does not only apply to #EdgeSub(Φ) but also to the more general problem of computing weighted linear combinations of subgraph counts. As a special case of such a linear combination, we introduce a parameterized variant of the Tutte Polynomial T^k_G of a graph G, to which many known combinatorial interpretations of values of the (classical) Tutte Polynomial can be extended. As an example, T^k_G(2,1) corresponds to the number of k-forests in the graph G. Our techniques allow us to completely understand the parameterized complexity of computing the evaluation of T^k_G at every pair of rational coordinates (x,y). In particular, our results give a new proof for the #W[1]-hardness of the problem of counting k-forests in a graph.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Theory of computation → Problems, reductions and completeness
  • Counting complexity
  • parameterized complexity
  • Tutte polynomial


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