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Counting Induced Subgraphs: An Algebraic Approach to #W[1]-hardness

Authors Julian Dörfler , Marc Roth , Johannes Schmitt , Philip Wellnitz

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ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Theory of computation → Problems, reductions and completeness
  • Mathematics of computing → Combinatorics
  • Mathematics of computing → Graph theory
Keywords
  • counting complexity
  • edge-transitive graphs
  • graph homomorphisms
  • induced subgraphs
  • parameterized complexity

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Abstract

We study the problem #IndSub(Phi) of counting all induced subgraphs of size k in a graph G that satisfy the property Phi. This problem was introduced by Jerrum and Meeks and shown to be #W[1]-hard when parameterized by k for some families of properties Phi including, among others, connectivity [JCSS 15] and even- or oddness of the number of edges [Combinatorica 17]. Very recently [IPEC 18], two of the authors introduced a novel technique for the complexity analysis of #IndSub(Phi), inspired by the "topological approach to evasiveness" of Kahn, Saks and Sturtevant [FOCS 83] and the framework of graph motif parameters due to Curticapean, Dell and Marx [STOC 17], allowing them to prove hardness of a wide range of properties Phi. In this work, we refine this technique for graph properties that are non-trivial on edge-transitive graphs with a prime power number of edges. In particular, we fully classify the case of monotone bipartite graph properties: It is shown that, given any graph property Phi that is closed under the removal of vertices and edges, and that is non-trivial for bipartite graphs, the problem #IndSub(Phi) is #W[1]-hard and cannot be solved in time f(k)* n^{o(k)} for any computable function f, unless the Exponential Time Hypothesis fails. This holds true even if the input graph is restricted to be bipartite and counting is done modulo a fixed prime. A similar result is shown for properties that are closed under the removal of edges only.

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Julian Dörfler, Marc Roth, Johannes Schmitt, and Philip Wellnitz. Counting Induced Subgraphs: An Algebraic Approach to #W[1]-hardness. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 26:1-26:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.MFCS.2019.26

Author Details

Julian Dörfler
  • Saarbrücken Graduate School of Computer Science, Saarland Informatics Campus (SIC), Germany
Marc Roth
  • Cluster of Excellence (MMCI), Saarland Informatics Campus (SIC), Saarbrücken, Germany
Johannes Schmitt
  • ETH Zürich, Switzerland
Philip Wellnitz
  • Max Planck Institute for Informatics, Saarland Informatics Campus (SIC), Saarbrücken, Germany

Funding

  • Schmitt, Johannes: The third author has received funding from the European Research Council (ERC) under the EU’s Horizon 2020 research and innovation programme (No 786580).

Acknowledgements

We are very grateful to Radu Curticapean and Holger Dell for fruitful discussions and valuable feedback on early drafts of this work.

References

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