Schloss Dagstuhl - Leibniz-Zentrum für Informatik GmbH Schloss Dagstuhl - Leibniz-Zentrum für Informatik GmbH scholarly article en Roth, Marc; Schmitt, Johannes; Wellnitz, Philip https://www.dagstuhl.de/lipics License: Creative Commons Attribution 4.0 license (CC BY 4.0)
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URN: urn:nbn:de:0030-drops-141778
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Detecting and Counting Small Subgraphs, and Evaluating a Parameterized Tutte Polynomial: Lower Bounds via Toroidal Grids and Cayley Graph Expanders

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Abstract

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.

BibTeX - Entry

@InProceedings{roth_et_al:LIPIcs.ICALP.2021.108,
  author =	{Roth, Marc and Schmitt, Johannes and Wellnitz, Philip},
  title =	{{Detecting and Counting Small Subgraphs, and Evaluating a Parameterized Tutte Polynomial: Lower Bounds via Toroidal Grids and Cayley Graph Expanders}},
  booktitle =	{48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
  pages =	{108:1--108:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-195-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{198},
  editor =	{Bansal, Nikhil and Merelli, Emanuela and Worrell, James},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2021/14177},
  URN =		{urn:nbn:de:0030-drops-141778},
  doi =		{10.4230/LIPIcs.ICALP.2021.108},
  annote =	{Keywords: Counting complexity, parameterized complexity, Tutte polynomial}
}

Keywords: Counting complexity, parameterized complexity, Tutte polynomial
Seminar: 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)
Issue date: 2021
Date of publication: 02.07.2021


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