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Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ICALP.2021.108
URN: urn:nbn:de:0030-drops-141778
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2021/14177/
Roth, Marc ;
Schmitt, Johannes ;
Wellnitz, Philip
Detecting and Counting Small Subgraphs, and Evaluating a Parameterized Tutte Polynomial: Lower Bounds via Toroidal Grids and Cayley Graph Expanders
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}
}
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Keywords: |
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Counting complexity, parameterized complexity, Tutte polynomial |
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Collection: |
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48th International Colloquium on Automata, Languages, and Programming (ICALP 2021) |
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Issue Date: |
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2021 |
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Date of publication: |
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02.07.2021 |
