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Twin-width and Polynomial Kernels

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Abstract

We study the existence of polynomial kernels, for parameterized problems without a polynomial kernel on general graphs, when restricted to graphs of bounded twin-width. Our main result is that a polynomial kernel for k -Dominating Set on graphs of twin-width at most 4 would contradict a standard complexity-theoretic assumption. The reduction is quite involved, especially to get the twin-width upper bound down to 4, and can be tweaked to work for Connected k -Dominating Set and Total k -Dominating Set (albeit with a worse upper bound on the twin-width). The k -Independent Set problem admits the same lower bound by a much simpler argument, previously observed [ICALP ’21], which extends to k -Independent Dominating Set, k -Path, k -Induced Path, k -Induced Matching, etc. On the positive side, we obtain a simple quadratic vertex kernel for Connected k -Vertex Cover and Capacitated k -Vertex Cover on graphs of bounded twin-width. Interestingly the kernel applies to graphs of Vapnik–Chervonenkis density 1, and does not require a witness sequence. We also present a more intricate \(O(k^{1.5})\) vertex kernel for Connected k -Vertex Cover. Finally we show that deciding if a graph has twin-width at most 1 can be done in polynomial time, and observe that most optimization/decision graph problems can be solved in polynomial time on graphs of twin-width at most 1.

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Notes

  1. See Sect. 2 for the relevant background on how to rule out a polynomial kernel

  2. All the subsequent results also hold for \(k\) -Independent Set.

  3. Sparse is an overloaded term; here we use it as not containing arbitrarily large bicliques as subgraphs.

  4. The definition of nowhere denseness being technical and unnecessary to the current paper, we refer the interested reader to [44]. Let us just mention that bounded-degree graphs, planar graphs, and proper (topological) minor-closed classes are all nowhere dense.

  5. We will not need a definition of expansion here. Bounded expansion classes are more general than topological-minor-free classes and less general than nowhere dense classes.

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Acknowledgements

We thank Noga Alon and Bart M. P. Jansen for independently asking whether \(k\) -Dominating Set admits a polynomial kernel on classes of bounded twin-width, an interesting question that led to our main result.

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Authors and Affiliations

  1. Univ Lyon, CNRS, ENS de Lyon, LIP UMR5668, Université Claude Bernard Lyon 1, Lyon, France

    Édouard Bonnet, Amadeus Reinald, Stéphan Thomassé & Rémi Watrigant

  2. CNRS UMR7243, LAMSADE, Université Paris-Dauphine, PSL University, Paris, France

    Eun Jung Kim

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Bonnet, É., Kim, E.J., Reinald, A. et al. Twin-width and Polynomial Kernels. Algorithmica 84, 3300–3337 (2022). https://doi.org/10.1007/s00453-022-00965-5

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