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Parameterized Inapproximability of Independent Set in H-Free Graphs

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Abstract

We study the Independent Set problem in H-free graphs, i.e., graphs excluding some fixed graph H as an induced subgraph. We prove several inapproximability results both for polynomial-time and parameterized algorithms. Halldórsson [SODA 1995] showed that for every \(\delta >0\) the Independent Set problem has a polynomial-time \((\frac{d-1}{2}+\delta )\)-approximation algorithm in \(K_{1,d}\)-free graphs. We extend this result by showing that \(K_{a,b}\)-free graphs admit a polynomial-time\({\mathcal {O}}(\alpha (G)^{1-1/a})\)-approximation, where \(\alpha (G)\) is the size of a maximum independent set in G. Furthermore, we complement the result of Halldórsson by showing that for some \(\gamma =\Theta (d/\log d),\) there is no polynomial-time \(\gamma \)-approximation algorithm for these graphs, unless NP  = ZPP. Bonnet et al. [Algorithmica 2020] showed that Independent Set parameterized by the size k of the independent set is W[1]-hard on graphs which do not contain (1) a cycle of constant length at least 4, (2) the star \(K_{1,4}\), and (3) any tree with two vertices of degree at least 3 at constant distance. We strengthen this result by proving three inapproximability results under different complexity assumptions for almost the same class of graphs (we weaken conditions (1) and (2) that G does not contain a cycle of constant length at least 5 or \(K_{1,5}\)). First, under the ETH, there is no \(f(k) \cdot n^{o(k/\log k)}\) algorithm for any computable function f. Then, under the deterministic Gap-ETH, there is a constant \(\delta >0\) such that no \(\delta \)-approximation can be computed in \(f(k) \cdot n^{O(1)}\) time. Also, under the stronger randomized Gap-ETH there is no such approximation algorithm with runtime \(f(k) \cdot n^{o(\sqrt{k})}\). Finally, we consider the parameterization by the excluded graph H, and show that under the ETH, Independent Set has no \(n^{o(\alpha (H))}\) algorithm in H-free graphs. Also, we prove that there is no \(d/k^{o(1)}\)-approximation algorithm for \(K_{1,d}\)-free graphs with runtime \(f(d,k) \cdot n^{{\mathcal {O}}(1)}\), under the deterministic Gap-ETH.

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Notes

  1. While this is stated under the randomized Gap-ETH in [9], a derandomization exists; see [9, Section 4.2.1].

  2. In the conference version of this paper [19] we mistakenly claimed our reduction excludes an algorithm with running time \(f(k) \cdot n^{o(k)}\).

  3. The result is implicit from [38, Theorem 2.1]: by setting \(t=2\) and using a straight-forward reduction from Label Cover to MCSI, where each of the \(\ell \) vertices of U is expanded into a colour class and an edge exists if the respective projected labels are the same for the unique (as \(t=2\)) shared neighbor in V.

  4. The possibilities for \(\{B_1,B_2\}\) are: \(\{C_{ij} D^1_{ij}\}\) or \(\{D^p_{ij}, D^{p+1}_{ij}\}\) for \(p < s\).

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Acknowledgements

We would like to thank to the anonymous reviewer, who suggested using gap amplification to obtain Theorem 6. We are also grateful to the other reviewer for pointing out the mistake in Theorem 5 in the conference version of our paper [19].

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Authors and Affiliations

  1. Faculty of Mathematics and Physics, Charles University, Prague, Czechia

    Pavel Dvořák & Andreas Emil Feldmann

  2. Faculty of Mathematics and Information Science, Warsaw University of Technology, Warsaw, Poland

    Paweł Rzążewski

  3. Institute of Informatics, University of Warsaw, Warsaw, Poland

    Paweł Rzążewski

  4. Department of Mathematics, IIT Delhi, New Delhi, India

    Ashutosh Rai

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Pavel Dvořák is now a visiting fellow at School of Technology and Computer Science, Tata Institute of Fundamental Research where Praladh Harsha, member of the editorial board, is a faculty. Other coauthors have no conflict of interest.

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An extended abstract of this paper was presented at WG 2020 [19].

P. Dvořák and A. E. Feldmann: Supported by Czech Science Foundation GAČR (Grant #19-27871X).

A. Rai: Supported by Center for Foundations of Modern Computer Science (Charles Univ. Project UNCE/SCI/004).

P. Rzążewski: Supported by Polish National Science Centre Grant No. 2018/31/D/ST6/00062.

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Dvořák, P., Feldmann, A.E., Rai, A. et al. Parameterized Inapproximability of Independent Set in H-Free Graphs. Algorithmica 85, 902–928 (2023). https://doi.org/10.1007/s00453-022-01052-5

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