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Improved Analysis of Highest-Degree Branching for Feedback Vertex Set

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Abstract

Recent empirical evaluations of exact algorithms for Feedback Vertex Set have demonstrated the efficiency of a highest-degree branching algorithm with a degree-based pruning. In this paper, we prove that this empirically fast algorithm runs in \(O(3.460^k n)\) time, where k is the solution size. This improves the previous best \(O(3.619^k n)\)-time deterministic algorithm obtained by Kociumaka and Pilipczuk (Inf Process Lett 114:556–560, 2014. https://doi.org/10.1016/j.ipl.2014.05.001).

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Notes

  1. The \(O^*(\cdot )\) notation hides factors polynomial in n. Note that for FVS, any \(O(f(k) n^{O(1)})\)-time FPT algorithm can be improved to \(O(f(k)k^{O(1)}+k^{O(1)}n)\) time by applying a linear-time polynomial-size kernel [16] as a preprocess. We can therefore focus only on the f(k) factor when comparing the running time. For this reason, in this paper, we use the \(O^*(\cdot )\) notation to describe the running time of FPT algorithms for FVS.

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Acknowledgements

A preliminary version of this paper has been presented at IPEC 2019 [17]. We would like to thank Yixin Cao for valuable discussions and to thank organizers of PACE challenge 2016 for motivating us to study FVS. This work was supported by JSPS KAKENHI Grants Number JP17K12643, JP16K16010, JP17K19960, and JP18H05291.

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  1. National Institute of Informatics, Tokyo, Japan

    Yoichi Iwata

  2. Kyoto University, Kyoto, Japan

    Yusuke Kobayashi

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  1. Yoichi Iwata
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  2. Yusuke Kobayashi
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Correspondence to Yoichi Iwata.

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A Program to Check Lemma 7

A Program to Check Lemma 7

We attach the source code of a python3 program to verify the inequalities (2)–(4) appeared in the proof of Lemma 7. The same source code is also available at https://github.com/wata-orz/FVS_analysis.

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Iwata, Y., Kobayashi, Y. Improved Analysis of Highest-Degree Branching for Feedback Vertex Set. Algorithmica 83, 2503–2520 (2021). https://doi.org/10.1007/s00453-021-00815-w

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