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A Simple Local Search Gives a PTAS for the Feedback Vertex Set Problem in Minor-Free Graphs

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Computing and Combinatorics (COCOON 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11653))

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Abstract

We show that a simple local search gives a PTAS for the Feedback Vertex Set (FVS) problem in minor-free graphs. An efficient PTAS in minor-free graphs was known for this problem by Fomin, Lokshtanov, Raman and Sauraubh [13]. However, their algorithm is a combination of many advanced algorithmic tools such as contraction decomposition framework introduced by Demaine and Hajiaghayi [10], Courcelle’s theorem [9] and the Robertson and Seymour decomposition [29]. In stark contrast, our local search algorithm is very simple and easy to implement. It keeps exchanging a constant number of vertices to improve the current solution until a local optimum is reached. Our main contribution is to show that the local optimum only differs the global optimum by \((1+\epsilon )\) factor.

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Notes

  1. 1.

    A polynomial-time approximation scheme for a minimization problem is an algorithm that, given a fixed constant \(\epsilon > 0\), runs in polynomial time and returns a solution within \(1 +\epsilon \) of optimal.

  2. 2.

    A PTAS is efficient if the running time is of the form \(2^{\mathrm {poly}(1/\epsilon )}n^{O(1)}\).

  3. 3.

    For k-means and k-median, the exchange graph is constructed from \(\textsc {L}\) and a nearly optimal solution \(\textsc {O}'\), which is obtained by removing some vertices of \(\textsc {O}\).

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Acknowledgement

We thank the anonymous reviewer who pointed out an error in our argument to bound the size of the exchange graph. This material is based upon work supported by the National Science Foundation under Grant No. CCF-1252833. This work was done while the first author was at Oregon State University.

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Authors and Affiliations

  1. Department of Computer Science, University of Victoria, Victoria, BC, Canada

    Hung Le

  2. School of Electrical Engineering and Computer Science, Oregon State University, Corvallis, OR, USA

    Baigong Zheng

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  1. Hung Le
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Correspondence to Hung Le .

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Editors and Affiliations

  1. The University of Texas at Dallas, Richardson, TX, USA

    Ding-Zhu Du

  2. Xidian University, Xi'an, China

    Zhenhua Duan

  3. Xidian University, Xi'an, China

    Cong Tian

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Le, H., Zheng, B. (2019). A Simple Local Search Gives a PTAS for the Feedback Vertex Set Problem in Minor-Free Graphs. In: Du, DZ., Duan, Z., Tian, C. (eds) Computing and Combinatorics. COCOON 2019. Lecture Notes in Computer Science(), vol 11653. Springer, Cham. https://doi.org/10.1007/978-3-030-26176-4_31

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  • DOI: https://doi.org/10.1007/978-3-030-26176-4_31

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