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Can Local Optimality Be Used for Efficient Data Reduction?

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Algorithms and Complexity (CIAC 2021)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12701))

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Abstract

An independent set S in a graph G is k-swap optimal if there is no independent set \(S'\) such that \(|S'|>|S|\) and \(|(S'\setminus S)\cup (S\setminus S')|\le k\). Motivated by applications in data reduction, we study whether we can determine efficiently if a given vertex v is contained in some k-swap optimal independent set or in all k-swap optimal independent sets. We show that these problems are NP-hard for constant values of k even on graphs with constant maximum degree. Moreover, we show that the problems are \(\mathrm {\Sigma ^{\text {P}}_{2}} \)-hard when k is not constant, even on graphs of constant maximum degree. We obtain similar hardness results for determining whether an edge is contained in a k-swap optimal max cut. Finally, we consider a certain type of edge-swap neighborhood for the Longest Path problem. We show that for a given edge we can decide in \(f(\varDelta +k)\cdot n^{\mathcal {O}(1)}\) time whether it is in some k-optimal path.

N. Morawietz—Supported by the Deutsche Forschungsgemeinschaft (DFG), project OPERAH, KO 3669/5-1.

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Notes

  1. 1.

    For even k, an independent set is k-swap optimal if and only if it is \((k-1)\)-swap optimal [10]. Thus, only odd values of k are interesting.

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Authors and Affiliations

  1. Fachbereich Mathematik und Informatik, Philipps-Universität Marburg, Marburg, Germany

    Christian Komusiewicz & Nils Morawietz

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  1. Christian Komusiewicz
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  2. Nils Morawietz
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Correspondence to Nils Morawietz .

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Editors and Affiliations

  1. Sapienza University of Rome, Rome, Italy

    Tiziana Calamoneri

  2. Sapienza University of Rome, Rome, Italy

    Federico Corò

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Komusiewicz, C., Morawietz, N. (2021). Can Local Optimality Be Used for Efficient Data Reduction?. In: Calamoneri, T., Corò, F. (eds) Algorithms and Complexity. CIAC 2021. Lecture Notes in Computer Science(), vol 12701. Springer, Cham. https://doi.org/10.1007/978-3-030-75242-2_25

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  • DOI: https://doi.org/10.1007/978-3-030-75242-2_25

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