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Greedy guarantees for minimum submodular cost submodular/non-submodular cover problem

Author

Listed:
  • Majun Shi

    (Xi’an Jiaotong University)

  • Zishen Yang

    (Xi’an Jiaotong University)

  • Wei Wang

    (Xi’an Jiaotong University)

Abstract

Minimum Submodular Cost Submodular Cover problem (MIN-SCSC) often occurs naturally in the areas of combinatorial optimization and particularly machine learning. It is well-known that the greedy algorithm proposed by Wan et al. yields a $$\rho H(\delta )$$ ρ H ( δ ) -approximation for an integer-valued submodular function f, where $$\rho $$ ρ is the curvature of submodular cost function c, $$\delta $$ δ is the maximum value of f over all singletons and $$H(\delta )$$ H ( δ ) is the $$\delta $$ δ -th harmonic number (Wan et al. in Comput Optim Appl 45(2):463–474). In this paper, we first extend MIN-SCSC to Minimum Submodular Cost Non-submodular Cover problem and analyze the performances of the widely used greedy algorithm for integer-valued and fraction-valued potential functions respectively. In addition, we also study MIN-SCSC with fraction-valued potential functions, with a new analysis of the performance ratio of the greedy algorithm, improving upon the result of Wan et al. (2010).

Suggested Citation

  • Majun Shi & Zishen Yang & Wei Wang, 2023. "Greedy guarantees for minimum submodular cost submodular/non-submodular cover problem," Journal of Combinatorial Optimization, Springer, vol. 45(1), pages 1-16, January.
    • Handle: RePEc:spr:jcomop:v:45:y:2023:i:1:d:10.1007_s10878-022-00941-3
    DOI: 10.1007/s10878-022-00941-3
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    References listed on IDEAS

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    1. Shi, Majun & Yang, Zishen & Wang, Wei, 2021. "Minimum non-submodular cover problem with applications," Applied Mathematics and Computation, Elsevier, vol. 410(C).
    2. Ding-Zhu Du & Ker-I Ko & Xiaodong Hu, 2012. "Design and Analysis of Approximation Algorithms," Springer Optimization and Its Applications, Springer, number 978-1-4614-1701-9, September.
    3. Wolsey, L.A., 1982. "An analysis of the greedy algorithm for the submodular set covering problem," LIDAM Reprints CORE 519, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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