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Approximation for maximizing monotone non-decreasing set functions with a greedy method

Author

Listed:
  • Zengfu Wang

    (Northwestern Polytechnical University
    The University of Melbourne)

  • Bill Moran

    (The University of Melbourne)

  • Xuezhi Wang

    (The University of Melbourne)

  • Quan Pan

    (Northwestern Polytechnical University)

Abstract

We study the problem of maximizing a monotone non-decreasing function $$f$$ f subject to a matroid constraint. Fisher, Nemhauser and Wolsey have shown that, if $$f$$ f is submodular, the greedy algorithm will find a solution with value at least $$\frac{1}{2}$$ 1 2 of the optimal value under a general matroid constraint and at least $$1-\frac{1}{e}$$ 1 - 1 e of the optimal value under a uniform matroid $$(\mathcal {M} = (X,\mathcal {I})$$ ( M = ( X , I ) , $$\mathcal {I} = \{ S \subseteq X: |S| \le k\}$$ I = { S ⊆ X : | S | ≤ k } ) constraint. In this paper, we show that the greedy algorithm can find a solution with value at least $$\frac{1}{1+\mu }$$ 1 1 + μ of the optimum value for a general monotone non-decreasing function with a general matroid constraint, where $$\mu = \alpha $$ μ = α , if $$0 \le \alpha \le 1$$ 0 ≤ α ≤ 1 ; $$\mu = \frac{\alpha ^K(1-\alpha ^K)}{K(1-\alpha )}$$ μ = α K ( 1 - α K ) K ( 1 - α ) if $$\alpha > 1$$ α > 1 ; here $$\alpha $$ α is a constant representing the “elemental curvature” of $$f$$ f , and $$K$$ K is the cardinality of the largest maximal independent sets. We also show that the greedy algorithm can achieve a $$1 - (\frac{\alpha + \cdots + \alpha ^{k-1}}{1+\alpha + \cdots + \alpha ^{k-1}})^k$$ 1 - ( α + ⋯ + α k - 1 1 + α + ⋯ + α k - 1 ) k approximation under a uniform matroid constraint. Under this unified $$\alpha $$ α -classification, submodular functions arise as the special case $$0 \le \alpha \le 1$$ 0 ≤ α ≤ 1 .

Suggested Citation

  • Zengfu Wang & Bill Moran & Xuezhi Wang & Quan Pan, 2016. "Approximation for maximizing monotone non-decreasing set functions with a greedy method," Journal of Combinatorial Optimization, Springer, vol. 31(1), pages 29-43, January.
  • Handle: RePEc:spr:jcomop:v:31:y:2016:i:1:d:10.1007_s10878-014-9707-3
    DOI: 10.1007/s10878-014-9707-3
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    References listed on IDEAS

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    1. G. L. Nemhauser & L. A. Wolsey, 1978. "Best Algorithms for Approximating the Maximum of a Submodular Set Function," Mathematics of Operations Research, INFORMS, vol. 3(3), pages 177-188, August.
    2. Jon Lee & Maxim Sviridenko & Jan Vondrák, 2010. "Submodular Maximization over Multiple Matroids via Generalized Exchange Properties," Mathematics of Operations Research, INFORMS, vol. 35(4), pages 795-806, November.
    3. Nemhauser, G.L. & Wolsey, L.A., 1978. "Best algorithms for approximating the maximum of a submodular set function," LIDAM Reprints CORE 343, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    4. Fisher, M.L. & Nemhauser, G.L. & Wolsey, L.A., 1978. "An analysis of approximations for maximizing submodular set functions," LIDAM Reprints CORE 341, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    5. Fisher, M.L. & Nemhauser, G.L. & Wolsey, L.A., 1978. "An analysis of approximations for maximizing submodular set functions - 1," LIDAM Reprints CORE 334, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    6. A.A. Ageev & M.I. Sviridenko, 2004. "Pipage Rounding: A New Method of Constructing Algorithms with Proven Performance Guarantee," Journal of Combinatorial Optimization, Springer, vol. 8(3), pages 307-328, September.
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    2. Ruiqi Yang & Dachuan Xu & Longkun Guo & Dongmei Zhang, 2021. "Sequence submodular maximization meets streaming," Journal of Combinatorial Optimization, Springer, vol. 41(1), pages 43-55, January.

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