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A fast and deterministic algorithm for Knapsack-constrained monotone DR-submodular maximization over an integer lattice

Author

Listed:
  • Suning Gong

    (Ocean University of China)

  • Qingqin Nong

    (Ocean University of China)

  • Shuyu Bao

    (Ocean University of China)

  • Qizhi Fang

    (Ocean University of China)

  • Ding-Zhu Du

    (University of Texas)

Abstract

We consider a knapsack-constrained maximization problem of a nonnegative monotone DR-submodular function f over a bounded integer lattice $$[\varvec{B}]$$ [ B ] in $${\mathbb {R}}_+^n$$ R + n , $$\max \{f({\varvec{x}}): {\varvec{x}}\in [\varvec{B}] \text {~and~} \sum _{i=1}^n {\varvec{x}}(i)c(i)\le 1\}$$ max { f ( x ) : x ∈ [ B ] and ∑ i = 1 n x ( i ) c ( i ) ≤ 1 } , where n is the cardinality of a ground set N and $$c(\cdot )$$ c ( · ) is a cost function defined on N. Soma and Yoshida [Math. Program., 172 (2018), pp. 539-563] present a $$(1-e^{-1}-O(\epsilon ))$$ ( 1 - e - 1 - O ( ϵ ) ) -approximation algorithm for this problem by combining threshold greedy algorithm with partial element enumeration technique. Although the approximation ratio is almost tight, their algorithm runs in $$O(\frac{n^3}{\epsilon ^3}\log ^3 \tau [\log ^3 \left\| \varvec{B}\right\| _\infty + \frac{n}{\epsilon }\log \left\| \varvec{B}\right\| _\infty \log \frac{1}{\epsilon c_{\min }}])$$ O ( n 3 ϵ 3 log 3 τ [ log 3 B ∞ + n ϵ log B ∞ log 1 ϵ c min ] ) time, where $$c_{\min }=\min _i c(i)$$ c min = min i c ( i ) and $$\tau $$ τ is the ratio of the maximum value of f to the minimum nonzero increase in the value of f. Besides, Ene and Nguy $$\tilde{\check{\text {e}}}$$ e ˇ ~ n [ arXiv:1606.08362 , 2016] indirectly give a $$(1-e^{-1}-O(\epsilon ))$$ ( 1 - e - 1 - O ( ϵ ) ) -approximation algorithm with $$O({(\frac{1}{\epsilon })}^{ O(1/\epsilon ^4)}n \log {\Vert \varvec{B}\Vert }_\infty \log ^2{(n \log {\Vert \varvec{B}\Vert }_\infty )})$$ O ( ( 1 ϵ ) O ( 1 / ϵ 4 ) n log ‖ B ‖ ∞ log 2 ( n log ‖ B ‖ ∞ ) ) time. But their algorithm is random. In this paper, we make full use of the DR-submodularity over a bounded integer lattice, carry forward the greedy idea in the continuous process and provide a simple deterministic rounding method so as to obtain a feasible solution of the original problem without loss of objective value. We present a deterministic algorithm and theoretically reduce its running time to a new record, $$O\big ((\frac{1}{\epsilon })^{O({1}/{\epsilon ^5})} \cdot n \log \frac{1}{c_{\min }} \log {\Vert \varvec{B}\Vert _\infty }\big )$$ O ( ( 1 ϵ ) O ( 1 / ϵ 5 ) · n log 1 c min log ‖ B ‖ ∞ ) , with the same approximate ratio.

Suggested Citation

  • Suning Gong & Qingqin Nong & Shuyu Bao & Qizhi Fang & Ding-Zhu Du, 2023. "A fast and deterministic algorithm for Knapsack-constrained monotone DR-submodular maximization over an integer lattice," Journal of Global Optimization, Springer, vol. 85(1), pages 15-38, January.
  • Handle: RePEc:spr:jglopt:v:85:y:2023:i:1:d:10.1007_s10898-022-01193-5
    DOI: 10.1007/s10898-022-01193-5
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    References listed on IDEAS

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    1. 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).
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    3. WOLSEY, Laurence A., 1982. "Maximising real-valued submodular functions: primal and dual heuristics for location problems," LIDAM Reprints CORE 486, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    4. Laurence A. Wolsey, 1982. "Maximising Real-Valued Submodular Functions: Primal and Dual Heuristics for Location Problems," Mathematics of Operations Research, INFORMS, vol. 7(3), pages 410-425, August.
    5. 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).
    6. Ariel Kulik & Hadas Shachnai & Tami Tamir, 2013. "Approximations for Monotone and Nonmonotone Submodular Maximization with Knapsack Constraints," Mathematics of Operations Research, INFORMS, vol. 38(4), pages 729-739, November.
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    8. repec:dgr:rugsom:99a17 is not listed on IDEAS
    9. 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).
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