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

We consider a knapsack-constrained maximization problem of a nonnegative monotone DR-submodular function f over a bounded integer lattice [B]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[\varvec{B}]$$\end{document} in R+n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}_+^n$$\end{document}, max{f(x):x∈[B]and∑i=1nx(i)c(i)≤1}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\max \{f({\varvec{x}}): {\varvec{x}}\in [\varvec{B}] \text {~and~} \sum _{i=1}^n {\varvec{x}}(i)c(i)\le 1\}$$\end{document}, where n is the cardinality of a ground set N and c(·)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c(\cdot )$$\end{document} is a cost function defined on N. Soma and Yoshida [Math. Program., 172 (2018), pp. 539-563] present a (1-e-1-O(ϵ))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1-e^{-1}-O(\epsilon ))$$\end{document}-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(n3ϵ3log3τ[log3B∞+nϵlogB∞log1ϵcmin])\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(\frac{n^3}{\epsilon ^3}\log ^3 \tau [\log ^3 \left\| \varvec{B}\right\| _\infty + \frac{n}{\epsilon }\log \left\| \varvec{B}\right\| _\infty \log \frac{1}{\epsilon c_{\min }}])$$\end{document} time, where cmin=minic(i)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_{\min }=\min _i c(i)$$\end{document} and τ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} is the ratio of the maximum value of f to the minimum nonzero increase in the value of f. Besides, Ene and Nguyeˇ~\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\check{\text {e}}}$$\end{document}n [arXiv:1606.08362, 2016] indirectly give a (1-e-1-O(ϵ))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1-e^{-1}-O(\epsilon ))$$\end{document}-approximation algorithm with O((1ϵ)O(1/ϵ4)nlog‖B‖∞log2(nlog‖B‖∞))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O({(\frac{1}{\epsilon })}^{ O(1/\epsilon ^4)}n \log {\Vert \varvec{B}\Vert }_\infty \log ^2{(n \log {\Vert \varvec{B}\Vert }_\infty )})$$\end{document} 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((1ϵ)O(1/ϵ5)·nlog1cminlog‖B‖∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O\big ((\frac{1}{\epsilon })^{O({1}/{\epsilon ^5})} \cdot n \log \frac{1}{c_{\min }} \log {\Vert \varvec{B}\Vert _\infty }\big )$$\end{document}, with the same approximate ratio.