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

Abstract

We consider a knapsack-constrained maximization problem of a nonnegative monotone DR-submodular function f over a bounded integer lattice $[\varvec{B}]$ in ${\mathbb {R}}_+^n$, $\max \{f({\varvec{x}}): {\varvec{x}}\in [\varvec{B}] \text {~and~} \sum _{i=1}^n {\varvec{x}}(i)c(i)\le 1\}$, where n is the cardinality of a ground set N and $c(\cdot )$ is a cost function defined on N. Soma and Yoshida [Math. Program., 172 (2018), pp. 539-563] present a $(1-e^{-1}-O(\epsilon ))$-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 $O(\frac{n^3}{\epsilon ^3}\log ^3 \tau [\log ^3 \left\| \varvec{B}\right\| _\infty + \frac{n}{\epsilon }\log \left\| \varvec{B}\right\| _\infty \log \frac{1}{\epsilon c_{\min }}])$ time, where $c_{\min }=\min _i c(i)$ and $\tau$ is the ratio of the maximum value of f to the minimum nonzero increase in the value of f. Besides, Ene and Nguy$\tilde{\check{\text {e}}}$n [arXiv:1606.08362, 2016] indirectly give a $(1-e^{-1}-O(\epsilon ))$-approximation algorithm with $O({(\frac{1}{\epsilon })}^{ O(1/\epsilon ^4)}n \log {\Vert \varvec{B}\Vert }_\infty \log ^2{(n \log {\Vert \varvec{B}\Vert }_\infty )})$ 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, $O\big ((\frac{1}{\epsilon })^{O({1}/{\epsilon ^5})} \cdot n \log \frac{1}{c_{\min }} \log {\Vert \varvec{B}\Vert _\infty }\big )$, with the same approximate ratio.