In this paper we consider the problem of maximizing a non-monotone and non-negative DR-submodular function on a bounded integer lattice $[\overrightarrow{B}]=\{(x_1,\dots ,x_n)\in {\mathbb{Z}}_{+}^n:0\le x_k\le B_k,\forall 1\le k\le n\}$ without any constraint, where $\overrightarrow{B}=(B_1,\dots ,B_n)\in {\mathbb{Z}}_{+}^n$. We design an algorithm for the problem and measure its performance by its approximation ratio and the number of value oracle queries it needs, where the latter one is the dominating term in the running time of an algorithm. It has been showed that, for the problem considered, any algorithm achieving an approximation ratio greater than $\frac{1}{2}$ requires an exponential number of value oracle queries. In the literature there are two algorithms that reach $\frac{1}{2}$ approximation guarantee. The first algorithm needs $O(n||B|{|}_{\infty })$ oracle queries. The second one reduces its number of oracle queries to $O(n\max \{1,\log ||\overrightarrow{B}|{|}_{\infty }\})$ but it needs large storage. In this paper we present a randomized approximation algorithm that has an approximation guarantee of $\frac{1}{2}$, calls $O(n\max \{1,\log ||\overrightarrow{B}|{|}_{\infty }\})$ oracle queries and does not need large storage, improving the results of the literature.

在本文中，我们考虑了在有界整数格上最大化非单调和非负的DR次模函数的问题 $[\overrightarrow{乙}]=\{（X_{\mathrm{1个}}，\dots ，X_{ñ}）\in {\mathbb{ž}}_{+}^{ñ}：0\le X_{ķ}\le {乙}_{ķ}，\forall \mathrm{1个}\le ķ\le ñ\}$ 没有任何限制 $\overrightarrow{乙}=（{乙}_{\mathrm{1个}}，\dots ，{乙}_{ñ}）\in {\mathbb{ž}}_{+}^{ñ}$。我们针对该问题设计了一种算法，并通过其逼近率和所需的oracle查询值的数量来衡量其性能，其中后者是算法运行时间的主要术语。已经表明，对于所考虑的问题，任何算法都可以实现大于$\frac{\mathrm{1个}}{2}$需要指数型oracle查询值。在文献中有两种算法可以达到$\frac{\mathrm{1个}}{2}$近似保证。第一种算法需要$Ø（ñ||乙|{|}_{\infty }）$oracle查询。第二个将其oracle查询数量减少为$Ø（ñ\mathrm{最高}\{\mathrm{1个}，\mathrm{日志}||\overrightarrow{乙}|{|}_{\infty }\}）$但它需要大容量存储空间。在本文中，我们提出了一种随机近似算法，该算法的近似保证为$\frac{\mathrm{1个}}{2}$，来电 $Ø（ñ\mathrm{最高}\{\mathrm{1个}，\mathrm{日志}||\overrightarrow{乙}|{|}_{\infty }\}）$ oracle查询并且不需要大量存储，从而改善了文献结果。