Maximizing non-monotone submodular functions is one of the most important problems in submodular optimization. Let \(\mathbf {B}=(B_1, B_2,\ldots , B_n)\in {\mathbb {Z}}_+^n\) be an integer vector and \([\mathbf { B}]=\{(x_1,\dots ,x_n) \in {\mathbb {Z}}_+^n: 0\le x_k \le B_k, \forall 1\le k\le n\}\) be the set of all non-negative integer vectors not greater than \(\mathbf {B}\). A function \(f:[\mathbf { B}] \rightarrow {\mathbb {R}}\) is said to be weak-submodular if \(f(\mathbf {x}+\delta \mathbf {1}_k)-f(\mathbf {x})\ge f(\mathbf {y}+\delta \mathbf {1}_k)-f(\mathbf {y})\) for any \(k\in \{1,\dots ,n\}\), any pair of \(\mathbf {x}, \mathbf {y}\in [\mathbf { B}]\) such that \(\mathbf {x}\le \mathbf {y}\) and \(x_k =y_k\), and any \(\delta \in {\mathbb {Z}}_+\) satisfying \(\mathbf {y}+\delta \mathbf {1}_k\in [\mathbf { B}]\). Here \(\mathbf {1}_k\) is the vector with the kth component equal to 1 and each of the others equals to 0. In this paper we consider the problem of maximizing a non-monotone and non-negative weak-submodular function on the bounded integer lattice without any constraint. We present an randomized algorithm with an approximation guarantee \(\frac{1}{2}\) for the problem.

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一个1/2逼近算法，用于最大化有界整数格上的非单调弱次模函数

最大化非单调子模函数是子模优化中最重要的问题之一。令\（\ mathbf {B} =（B_1，B_2，\ ldots，B_n）\在{\ mathbb {Z}} _ + ^ n \）中为整数向量，而\（[\ mathbf {B}] = \ {（x_1，\ dots，x_n）\ in {\ mathbb {Z}} _ + ^ n：0 \ le x_k \ le B_k，\ forall 1 \ le k \ le n \} \）是所有非-不大于\（\ mathbf {B} \）的负整数向量。如果\（f（\ mathbf {x} + \ delta \ mathbf {1} _k，则函数\（f：[\ mathbf {B}] \ rightarrow {\ mathbb {R}} \）被称为弱次模）-f（\ mathbf {x}）\ ge f（\ mathbf {y} + \ delta \ mathbf {1} _k）-f（\ mathbf {y}）\）对于任何\（k \ in \ {1 ，\ dots，n \} \），在[\ mathbf {B}] \中的任意一对\（\ mathbf {x}，\ mathbf {y} \这样\（\ mathbf {x} \ le \ mathbf {y} \）和\（x_k = y_k \），以及任何满足（{\ mathbbf {z}} _ + \）的\（\ delta \ mathbf {y} + \ delta \ mathbf {1} _k \ in [\ mathbf {B}] \）中。这里\（\ mathbf {1} _k \）是第k个分量等于1且每个其他分量等于0的向量。在本文中，我们考虑最大化非单调且非负的弱向量的问题。有界整数格上的子模函数，没有任何约束。我们针对该问题提出了一种具有近似保证\（\ frac {1} {2} \）的随机算法。