Using a short proof, we show that every set function f can be decomposed into the difference of two monotone increasing and strictly submodular functions g and h, i.e., \(f=g-h\), and every set function f can also be decomposed into the difference of two monotone increasing and strictly supermodular functions g and h, i.e., \(f=g-h\).

中文翻译：

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集函数优化中DS分解的更强版本的简短证明

使用一个简短的证明，我们表明每个集合函数f都可以分解为两个单调递增函数和严格的子模函数g和h的差，即\（f = gh \），每个集合函数f也可以分解为两个单调递增函数和严格的超模函数g和h之差，即\（f = gh \）。