We define the supermodular rank of a function on a lattice. This is the smallest number of terms needed to decompose it into a sum of supermodular functions. The supermodular summands are defined with respect to different partial orders. We characterize the maximum possible value of the supermodular rank and describe the functions with fixed supermodular rank. We analogously define the submodular rank. We use submodular decompositions to optimize set functions. Given a bound on the submodular rank of a set function, we formulate an algorithm that splits an optimization problem into submodular subproblems. We show that this method improves the approximation ratio guarantees of several algorithms for monotone set function maximization and ratio of set functions minimization, at a computation overhead that depends on the submodular rank.

翻译：我们定义了格上函数的超模秩，即将其分解为超模函数之和所需的最少项数。这些超模加项相对于不同的偏序关系定义。我们刻画了超模秩的最大可能取值，并描述了具有固定超模秩的函数。类似地，我们定义了次模秩。我们利用次模分解来优化集合函数。给定集合函数次模秩的上界，我们提出一种算法，将优化问题拆解为次模子问题。我们证明，该方法能够改善若干单调集合函数最大化与集合函数比值最小化算法的近似比保证，其计算开销取决于次模秩。