Supermodular Rank: Set Function Decomposition and Optimization

The paper introduces the concept of supermodular rank, defined as the minimal number of supermodular components required to express a function on a lattice. It characterizes the maximum possible rank and the structure of functions with a fixed rank, and analogously defines submodular rank. Leveraging submodular decompositions, the authors propose an algorithm that partitions a set‑function optimization problem into submodular subproblems, thereby improving approximation guarantees for monotone maximization and ratio minimization while incurring a computational cost tied to the submodular rank.

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