Supermodular Rank: Set Function Decomposition and Optimization
This work addresses optimization challenges in set functions for researchers in combinatorial optimization and machine learning, offering incremental improvements in approximation algorithms.
The paper tackles the problem of decomposing set functions into supermodular or submodular components, defining supermodular rank as the minimal number of terms, and uses this to optimize set functions by splitting problems into submodular subproblems, improving approximation ratio guarantees for monotone set function maximization and ratio minimization with computational overhead dependent on rank.
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.