Polynomial-size constant-factor universal families for nonnegative submodular functions

Establish whether there exists a polynomial-size universal family of subsets of an n-element ground set that approximates the maximum of every nonnegative submodular function within a fixed constant factor c>0.

Background

The paper studies universal families of predetermined subsets that can be queried non-adaptively to approximate the maximum value of an unknown nonnegative submodular function. It constructs logarithmic- and subpolynomial-size families with approximation factors that vanish as n grows, and it proves that polynomial-size families achieving a constant factor exist for polynomially representable subclasses of nonnegative submodular functions.

For the full class of nonnegative submodular functions, however, the paper does not determine whether polynomially many predetermined sets suffice for a constant-factor approximation. The lower bounds for pairwise-independent and k-wise-independent distributions show limitations of important candidate constructions but do not resolve the existence of an arbitrary polynomial-size universal family.