Polynomial-time maximization of ordinally weak-concave functions with a value oracle

Determine whether there exists a polynomial-time algorithm that, given only function evaluation oracle access to an ordinally weak-concave function u: 2^E -> R (as defined in Definition 1.2), computes a global maximizer over all subsets of the finite ground set E; furthermore, ascertain this polynomial-time solvability even for some special subclass of ordinally weak-concave functions.

Background

The paper introduces ordinal weak-concavity (a relaxation of ordinal concavity) for set functions u: 2^E -> R and develops several structural results, including a characterization via M-convexity of maximizer sets on intervals and a local-to-global optimality property. These enable a simple hill-climbing method (via neighborhood search) to find a global maximizer, but no polynomial-time bound is provided for arbitrary ordinally weak-concave functions under a value oracle model.

In contrast, for ordinally concave functions the authors present a dedicated algorithm that finds a maximizer using O(|E|^2) function evaluations. This contrast motivates the explicit open question of whether maximization of ordinally weak-concave functions is tractable in polynomial time under oracle access, or at least for some identifiable special subclasses.