Preserve α while improving β in monotone + non-monotone DR-submodular maximization

Determine whether there exists an algorithm for maximizing F(x)=G(x)+H(x) over a solvable down-closed convex polytope P⊂[0,1]^n, where G is non-negative, monotone, DR-submodular and H is non-negative, DR-submodular, that achieves an approximation guarantee of the form F(x)≥α·G(o)+β·H(o)−err with α=1−1/e and β strictly larger than 1/e, without deteriorating α below 1−1/e.

Background

The paper studies maximizing F=G+H, the sum of a monotone DR-submodular function G and a non-monotone DR-submodular function H, over a down-closed convex polytope P. The authors believe a Frank–Wolfe-like approach can achieve α=1−1/e and β=1/e, with α being optimal. They note that better β values are known with more involved algorithms.

However, the authors explicitly state uncertainty about whether β can be improved without lowering α, making this trade-off an open question within their proposed framework.