  We study a pair of budget- and performance-constrained weak submodular
maximization problems. For computational efficiency, we explore the use of
stochastic greedy algorithms which limit the search space via random sampling
instead of the standard greedy procedure which explores the entire feasible
search space. We propose a pair of stochastic greedy algorithms, namely,
Modified Randomized Greedy (MRG) and Dual Randomized Greedy (DRG) to
approximately solve the budget- and performance-constrained problems,
respectively. For both algorithms, we derive approximation guarantees that hold
with high probability. We then examine the use of DRG in robust optimization
problems wherein the objective is to maximize the worst-case of a number of
weak submodular objectives and propose the Randomized Weak Submodular
Saturation Algorithm (Random-WSSA). We further derive a high-probability
guarantee for when Random-WSSA successfully constructs a robust solution.
Finally, we showcase the effectiveness of these algorithms in a variety of
relevant uses within the context of Earth-observing LEO constellations which
estimate atmospheric weather conditions and provide Earth coverage.
