  Chance constraints are frequently used to limit the probability of constraint
violations in real-world optimization problems where the constraints involve
stochastic components. We study chance-constrained submodular optimization
problems, which capture a wide range of optimization problems with stochastic
constraints. Previous studies considered submodular problems with stochastic
knapsack constraints in the case where uncertainties are the same for each item
that can be selected. However, uncertainty levels are usually variable with
respect to the different stochastic components in real-world scenarios, and
rigorous analysis for this setting is missing in the context of submodular
optimization. This paper provides the first such analysis for this case, where
the weights of items have the same expectation but different dispersion. We
present greedy algorithms that can obtain a high-quality solution, i.e., a
constant approximation ratio to the given optimal solution from the
deterministic setting. In the experiments, we demonstrate that the algorithms
perform effectively on several chance-constrained instances of the maximum
coverage problem and the influence maximization problem.
