Optimizing Chance-Constrained Submodular Problems with Variable Uncertainties

TL;DR

This paper introduces new greedy algorithms for chance-constrained submodular optimization problems with variable uncertainties, providing theoretical guarantees and demonstrating effectiveness on coverage and influence maximization tasks.

Contribution

It is the first to analyze and develop algorithms for submodular problems with stochastic constraints where uncertainties vary across items, despite having the same expected value.

Findings

Algorithms achieve constant approximation ratios.

Effective performance on maximum coverage instances.

Successful application to influence maximization.

Abstract

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.