Optimizing Chance-Constrained Submodular Problems with Variable Uncertainties

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

Maximization of Non-Monotone Submodular Functions

A litany of questions from a wide variety of scientific disciplines can be cast as non-monotone submodular maximization problems. Since this class of problems includes max-cut, it is NP-hard. Thus, general purpose algorithms for the class tend to be approximation algorithms. For unconstrained problem instances, one recent innovation in this vein includes an algorithm of Buchbinder et al. (2012) that guarantees a ½ - approximation to the maximum. Building on this, for problems subject to cardinality constraints, Buchbinderet al. (2014) o_er guarantees in the range [0:356; ½ + o(1)]. Earlier work has the best approximation factors for more complex constraints and settings. For constraints that can be characterized as a solvable polytope, Chekuri et al. (2011) provide guarantees. For the online secretary setting, Gupta et al. (2010) provide guarantees. In sum, the current body of work on non-monotone submodular maximization lays strong foundations. However, there remains ample room for future algorithm development