Faster Guarantees of Evolutionary Algorithms for Maximization of Monotone Submodular Functions

In this paper, the monotone submodular maximization problem (SM) is studied. SM is to find a subset of size kappa from a universe of size n that maximizes a monotone submodular objective function f. We show using a novel analysis that the Pareto optimization algorithm achieves a worst-case ratio of (1-ε)(1-1/e) in expectation for every cardinality constraint kappa < P, where P≤ n+1 is an input, in O(nPln(1/ε)) queries of f. In addition, a novel evolutionary algorithm called the biased Pareto optimization algorithm, is proposed that achieves a worst-case ratio of (1-ε)(1-1/e) in expectation for every cardinality constraint kappa < P in O(nln(P)ln(1/ε)) queries of f. Further, the biased Pareto optimization algorithm can be modified in order to achieve a worst-case ratio of (1-ε)(1-1/e) in expectation for cardinality constraint kappa in O(nln(1/ε)) queries of f. An empirical evaluation corroborates our theoretical analysis of the algorithms, as the algorithms exceed the stochastic greedy solution value at roughly when one would expect based upon our analysis.