Faster Guarantees of Evolutionary Algorithms for Maximization of Monotone Submodular Functions

Abstract

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 $(1-\epsilon )(1-1/e)$ in expectation for every cardinality constraint $\kappa <P$, where $P\le n+1$ is an input, in $O(nP\ln (1/\epsilon ))$ queries of $f$. In addition, a novel evolutionary algorithm called the biased Pareto optimization algorithm, is proposed that achieves a worst-case ratio of $(1-\epsilon )(1-1/e)$ in expectation for every cardinality constraint $\kappa <P$ in $O(n\ln (P)\ln (1/\epsilon ))$ queries of $f$. Further, the biased Pareto optimization algorithm can be modified in order to achieve a worst-case ratio of $(1-\epsilon )(1-1/e)$ in expectation for cardinality constraint $\kappa$ in $O(n\ln (1/\epsilon ))$ 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.

arXiv commentTo be presented as a full paper at IJCAI 2021