Maximizing Submodular Functions under Matroid Constraints by Multi-objective Evolutionary Algorithms

Many combinatorial optimization problems have underlying goal functions that are submodular. The classical goal is to find a good solution for a given submodular function f under a given set of constraints. In this paper, we investigate the runtime of a multi-objective evolutionary algorithm called GSEMO until it has obtained a good approximation for submodular functions. For the case of monotone submodular functions and uniform cardinality constraints we show that GSEMO achieves a (1 − 1/e)-approximation in expected time O(n (logn + k)), where k is the value of the given constraint. For the case of non-monotone submodular functions with k matroid intersection constraints, we show that GSEMO achieves a 1/(k + 2 + 1/k + ε)-approximation in expected time O(n log(n)/ε).