Maximizing Submodular Functions under Matroid Constraints by 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 simple single objective evolutionary algorithm called (1 + 1) EA and a multiobjective evolutionary algorithm called GSEMO until they have obtained a good approximation for submodular functions. For the case of monotone submodular functions and uniform cardinality constraints, we show that the GSEMO achieves a (1 - 1/e)-approximation in expected polynomial time. For the case of monotone functions where the constraints are given by the intersection of K ≥ 2 matroids, we show that the (1 + 1) EA achieves a (1/k + δ)-approximation in expected polynomial time for any constant δ > 0. Turning to nonmonotone symmetric submodular functions with k ≥ 1 matroid intersection constraints, we show that the GSEMO achieves a 1/((k + 2)(1 + ε))-approximation in expected time O(n(k + 6)log(n)/ε.