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Evolutionary Computation

Winter 2015, Vol. 23, No. 4, Pages 543-558
(doi: 10.1162/EVCO_a_00159)
© 2015 Massachusetts Institute of Technology
Maximizing Submodular Functions under Matroid Constraints by Evolutionary Algorithms
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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 ≥ 2 matroids, we show that the (1 +1) EA achieves a (1/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 .