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...

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Bibliographic Details
Published in:Evolutionary computation Vol. 23; no. 4; pp. 543 - 558
Main Authors: Friedrich, Tobias, Neumann, Frank
Format: Journal Article
Language:English
Published: United States 01.12.2015
Subjects:
Algorithms
Biological Evolution
Computational Biology
Computer Simulation
Humans
Models, Statistical
multiobjective optimization
approximation
maximum cut
Submodular functions
matroid constraints
runtime
hypervolume indicator
theory
ISSN:1530-9304
Online Access:Get more information
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Summary: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.
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ISSN:1530-9304
DOI:10.1162/EVCO_a_00159