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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Vydáno v:Evolutionary computation Ročník 23; číslo 4; s. 543 - 558
Hlavní autoři: Friedrich, Tobias, Neumann, Frank
Médium: Journal Article
Jazyk:angličtina
Vydáno: United States 01.12.2015
Témata:
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
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Shrnutí: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.
Bibliografie:ObjectType-Article-1
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ISSN:1530-9304
DOI:10.1162/EVCO_a_00159