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

Abstract

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 \(\mathcal{O}(n^2\,(\log n+k))\), 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 \(\mathcal{O}(n^{k+5}\log (n) / \varepsilon )\).KeywordsCombinatorial Optimization ProblemSubmodular FunctionOracle AccessEvolutionary Multiobjective OptimizationHypervolume IndicatorThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.