Symmetry and Approximability of Submodular Maximization Problems

Abstract— A number of recent results on optimization problems involving submodular functions have made use of the ”multilinear relaxation” of the problem [3], [8], [24], [14], [13]. We present a general approach to deriving inapproximability results in the value oracle model, based on the notion of ”symmetry gap”. Our main result is that for any fixed instance that exhibits a certain ”symmetry gap” in its multilinear relaxation, there is a naturally related class of instances for which a better approximation factor than the symmetry gap would require exponentially many oracle queries. This unifies several known hardness results for submodular maximization, e.g. the optimality of (1 − 1/e)-approximation for monotone submodular maximization under a cardinality constraint [20], [7], and the impossibility of (1 2 + )-approximation for unconstrained (non-monotone) submodular maximization [8]. It follows from our result that (1 2 + )-approximation is also impossible for non...

• Type: Conference
• Year: 2009
• Where: FOCS
• Authors: Jan Vondrák
• Keywords: FOCS 2009, Monotone Submodular Maximization, Submodular Maximization, Theoretical Computer Science, ”symmetry Gap