For a-reduced submodular functions the supremum λ of singleton ratios satisfies Ω(n/log n) ≤ λ ≤ doubly exponential in n.
Learning with Submodular Functions: A Convex Optimization Perspective
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On the Supremum of Singleton Ratios in Submodular Functions
For a-reduced submodular functions the supremum λ of singleton ratios satisfies Ω(n/log n) ≤ λ ≤ doubly exponential in n.