Well-layered maps—A class of greedily optimizable set functions Original Research Article

Given a finite set E of cardinality, say, N and a map ƒ:P(E) →View the MathML source ≔ View the MathML source∪{−∞}, we define a sequence e1, …, eN of elements of E to be ƒ-greedy if ei ∉ {e1, …, ei−1} and View the MathML source for all i ∈ {1, …, N} and all e ∈ E⧸{e1, …, ei−1}. In addition, for any map η : E →View the MathML source we put View the MathML source for all x ⊆ E, and we define ƒ to be a well-layered map if for every η : E →View the MathML source, every View the MathML source-greedy sequence e1, …, eN ∈ E and every layer Pi(E) ≔ {y ⊆ E | #y = i} (i = 0, …, N) we have View the MathML source. In this note, we characterize well-layered maps by some appropriate quantified version of the greedoid exchange condition or—equivalently—by some appropriate greedoidal version of the quantitative relations defining valuated matroids and valuated Δ-matroids, respectively.