Rewarding Maps: On Greedy Optimization of Set Functions

Abstract

Given a set function, that is, a map ƒ: P ( E) → R ≔ R ∪ {−∞} from the set P ( E) of subsets of a finite set E into the reals including −∞, the standard greedy algorithm (GA) for optimizing ƒ starts with the empty set and then proceeds by enlarging this set greedily, element by element. A set function ƒ is said to be tractable if in this way a sequence x 0 ≔ ∅, x 1, ., x N ≔ E ( N ≔ # E) of subsets with max(ƒ) ∈ {ƒ( x 0), ƒ( x 1), ., ƒ( x N )} will always be found. In this note, we will reinterpret and transcend the traditions of classical GA-theory (cf., e.g., [KLS]) by establishing necessary and sufficient conditions for a set function ƒ not just to be tractable as it stands, but to give rise to a whole family of tractable set functions ƒ(η) : P ( E) → R : x ↦ ƒ( x) + Σ e ∈ x η( e), where η runs through all real valued weighting schemes η : E → R , in which case ƒ will be called rewarding. In addition, we will characterize two important subclasses of rewarding maps, viz. truncatably rewarding (or well-layered) maps, that is, set functions ƒ such that [formula] is rewarding for every i = 1, ., N, and matroidal maps, that is, set functions ƒ such that for every η : E → R and every ƒ eta-greedy sequence x 0, x 1, ., x N as above, one has max(ƒ η) = ƒ η( x i ) for the unique i ∈ {0, ., N} with ƒ η( x 0) < ƒ η( x 1) < ··· < ƒ η( x i ) ≥ ƒ η( x i + 1 ).