Are Gross Substitutes a Substitute for Submodular Valuations?

Abstract

The class of gross substitutes (GS) set functions plays a central role in Economics and Computer Science. GS belongs to the hierarchy of complement free valuations introduced by Lehmann, Lehmann and Nisan, along with other prominent classes: GS ⊊ Submodular ⊊ XOS ⊊ Subadditive. The GS class has always been more enigmatic than its counterpart classes, both in its definition and in its relation to the other classes. For example, while it is well understood how closely the Submodular, XOS and Subadditive classes (point-wise) approximate one another, approximability of these classes by GS remained wide open. In particular, the largest gap known between Submodular and GS valuations was some constant ratio smaller than 2.Our main result is the existence of a submodular valuation (one that is also budget additive) that cannot be approximated by GS within a ratio better than Ω( log m log log m ), where m is the number of items. En route, we uncover a new symmetrization operation that preserves GS, which may be of independent interest.We show that our main result is tight with respect to budget additive valuations. However, whether GS approximates general submodular valuations within a poly-logarithmic factor remains open, even in the special case of concave of GS valuations (a subclass of Submodular containing budget additive). For concave of Rado valuations (Rado is a significant subclass of GS, containing, e.g., weighted matroid rank functions and OXS), we show approximability by GS within an O (log 2 m) factor. CCS Concepts: • Theory of computation → Algorithmic mechanism design.

Fair Division of Indivisible Goods for a Class of Concave Valuations

“…We note that CASC is more general than ASC by setting c i ≥ j∈M k j =1 u i,j, . They are a special case of submodular functions but incomparable to gross substitutes functions, which are studied frequently especially in the special case of capped additive functions; see Dobzinski et al (2021), Feldman et al (2016), , Roughgarden and Talgam-Cohen (2015).…”

Constrained Submodular Maximization via a Nonsymmetric Technique

“…Submodular maximization has also been studied in various computational models, such as online and secretary (random arrival) settings [8,15,19,45], streaming [2,21,41,44,59,71], parallel computation [6,7,23,24] and distributed computing [74,83]. Some works also took a game theoretic perspective on submodular optimization [3,32,77].…”