Are Gross Substitutes a Substitute for Submodular Valuations?
Authors: 
Shahar Dobzinski, Uriel Feige, and Michal FeldmanAuthors Info & Claims
Shahar Dobzinski, Uriel Feige, and Michal FeldmanAuthors Info & ClaimsJuly 2021
Pages 390 - 408
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 $Ømega(łog m/łogłog 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(łog2m) factor.
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- Are Gross Substitutes a Substitute for Submodular Valuations?
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July 2021
950 pages
ISBN:9781450385541
DOI:10.1145/3465456
- General Chair:
- Péter Biró,
- Program Chairs:
- Shuchi Chawla,
- Federico Echenique
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Published: 18 July 2021
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