A unifying hierarchy of valuations with complements and substitutes

Uriel Feige, Michal Feldman, Nicole Immorlica, Rani Izsak, Brendan Lucier, Vasilis Syrgkanis

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

We introduce a new hierarchy over monotone set functions, that we refer to as MPH (Maximum over Positive Hyper-graphs). Levels of the hierarchy correspond to the degree of complementarity in a given function. The highest level of the hierarchy, MPH-m (where m is the total number of items) captures all monotone functions. The lowest level, MPH-l, captures all monotone submodular functions, and more generally, the class of functions known as XOS. Every monotone function that has a positive hypergraph representation of rank κ (in the sense defined by Abraham, Babaioff, Dughmi and Roughgarden [EC 2012]) is in MPH-κ. Every monotone function that has supermodular degree κ (in the sense defined by Feige and Izsak [ITCS 2013]) is in MPH-( κ+1). In both cases, the converse direction does not hold, even in an approximate sense. We present additional results that demonstrate the expressiveness power of MPH-κ. One can obtain good approximation ratios for some natural optimization problems, provided that functions are required to lie in low levels of the MPH hierarchy. We present two such applications. One shows that the maximum welfare problem can be approximated within a ratio of κ + 1 if all players hold valuation functions in MPH-κ. The other is an upper bound of 2 κ on the price of anarchy of simultaneous first price auctions.

Original languageEnglish
Title of host publicationProceedings of the 29th AAAI Conference on Artificial Intelligence, AAAI 2015 and the 27th Innovative Applications of Artificial Intelligence Conference, IAAI 2015
PublisherAI Access Foundation
Pages872-878
Number of pages7
ISBN (Electronic)9781577357001
StatePublished - 1 Jun 2015
Event29th AAAI Conference on Artificial Intelligence, AAAI 2015 and the 27th Innovative Applications of Artificial Intelligence Conference, IAAI 2015 - Austin, United States
Duration: 25 Jan 201530 Jan 2015

Publication series

NameProceedings of the National Conference on Artificial Intelligence
Volume2

Conference

Conference29th AAAI Conference on Artificial Intelligence, AAAI 2015 and the 27th Innovative Applications of Artificial Intelligence Conference, IAAI 2015
Country/TerritoryUnited States
CityAustin
Period25/01/1530/01/15

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