TY - GEN

T1 - A unifying hierarchy of valuations with complements and substitutes

AU - Feige, Uriel

AU - Feldman, Michal

AU - Immorlica, Nicole

AU - Izsak, Rani

AU - Lucier, Brendan

AU - Syrgkanis, Vasilis

N1 - Publisher Copyright: Copyright © 2015, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved.

PY - 2015/6/1

Y1 - 2015/6/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84959924889&partnerID=8YFLogxK

M3 - منشور من مؤتمر

T3 - Proceedings of the National Conference on Artificial Intelligence

SP - 872

EP - 878

BT - Proceedings of the 29th AAAI Conference on Artificial Intelligence, AAAI 2015 and the 27th Innovative Applications of Artificial Intelligence Conference, IAAI 2015

PB - AI Access Foundation

T2 - 29th AAAI Conference on Artificial Intelligence, AAAI 2015 and the 27th Innovative Applications of Artificial Intelligence Conference, IAAI 2015

Y2 - 25 January 2015 through 30 January 2015

ER -