Sketching valuation functions

Ashwinkumar Badanidiyuru*, Shahar Dobzinski, Hu Fu, Robert Kleinberg, Noam Nisan, Tim Roughgarden

*Corresponding author for this work

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

56 Scopus citations

Abstract

Motivated by the problem of querying and communicating bidders' valuations in combinatorial auctions, we study how well different classes of set functions can be sketched. More formally, let f be a function mapping subsets of some ground set [n] to the non-negative real numbers. We say that f′ is an α-sketch of f if for every set S, the value f′(S) lies between f(S)/α and f(S), and f′ can be specified by poly(n) bits. We show that for every subadditive function f there exists an α-sketch where α = n1/2·O(polylog(n)). Furthermore, we provide an algorithm that finds these sketches with a polynomial number of demand queries. This is essentially the best we can hope for since: 1. We show that there exist subadditive functions (in fact, XOS functions) that do not admit an o(n 1/2) sketch. (Balcan and Harvey [3] previously showed that there exist functions belonging to the class of substitutes valuations that do not admit an O(n1/3) sketch.) 2. We prove that every deterministic algorithm that accesses the function via value queries only cannot guarantee a sketching ratio better than n1-ε. We also show that coverage functions, an interesting subclass of submodular functions, admit arbitrarily good sketches. Finally, we show an interesting connection between sketching and learning. We show that for every class of valuations, if the class admits an α-sketch, then it can be α-approximately learned in the PMAC model of Balcan and Harvey. The bounds we prove are only information-theoretic and do not imply the existence of computationally efficient learning algorithms in general.

Original languageAmerican English
Title of host publicationProceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012
PublisherAssociation for Computing Machinery
Pages1025-1035
Number of pages11
ISBN (Print)9781611972108
DOIs
StatePublished - 2012
Event23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012 - Kyoto, Japan
Duration: 17 Jan 201219 Jan 2012

Publication series

NameProceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

Conference

Conference23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012
Country/TerritoryJapan
CityKyoto
Period17/01/1219/01/12

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