When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ITCS.2017.33
URN: urn:nbn:de:0030-drops-81619
URL: https://drops.dagstuhl.de/opus/volltexte/2017/8161/
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### Testing Submodularity and Other Properties of Valuation Functions

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### Abstract

We show that for any constant \epsilon > 0 and p \ge 1, it is possible to distinguish functions f : \{0,1\}^n \to [0,1] that are submodular from those that are \epsilon-far from every submodular function in \ell_p distance with a constant number of queries. More generally, we extend the testing-by-implicit-learning framework of Diakonikolas et al.(2007) to show that every property of real-valued functions that is well-approximated in \ell_2 distance by a class of k-juntas for some k = O(1) can be tested in the \ell_p-testing model with a constant number of queries. This result, combined with a recent junta theorem of Feldman and \Vondrak (2016), yields the constant-query testability of submodularity. It also yields constant-query testing algorithms for a variety of other natural properties of valuation functions, including fractionally additive (XOS) functions, OXS functions, unit demand functions, coverage functions, and self-bounding functions.

### BibTeX - Entry

@InProceedings{blais_et_al:LIPIcs:2017:8161,
author =	{Eric Blais and Abhinav Bommireddi},
title =	{{Testing Submodularity and Other Properties of Valuation Functions}},
booktitle =	{8th Innovations in Theoretical Computer Science Conference (ITCS 2017)},
pages =	{33:1--33:17},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-029-3},
ISSN =	{1868-8969},
year =	{2017},
volume =	{67},