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When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.CCC.2016.17
URN: urn:nbn:de:0030-drops-58383
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2016/5838/
Lee, Troy ;
Prakash, Anupam ;
de Wolf, Ronald ;
Yuen, Henry
On the Sum-of-Squares Degree of Symmetric Quadratic Functions
Abstract
We study how well functions over the boolean hypercube of the form f_k(x)=(|x|-k)(|x|-k-1) can be approximated by sums of squares of low-degree polynomials, obtaining good bounds for the case of approximation in l_{infinity}-norm as well as in l_1-norm. We describe three complexity-theoretic applications: (1) a proof that the recent breakthrough lower bound of Lee, Raghavendra, and Steurer [Lee/Raghavendra/Steurer, STOC 2015] on the positive semidefinite extension complexity of the correlation and TSP polytopes cannot be improved further by showing better sum-of-squares degree lower bounds on l_1-approximation of f_k; (2) a proof that Grigoriev's lower bound on the degree of Positivstellensatz refutations for the knapsack problem is optimal, answering an open question from [Grigoriev, Comp. Compl. 2001]; (3) bounds on the query complexity of quantum algorithms whose expected output approximates such functions.
BibTeX - Entry
@InProceedings{lee_et_al:LIPIcs:2016:5838,
author = {Troy Lee and Anupam Prakash and Ronald de Wolf and Henry Yuen},
title = {{On the Sum-of-Squares Degree of Symmetric Quadratic Functions}},
booktitle = {31st Conference on Computational Complexity (CCC 2016)},
pages = {17:1--17:31},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-008-8},
ISSN = {1868-8969},
year = {2016},
volume = {50},
editor = {Ran Raz},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2016/5838},
URN = {urn:nbn:de:0030-drops-58383},
doi = {10.4230/LIPIcs.CCC.2016.17},
annote = {Keywords: Sum-of-squares degree, approximation theory, Positivstellensatz refutations of knapsack, quantum query complexity in expectation, extension complexity}
}
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Keywords: |
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Sum-of-squares degree, approximation theory, Positivstellensatz refutations of knapsack, quantum query complexity in expectation, extension complexity |
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Collection: |
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31st Conference on Computational Complexity (CCC 2016) |
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Issue Date: |
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2016 |
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Date of publication: |
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19.05.2016 |
