License
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://drops.dagstuhl.de/opus/volltexte/2016/5838/
Go to the corresponding LIPIcs Volume Portal


Lee, Troy ; Prakash, Anupam ; de Wolf, Ronald ; Yuen, Henry

On the Sum-of-Squares Degree of Symmetric Quadratic Functions

pdf-format:
17.pdf (0.7 MB)


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}
}

Keywords: Sum-of-squares degree, approximation theory, Positivstellensatz refutations of knapsack, quantum query complexity in expectation, extension complexity
Seminar: 31st Conference on Computational Complexity (CCC 2016)
Issue Date: 2016
Date of publication: 18.05.2016


DROPS-Home | Fulltext Search | Imprint Published by LZI