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Title: On the sum-of-squares degree of symmetric quadratic functions
Authors: de Wolf, Ronald
Yuen, Henry
Lee, Troy
Prakash, Anupam
Keywords: Sum-of-squares Degree
Approximation Theory
Issue Date: 2016
Source: Lee, T., Prakash, A., de Wolf, R., & Yuen, H. (2016). On the sum-of-squares degree of symmetric quadratic functions. Leibniz International Proceedings in Informatics, 50, 17-. doi:10.4230/LIPIcs.CCC.2016.17
Series/Report no.: Leibniz International Proceedings in Informatics
Abstract: We study how well functions over the boolean hypercube of the form f_k(x)=(lxl-k)(lxl-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.
DOI: 10.4230/LIPIcs.CCC.2016.17
Rights: © 2016 The Author(s) (Leibniz International Proceedings in Informatics). Licensed under Creative Commons License CC-BY.
Fulltext Permission: open
Fulltext Availability: With Fulltext
Appears in Collections:SPMS Journal Articles

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