eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2016-05-19
17:1
17:31
10.4230/LIPIcs.CCC.2016.17
article
On the Sum-of-Squares Degree of Symmetric Quadratic Functions
Lee, Troy
Prakash, Anupam
de Wolf, Ronald
Yuen, Henry
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol050-ccc2016/LIPIcs.CCC.2016.17/LIPIcs.CCC.2016.17.pdf
Sum-of-squares degree
approximation theory
Positivstellensatz refutations of knapsack
quantum query complexity in expectation
extension complexity