Lee, Troy
[Author];
Prakash, Anupam
[Author];
de Wolf, Ronald
[Author];
Yuen, Henry
[Author]
;
Troy Lee and Anupam Prakash and Ronald de Wolf and Henry Yuen
[Contributor]
On the Sum-of-Squares Degree of Symmetric Quadratic Functions
Footnote:
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Description:
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.