# On the sum-of-squares degree of symmetric quadratic functions

@inproceedings{Lee2016OnTS, title={On the sum-of-squares degree of symmetric quadratic functions}, author={Troy Lee and Anupam Prakash and Ronald de Wolf and Henry S. Yuen}, booktitle={Computational Complexity Conference}, year={2016} }

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 $\ell_{\infty}$-norm as well as in $\ell_1$-norm. We describe three complexity-theoretic applications: (1) a proof that the recent breakthrough lower bound of Lee, Raghavendra, and Steurer on the positive semidefinite extension complexity of the correlation and TSP polytopes cannot be…

## 12 Citations

Symmetric sums of squares over k-subset hypercubes

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A variant of the Gatermann–Parrilo symmetry-reduction method tailored to the authors' setting that allows for several simplifications and a connection to flag algebras is developed, and every symmetric polynomial that has a sos expression of a fixed degree also has a succinct sosexpression whose size depends only on the degree and not on the number of variables.

SoS Certification for Symmetric Quadratic Functions and Its Connection to Constrained Boolean Hypercube Optimization

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It is proved that the SoS rank for SQFs is at most $O(\sqrt{nk}\log(n)$ and connected to two constrained Boolean hypercube optimization problems.

Sum-Of-Squares Bounds via Boolean Function Analysis

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A method for proving bounds on the SoS rank based on Boolean Function Analysis and Approximation Theory is introduced and applied to improve upon existing results, thus making progress towards answering several open questions.

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- Mathematics, Computer ScienceComputational Complexity Conference
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It is shown that any real-valued harmonic multilinear polynomial on the slice whose degree is o(n) has approximately the same distribution under the slice and cube measures.

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This paper proves a tight sum of squares lower bound for the following Turan type problem: Minimize the number of triangles in a graph $G$ with a fixed edge density.

Partial Boolean Functions With Exact Quantum Query Complexity One

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It is shown that the number of all n-bit partial Boolean functions that depend on k bits and can be computed exactly by a 1-query quantum algorithm is not bigger than an upper bound depending on n and k.

A ug 2 01 6 Exact quantum query complexity of EXACT nk , l

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The minimum number of queries for an exact quantum algorithm computing the function f is denoted by QE(f), and the following n bit function with 0 ≤ k ≤ l ≤ n is considered.

Optimal one-shot quantum algorithm for EQUALITY and AND

- Computer ScienceBalt. J. Mod. Comput.
- 2016

It is shown that the lowest possible error probability for $AND_n$ and $EQUALITY_{n+1}$ is $1/2-n/(n^2+1)$.

Lower Bounds for Interactive Compression and Linear Programs

- Computer Science
- 2018

A new technique, the notion of fooling distributions, is introduced to prove that information can be exponentially smaller than communication, and a tight non-negative rank lower bound is obtained for a family of matrices known as lopsided unique disjointness.

Exact Quantum Query Complexity of \text EXACT_k, l^n

- Computer Science, MathematicsSOFSEM
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An optimal algorithm (for some cases), and a nontrivial general lower and upper bound on the minimum number of queries to the black box.

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