# Questions tagged [sum-of-squares]

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### Complexity of (Graph) Ramsey Theorem in Sum-of-Squares Proof System

(One formulation of) Ramsey's theorem states that any colouring of edges of the complete graph with $4^n$ vertices with two colours will contain a monochromatic clique of size $n$. I am new to proof ...
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### Sum-of-Squares Certificates

We say that $f$ has a degree $2d$ sum-of-squares certificate if $f=\sum_{i=1}^r (g_i(x))^2$, where for each $i\in[r]$, we have that $g_i$ is a polynomial of degree at most $d$. Thus showing that $f$ ...
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### Are those two Sum-Of-Squares approach for unconstrained polynomial optimization related? [closed]

This is a crosspost of mathoverflow/345282 I found 2 approaches to solve an unconstrained polynomial optimization problem using the Lasserre / SOS hierarchy: $$\inf_{x\in\mathbb{R}^n}\quad p(x)$$ ...
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### Testing emptiness property complexity in Sum of Squares Proof systems

Take the set $$\mathcal T=\{f_1(x_1,\dots,x_n)=\dots=f_m(x_1,\dots,x_n)=0, h_1(x_1,\dots,x_n)\geq a_1,\dots,h_t(x_1,\dots,x_n)\geq a_t\}$$ where h_1(x_1,\dots,x_n),\dots,h_t(x_1,\dots,x_n)\in\mathbb ...
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1 vote
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### Some questions about the Ryan O'Donnel and Yuan Zhou's paper "Approximability and proof complexity"

My question is particularly about the set-up in section $8$ (Analysis of the KV Max-Cut instances") of the paper, https://arxiv.org/pdf/1211.1958.pdf. What they call the Khot-Vishnoi UG instance ...
• 1,443
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### Can the Lasserre relaxation be defined over the reals?

If one wants to say minimize a function $f : \{-1,1\}^n \rightarrow \mathbb{R}$ on its domain then a degree$-d$ Lasserre relaxation of it would be to solve the problem of $\min \mathbb{E}_\mu [f(x)]$ ...
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### SOS and the small set expansion property

For what graphs do we know that their small set expansion property has a low degree SOS proof? Is this known to be true for say the complete graphs? A terminology issue about what is low degree" :...
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### Is there a relationship between the probabilistic interepretation of Sherali-Adams SDP hierarchy and the Lasserre SDP hierarchy?

Firstly note this paper http://ttic.uchicago.edu/~madhurt/Papers/reductions.pdf where a Lasserre SDP is being setup for the independent set probblem at the bottom of page 4 where the author says says, ...
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### SOS hardness of $Max-2-Lin(\mathbb{Z}_2)$?

Do we know of instances of $Max-2-Lin(\mathbb{Z}_2)$ which have a integrality gaps w.r.t to high degree (> 4) SOS relaxations? Or if we specialize to Max-CUT do we know of graphs whose Max-CUT ...
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1 vote
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### Numerical precision in sum-of-squares method?

I have been reading a bit about the sum-of-squares method (SOS) from the survey of Barak & Steurer and the lecture notes of Barak. In both cases they sweep issues of numerical accuracy under the ...
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### Sum-of-squares proof system

Recently I have seen several articles on arxiv that refer to a proof system called sum-of-squares. Can someone explain what is a sum-of-squares proof and why such proofs are important/interesting? ...
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