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See the paper by Julia Chuzhoy and myself on Treewidth sparsifiers.
We show that one can obtain a subgraph of degree at most 3 with treewidth $\Omega(k/polylog(k))$ where $k$ is the treewidth of $G$.
https://arxiv.org/abs/1410.1016 The proof is shorter than the one
for grid minors but it is still not that that easy and builds on several previous
tools.
...
11
It's not really needed, so much as it is a matter of convention and utility. Of course, depending on your aims and your specific problem, it is completely reasonable to consider arithmetic circuits of polynomial size regardless of degree. This class is often denoted $\mathsf{VP}_{nb}$ (for "Non-degree-Bounded"), or sometimes "algebraic $\mathsf{P/poly}$ or $\...
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Here's an algorithm that beats the trivial attempts.
The following is a known fact (Exercise 1.12 in O'Donnell's book) : If $f:\{-1,1\}^n\to\{-1,1\}$ is a Boolean function which has degree $\le d$ as a polynomial, then every Fourier coefficient of $f$, $\hat{f}(S)$ is an integer multiple of $2^{-d}$.
Using Cauchy-Schwarz and Parseval one gets that there are ...
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