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Revision #1 to TR20-170 | 18th July 2021 03:46

#### High Dimensional Expanders: Eigenstripping, Pseudorandomness, and Unique Games

Revision #1
Authors: Mitali Bafna, Max Hopkins, Tali Kaufman, Shachar Lovett
Accepted on: 18th July 2021 03:46
Keywords:

Abstract:

Higher order random walks (HD-walks) on high dimensional expanders (HDX) have seen an incredible amount of study and application since their introduction by Kaufman and Mass (ITCS 2016), yet their broader combinatorial and spectral properties remain poorly understood. We develop a combinatorial characterization of the spectral structure of HD-walks on two-sided local-spectral expanders (Dinur and Kaufman FOCS 2017), which offer a broad generalization of the well-studied Johnson and Grassmann graphs. Our characterization, which shows that the spectra of HD-walks lie tightly concentrated in a few combinatorially structured strips, leads to novel structural theorems such as a tight $\ell_2$-characterization of edge-expansion, as well as to a new understanding of local-to-global graph algorithms on HDX.

Towards the latter, we introduce a novel spectral complexity measure called \textit{Stripped Threshold Rank}, and show how it can replace the (much larger) threshold rank as a parameter controlling the performance of algorithms on structured objects. Combined with a sum-of-squares proof for the former $\ell_2$-characterization, we give a concrete application of this framework to algorithms for unique games on HD-walks, where in many cases we improve the state of the art (Barak, Raghavendra, and Steurer FOCS 2011, and Arora, Barak, and Steurer JACM 2015) from nearly-exponential to polynomial time (e.g.\ for sparsifications of Johnson graphs or of slices of the $q$-ary hypercube). Our characterization of expansion also holds an interesting connection to hardness of approximation, where an $\ell_\infty$-variant for the Grassmann graphs was recently used to resolve the 2-2 Games Conjecture (Khot, Minzer, and Safra FOCS 2018). We give a reduction from a related $\ell_\infty$-variant to our $\ell_2$-characterization, but it loses factors in the regime of interest for hardness where the gap between $\ell_2$ and $\ell_\infty$ structure is large. Nevertheless, our results open the door for further work on the use of HDX in hardness of approximation and their general relation to unique games.

Changes to previous version:

New version includes UG algorithm for HD-walks over all two-sided local-spectral expanders (previously only over the complete complex). It also includes tighter structural theorems and simplified proofs.

### Paper:

TR20-170 | 9th November 2020 00:07

#### High Dimensional Expanders: Random Walks, Pseudorandomness, and Unique Games

TR20-170
Authors: Max Hopkins, Tali Kaufman, Shachar Lovett
Publication: 11th November 2020 19:13