Contemplating the recently announced 1-local expanders of Viola and Wigderson (ECCC, TR16-129, 2016), one may observe that weaker constructs are well know. For example, one may easily obtain a 4-regular $N$-vertex graph with spectral gap that is $\Omega(1/\log^2 N)$, and similarly a $O(1)$-regular $N$-vertex graph with spectral gap $1/\tildeO(\log N)$.
Starting from a generic candidate for a 1-local expander, we formulate a natural problem regarding coordinated random walks (CRW) on the corresponding relocation graph (which has size that is logarithmic in the size of the candidate 1-local graph), and observe that
any solution to the CRW problem yields 1-local expanders,
and
any constant-size expanding set of generators for the symmetric group yields a solution to the CRW problem.
This yields an alternative construction and different analysis than the one used by Viola and Wigderson.
Furthermore, we show that solving the CRW problem is equivalent to constructing 1-local expanders.
Fixing various minor errors and adding clarifications.
Contemplating the recently announced 1-local expanders of Viola and Wigderson (ECCC, TR16-129, 2016), one may observe that weaker constructs are well know. For example, one may easily obtain a 4-regular $N$-vertex graph with spectral gap that is $\Omega(1/\log^2 N)$, and similarly a $O(1)$-regular $N$-vertex graph with spectral gap $1/\tildeO(\log N)$.
Starting from a generic candidate for a 1-local expander, we formulate a natural problem regarding coordinated random walks (CRW) on the corresponding relocation graph (which has size that is logarithmic in the size of the candidate 1-local graph), and observe that
any solution to the CRW problem yields 1-local expanders,
and
any constant-size expanding set of generators for the symmetric group yields a solution to the CRW problem.
This yields an alternative construction and different analysis than the one used by Viola and Wigderson.
Furthermore, we show that solving the CRW problem is equivalent to constructing 1-local expanders.
