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We prove that for every D ∈ N, and large enough constant d ∈ N, with high probability over the choice of G ∼ G(n,d/n), the Erdos-Renyi random graph distribution, the canonical degree 2D Sum-of-Squares relaxation fails to certify that the largest ...

We introduce a relaxation for homomorphism problems that combines semidefinite programming with linear Diophantine equations, and propose a framework for the analysis of its power based on the spectral theory of association schemes. We use this framework ...

Relaxations for the constraint satisfaction problem (CSP) include bounded width, linear program (LP), semidefinite program (SDP), affine integer program (AIP), and the combined LP+AIP of Brakensiek, Guruswami, Wrochna, and Živný (SICOMP 2020). Tightening ...

We introduce and study swap cosystolic expansion, a new expansion property of simplicial complexes. We prove lower bounds for swap coboundary expansion of spherical buildings and use them to lower bound swap cosystolic expansion of the LSV Ramanujan ...

Let X be a family of k-element subsets of [n] and let {f_s:s→Σ : s∈ X} be an ensemble of local functions, each defined over a subset s⊂ [n]. Is there a global function G:[n]→Σ such that f_s = G|_s for all s∈ X ? An agreement test is a randomized property ...

A d-dimensional simplicial complex X is said to support a direct product tester if any locally consistent function defined on its k-faces (where k≪ d) necessarily come from a function over its vertices. More precisely, a direct product tester has a ...