Sum of Squares Lower Bounds from Pairwise Independence

We prove that for every $\epsilon>0$ and predicate $P:{0,1}^k\rightarrow {0,1}$ that supports a pairwise independent distribution, there exists an instance $\mathcal{I}$ of the $\mathsf{Max}P$ constraint satisfaction problem on $n$ variables such that no assignment can satisfy more than a $\tfrac{|P^{-1}(1)|}{2^k}+\epsilon$ fraction of $\mathcal{I}$’s constraints but the degree $\Omega(n)$ Sum of Squares semidefinite programming hierarchy cannot certify that $\mathcal{I}$ is unsatisfiable. Similar results were previously only known for weaker hierarchies.

💡 Research Summary

The paper “Sum of Squares Lower Bounds from Pairwise Independence” establishes strong integrality‑gap results for the Sum‑of‑Squares (SOS) semidefinite programming hierarchy, a powerful convex relaxation framework for constraint satisfaction problems (CSPs). The authors focus on Boolean predicates P : {0,1}^k → {0,1} that admit a pairwise independent distribution μ supported on the satisfying assignments P^{-1}(1). For any ε>0 they construct, for sufficiently large n, a Max‑P instance I on n variables with the following two properties:

1. Soundness – Every assignment satisfies at most |P^{-1}(1)|/2^k + ε fraction of the constraints. This is achieved by taking a random k‑uniform hypergraph with m=Θ(n) hyperedges (each hyperedge corresponds to a k‑tuple of literals). By choosing m large enough and pruning a negligible fraction of edges, the hypergraph has girth Ω(log n) and strong expansion, guaranteeing that for any fixed assignment the induced distribution on the k‑tuples of literals is ε‑close to uniform (Chernoff bound + union bound).

2. SOS Completeness – The degree‑d SOS relaxation, with d=Ω(n), cannot certify that the instance is unsatisfiable; its optimum value is 1. To prove this, the authors explicitly construct a degree‑d pseudo‑expectation operator ˜E that satisfies all SOS constraints. For each variable set S with |S|≤d they define an actual probability distribution ν_S over {±1}^S. ν_S is built by “growing” from a root clause using the pairwise independent distribution μ; because μ’s marginals are uniform, the order of traversal does not affect the resulting distribution. For a general S, they first take its minimal “closed” superset cl(S) (a set that contains whole clauses or intersects each clause in at most one variable) and define ν_S as the projection of ν_{cl(S)}. This construction guarantees local consistency: if S⊆U then the projection of ν_U onto S equals ν_S, and each clause contained in S has marginal μ.

The pseudo‑expectation is then defined on the basis functions χ_S(x)=∏_{i∈S}x_i by ˜E