# Approximating Constraint Satisfaction Problems on High-Dimensional Expanders

## TL;DR

This paper demonstrates that high-dimensional expanders enable efficient approximation of certain constraint satisfaction problems (CSPs) using sum-of-squares hierarchies, contrasting with the hardness of random instances.

## Contribution

It extends approximation results for 2-CSPs to high-dimensional expanders for k-CSPs, introducing a threshold-rank notion and analyzing spectral expansion parameters.

## Key findings

- High-dimensional expanders allow approximation of MAX k-CSPs with small spectral parameter γ.
- Sum-of-squares hierarchy levels needed depend on γ and threshold-rank.
- High-dimensional expanders with small γ have threshold rank 1, enabling efficient approximation.

## Abstract

We consider the problem of approximately solving constraint satisfaction problems with arity $k > 2$ ($k$-CSPs) on instances satisfying certain expansion properties, when viewed as hypergraphs. Random instances of $k$-CSPs, which are also highly expanding, are well-known to be hard to approximate using known algorithmic techniques (and are widely believed to be hard to approximate in polynomial time). However, we show that this is not necessarily the case for instances where the hypergraph is a high-dimensional expander. We consider the spectral definition of high-dimensional expansion used by Dinur and Kaufman [FOCS 2017] to construct certain primitives related to PCPs. They measure the expansion in terms of a parameter $\gamma$ which is the analogue of the second singular value for expanding graphs. Extending the results by Barak, Raghavendra and Steurer [FOCS 2011] for 2-CSPs, we show that if an instance of MAX k-CSP over alphabet $[q]$ is a high-dimensional expander with parameter $\gamma$, then it is possible to approximate the maximum fraction of satisfiable constraints up to an additive error $\epsilon$ using $q^{O(k)} \cdot (k/\epsilon)^{O(1)}$ levels of the sum-of-squares SDP hierarchy, provided $\gamma \leq \epsilon^{O(1)} \cdot (1/(kq))^{O(k)}$. Based on our analysis, we also suggest a notion of threshold-rank for hypergraphs, which can be used to extend the results for approximating 2-CSPs on low threshold-rank graphs. We show that if an instance of MAX k-CSP has threshold rank $r$ for a threshold $\tau = (\epsilon/k)^{O(1)} \cdot (1/q)^{O(k)}$, then it is possible to approximately solve the instance up to additive error $\epsilon$, using $r \cdot q^{O(k)} \cdot (k/\epsilon)^{O(1)}$ levels of the sum-of-squares hierarchy. As in the case of graphs, high-dimensional expanders (with sufficiently small $\gamma$) have threshold rank 1 according to our definition.