# Sherali--Adams Strikes Back

## TL;DR

This paper demonstrates that a certain level of the Sherali--Adams linear programming hierarchy can effectively certify bounds on Max-Cut in graphs with specific spectral properties, challenging prior beliefs about LP hierarchies' limitations.

## Contribution

It shows that Sherali--Adams hierarchy can efficiently refute Max-Cut and random CSP instances in graphs with bounded eigenvalues, outperforming previous spectral and SDP methods.

## Key findings

- Sherali--Adams certifies Max-Cut at most 50.1% in certain graphs.
- Constant-round Sherali--Adams refutes random Boolean k-CSPs with many constraints.
- Results contrast with the belief that LP hierarchies perform poorly for Max-Cut.

## Abstract

Let $G$ be any $n$-vertex graph whose random walk matrix has its nontrivial eigenvalues bounded in magnitude by $1/\sqrt{\Delta}$ (for example, a random graph $G$ of average degree~$\Theta(\Delta)$ typically has this property). We show that the $\exp\Big(c \frac{\log n}{\log \Delta}\Big)$-round Sherali--Adams linear programming hierarchy certifies that the maximum cut in such a~$G$ is at most $50.1\%$ (in fact, at most $\tfrac12 + 2^{-\Omega(c)}$). For example, in random graphs with $n^{1.01}$ edges, $O(1)$ rounds suffice; in random graphs with $n \cdot \text{polylog}(n)$ edges, $n^{O(1/\log \log n)} = n^{o(1)}$ rounds suffice. Our results stand in contrast to the conventional beliefs that linear programming hierarchies perform poorly for \maxcut and other CSPs, and that eigenvalue/SDP methods are needed for effective refutation. Indeed, our results imply that constant-round Sherali--Adams can strongly refute random Boolean $k$-CSP instances with $n^{\lceil k/2 \rceil + \delta}$ constraints; previously this had only been done with spectral algorithms or the SOS SDP hierarchy.