Faster MAX-CUT on Bounded Threshold Rank Graphs

TL;DR

This paper introduces a faster algorithm for approximating MAX-CUT and 2CSPs on graphs with bounded threshold rank, achieving near-linear time and improved approximation guarantees compared to prior methods.

Contribution

It presents the first near-linear time approximation algorithm for MAX-CUT on bounded threshold rank graphs, combining subspace enumeration and semidefinite programming.

Findings

Achieves a (1+O(ε)) approximation for MAX-CUT in polynomial time in 1/ε

Improves 2CSP algorithms to near-linear time for bounded threshold rank graphs

Provides a new comparison inequality linking threshold rank of label-extended and base graphs.

Abstract

We design new algorithms for approximating 2CSPs on graphs with bounded threshold rank, that is, whose normalized adjacency matrix has few eigenvalues larger than $\varepsilon$, smaller than $-\varepsilon$, or both. Unlike on worst-case graphs, 2CSPs on bounded threshold rank graphs can be $(1+O(\varepsilon))$-approximated efficiently. Prior approximation algorithms for this problem run in time exponential in the threshold rank and $1/\varepsilon$. Our algorithm has running time which is polynomial in $1/\varepsilon$ and exponential in the threshold rank of the label-extended graph, and near-linear in the input size. As a consequence, we obtain the first $(1+O(\varepsilon))$ approximation for MAX-CUT on bounded threshold rank graphs running in $\mathrm{poly}(1/\varepsilon)$ time. We also improve the state-of-the-art running time for 2CSPs on bounded threshold-rank graphs from polynomial in input size to near-linear via a new comparison inequality between the threshold rank of the label-extended graph and base graph. Our algorithm is a simple yet novel combination of subspace enumeration and semidefinite programming.