Faster MAX-CUT on Bounded Threshold Rank Graphs

2025-11-14

AI Summary

This paper addresses the MAX-CUT problem (and more generally, binary constraint satisfaction problems) on bounded-threshold-rank graphs. Prior algorithms incurred exponential dependence on both the threshold rank (r) and the accuracy parameter (varepsilon). We propose a novel algorithm integrating subspace enumeration, semidefinite programming, and spectral graph theory. First, we establish a new comparison inequality linking the threshold rank of the label-extended graph to structural properties of the base graph. Then, we design an efficient solution framework based on enumeration over low-dimensional subspaces. Our algorithm runs in nearly linear time in the input size and achieves a ((1+O(varepsilon)))-approximation in (mathrm{poly}(1/varepsilon)) time. Crucially, it reduces the dependence on both (r) and (1/varepsilon) from exponential to polynomial—marking the first such result—and significantly improves the efficiency of high-accuracy approximation for this class of graphs.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Graph-based Machine Learning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

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.

Problem

• Designing faster approximation algorithms for 2CSPs on bounded threshold rank graphs
• Achieving polynomial time approximation for MAX-CUT on bounded threshold rank graphs
• Improving running time from polynomial to near-linear for 2CSP approximations

Innovation

• Polynomial time approximation for bounded threshold rank graphs
• Combines subspace enumeration with semidefinite programming
• Near-linear runtime via label-extended graph comparison