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Faster MAX-CUT on Bounded Threshold Rank Graphs

Prashanti Anderson, Samuel B. Hopkins, Amit Rajaraman, David Steurer

TL;DR

This work shows that 2CSPs on graphs with bounded threshold rank admit near-linear time, near-optimal approximations, notably a $(1+O(\varepsilon))$-approximation for MAX-CUT and for general $q$-ary 2CSPs, with runtimes exponential only in the threshold-rank parameter. The authors combine subspace enumeration with simple SDPs, and introduce a rank-inequality linking the label-extended graph to the base graph to achieve near-linear time via fast SDP solvers. A central contribution is bounding the threshold rank of the label-extended graph by the base graph's threshold rank, thereby enabling efficient CSP algorithms beyond dense or expanders. The results broaden the tractable landscape for CSPs by exploiting spectral structure, and offer practical implications for fast approximation on graphs with limited negative eigenvalues and related CSP instances.

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.

Faster MAX-CUT on Bounded Threshold Rank Graphs

TL;DR

This work shows that 2CSPs on graphs with bounded threshold rank admit near-linear time, near-optimal approximations, notably a -approximation for MAX-CUT and for general -ary 2CSPs, with runtimes exponential only in the threshold-rank parameter. The authors combine subspace enumeration with simple SDPs, and introduce a rank-inequality linking the label-extended graph to the base graph to achieve near-linear time via fast SDP solvers. A central contribution is bounding the threshold rank of the label-extended graph by the base graph's threshold rank, thereby enabling efficient CSP algorithms beyond dense or expanders. The results broaden the tractable landscape for CSPs by exploiting spectral structure, and offer practical implications for fast approximation on graphs with limited negative eigenvalues and related CSP instances.

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 , smaller than , or both. Unlike on worst-case graphs, 2CSPs on bounded threshold rank graphs can be -approximated efficiently. Prior approximation algorithms for this problem run in time exponential in the threshold rank and . Our algorithm has running time which is polynomial in 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 approximation for MAX-CUT on bounded threshold rank graphs running in 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.

Paper Structure

This paper contains 26 sections, 13 theorems, 24 equations, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our Contributions.
  3. Our Techniques, in Brief
  4. Results
  5. Quadratic Optimization over the Hypercube.
  6. MAX-CUT on Low Threshold Rank Graphs.
  7. General 2CSPs.
  8. Which Graphs Have Small Threshold Rank?
  9. Techniques
  10. Algorithm for Boolean Quadratic Optimization and MAX-CUT.
  11. Algorithm for General 2CSPs, and Threshold Ranks of Label-Extended Graphs
  12. From Polynomial to Near-Linear Time
  13. Related Work
  14. SDP-based algorithms.
  15. Subspace enumeration algorithms.
  16. ...and 11 more sections

Key Result

Theorem 1.1

Let $A \in \mathbb{R}^{n \times n}$ be an arbitrary symmetric matrix, scaled to have $\left\|{A}\right\|_{\mathrm{op}} \leqslant 1$, and let $\mathsf{OPT} = \max_{x \in \{\pm 1\}^n} x^\top A x$. Set $k = \operatorname{rank}_{\geqslant \varepsilon}(A)$. Then, there is an algorithm running in time $\m

Theorems & Definitions (27)

  • Theorem 1.1: Quadratic optimization
  • Remark 1.2
  • Theorem 1.3: $(1+O(\eps))$ Approximation for MAX-CUT in $\poly(1/\eps)$ Time
  • Theorem 1.4
  • Definition 2.1: Threshold Rank
  • Definition 2.2: Label Extended Graph
  • Definition 2.3: Pseudodistribution
  • Definition 2.4: Pseudoexpectation
  • Definition 2.6: Constrained pseudodistributions
  • Theorem 2.7: Efficient optimization over pseudodistributions of degree $2$
  • ...and 17 more