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Hermitian Laplacians and a Cheeger Inequality for the Max-2-Lin Problem

Authors Huan Li, He Sun, Luca Zanetti

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ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Theory of computation → Approximation algorithms analysis
  • Mathematics of computing → Spectra of graphs
Keywords
  • Spectral methods
  • Hermitian Laplacians
  • the Max-2-Lin problem
  • Unique Games

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Abstract

We study spectral approaches for the MAX-2-LIN(k) problem, in which we are given a system of m linear equations of the form x_i - x_j is equivalent to c_{ij} mod k, and required to find an assignment to the n variables {x_i} that maximises the total number of satisfied equations. 
We consider Hermitian Laplacians related to this problem, and prove a Cheeger inequality that relates the smallest eigenvalue of a Hermitian Laplacian to the maximum number of satisfied equations of a MAX-2-LIN(k) instance I. We develop an O~(kn^2) time algorithm that, for any (1-epsilon)-satisfiable instance, produces an assignment satisfying a (1 - O(k)sqrt{epsilon})-fraction of equations. We also present a subquadratic-time algorithm that, when the graph associated with I is an expander, produces an assignment satisfying a (1- O(k^2)epsilon)-fraction of the equations. Our Cheeger inequality and first algorithm can be seen as generalisations of the Cheeger inequality and algorithm for MAX-CUT developed by Trevisan.

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Huan Li, He Sun, and Luca Zanetti. Hermitian Laplacians and a Cheeger Inequality for the Max-2-Lin Problem. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 71:1-71:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.ESA.2019.71

Author Details

Huan Li
  • School of Computer Science, Fudan University, China
He Sun
  • School of Informatics, University of Edinburgh, United Kingdom
Luca Zanetti
  • Department of Computer Science and Technology, University of Cambridge, United Kingdom

Funding

  • Zanetti, Luca: Supported by the ERC Starting Grant "DYNAMIC MARCH".

Acknowledgements

We are very grateful to Mihai Cucuringu for valuable discussions on Hermitian Laplacian matrices and their applications. We would also like to thank Chris Heunen for some fruitful conversations on topics closely related to this paper.

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