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Rounding via Low Dimensional Embeddings

Authors Mark Braverman, Dor Minzer

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Author Details

Mark Braverman
  • Department of Computer Science, Princeton University, NJ, USA
Dor Minzer
  • Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA

Funding

  • Braverman, Mark: Research supported in part by the NSF Alan T. Waterman Award, Grant No. 1933331, a Packard Fellowship in Science and Engineering, and the Simons Collaboration on Algorithms and Geometry.
  • Minzer, Dor: Supported by a Sloan Research Fellowship.

Cite AsGet BibTex

Mark Braverman and Dor Minzer. Rounding via Low Dimensional Embeddings. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 26:1-26:30, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.ITCS.2023.26

Abstract

A regular graph G = (V,E) is an (ε,γ) small-set expander if for any set of vertices of fractional size at most ε, at least γ of the edges that are adjacent to it go outside. In this paper, we give a unified approach to several known complexity-theoretic results on small-set expanders. In particular, we show: 1) Max-Cut: we show that if a regular graph G = (V,E) is an (ε,γ) small-set expander that contains a cut of fractional size at least 1-δ, then one can find in G a cut of fractional size at least 1-O(δ/(εγ⁶)) in polynomial time. 2) Improved spectral partitioning, Cheeger’s inequality and the parallel repetition theorem over small-set expanders. The general form of each one of these results involves square-root loss that comes from certain rounding procedure, and we show how this can be avoided over small set expanders. Our main idea is to project a high dimensional vector solution into a low-dimensional space while roughly maintaining 𝓁₂² distances, and then perform a pre-processing step using low-dimensional geometry and the properties of 𝓁₂² distances over it. This pre-processing leverages the small-set expansion property of the graph to transform a vector valued solution to a different vector valued solution with additional structural properties, which give rise to more efficient integral-solution rounding schemes.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Discrete optimization
Keywords
  • Parallel Repetition
  • Small Set Expanders
  • Semi-Definite Programs

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