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Sparse High Dimensional Expanders via Local Lifts

Authors Inbar Ben Yaacov , Yotam Dikstein , Gal Maor

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Author Details

Inbar Ben Yaacov
  • Weizmann Institute of Science, Rehovot, Israel
Yotam Dikstein
  • Institute for Advanced Study, Princeton, NJ, USA
Gal Maor
  • Tel Aviv University, Tel Aviv, Israel

Funding

  • Ben Yaacov, Inbar: This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 819702) and from the Simons Foundation Collaboration on the Theory of Algorithmic Fairness.
  • Dikstein, Yotam: This material is based upon work supported by the National Science Foundation under Grant No. DMS-1926686.
  • Maor, Gal: Supported by Gil Cohen’s ERC grant 949499.

Cite AsGet BibTex

Inbar Ben Yaacov, Yotam Dikstein, and Gal Maor. Sparse High Dimensional Expanders via Local Lifts. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 68:1-68:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2024.68

Abstract

High dimensional expanders (HDXs) are a hypergraph generalization of expander graphs. They are extensively studied in the math and TCS communities due to their many applications. Like expander graphs, HDXs are especially interesting for applications when they are bounded degree, namely, if the number of edges adjacent to every vertex is bounded. However, only a handful of constructions are known to have this property, all of which rely on algebraic techniques. In particular, no random or combinatorial construction of bounded degree high dimensional expanders is known. As a result, our understanding of these objects is limited. The degree of an i-face in an HDX is the number of (i+1)-faces that contain it. In this work we construct complexes whose higher dimensional faces have bounded degree. This is done by giving an elementary and deterministic algorithm that takes as input a regular k-dimensional HDX X and outputs another regular k-dimensional HDX X̂ with twice as many vertices. While the degree of vertices in X̂ grows, the degree of the (k-1)-faces in X̂ stays the same. As a result, we obtain a new "algebra-free" construction of HDXs whose (k-1)-face degree is bounded. Our construction algorithm is based on a simple and natural generalization of the expander graph construction by Bilu and Linial [Yehonatan Bilu and Nathan Linial, 2006], which build expander graphs using lifts coming from edge signings. Our construction is based on local lifts of high dimensional expanders, where a local lift is a new complex whose top-level links are lifts of the links of the original complex. We demonstrate that a local lift of an HDX is also an HDX in many cases. In addition, combining local lifts with existing bounded degree constructions creates new families of bounded degree HDXs with significantly different links than before. For every large enough D, we use this technique to construct families of bounded degree HDXs with links that have diameter ≥ D.

Subject Classification

ACM Subject Classification
  • Theory of computation → Expander graphs and randomness extractors
Keywords
  • High Dimensional Expanders
  • HDX
  • Spectral Expansion
  • Lifts
  • Covers
  • Explicit Constructions
  • Randomized Constructions
  • Deterministic Constructions

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