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Two-Sided Lossless Expanders in the Unbalanced Setting

Eshan Chattopadhyay, Mohit Gurumukhani, Noam Ringach, Yunya Zhao

TL;DR

This work provides the first explicit construction of two-sided lossless expanders in the unbalanced bipartite setting, anchored by the KT graph built from multiplicity codes. It proves that the KT graph exhibits both left-to-right and right-to-left lossless expansion, and it establishes tightness for right expansion in the regime $n<2s+2$. By taking the bipartite half of the KT graph, the authors also obtain a high-degree, non-bipartite lossless expander with explicit parameters and a free group action, broadening applications to non-bipartite graphs. The results close a gap in unbalanced expander theory and provide explicit graphs with provable expansion properties, potentially enabling new coding-theoretic and combinatorial constructions. Overall, the paper delivers a complete, explicit framework for unbalanced two-sided lossless expanders and their non-bipartite counterparts, with tight, implementable parameter regimes.

Abstract

We present the first explicit construction of two-sided lossless expanders in the unbalanced setting (bipartite graphs that have polynomially many more nodes on the left than on the right). Prior to our work, all known explicit constructions in the unbalanced setting achieved only one-sided lossless expansion. Specifically, we show that the one-sided lossless expanders constructed by Kalev and Ta-Shma (RANDOM'22) -- that are based on multiplicity codes introduced by Kopparty, Saraf, and Yekhanin (STOC'11) -- are, in fact, two-sided lossless expanders. Moreover, we show that our result is tight, thus completely characterizing the graph of Kalev and Ta-Shma. Using our unbalanced bipartite expander, we easily obtain lossless (non-bipartite) expander graphs on $N$ vertices with polynomial degree $\ll N$ and expanding sets of size $N^{0.49}$.

Two-Sided Lossless Expanders in the Unbalanced Setting

TL;DR

This work provides the first explicit construction of two-sided lossless expanders in the unbalanced bipartite setting, anchored by the KT graph built from multiplicity codes. It proves that the KT graph exhibits both left-to-right and right-to-left lossless expansion, and it establishes tightness for right expansion in the regime . By taking the bipartite half of the KT graph, the authors also obtain a high-degree, non-bipartite lossless expander with explicit parameters and a free group action, broadening applications to non-bipartite graphs. The results close a gap in unbalanced expander theory and provide explicit graphs with provable expansion properties, potentially enabling new coding-theoretic and combinatorial constructions. Overall, the paper delivers a complete, explicit framework for unbalanced two-sided lossless expanders and their non-bipartite counterparts, with tight, implementable parameter regimes.

Abstract

We present the first explicit construction of two-sided lossless expanders in the unbalanced setting (bipartite graphs that have polynomially many more nodes on the left than on the right). Prior to our work, all known explicit constructions in the unbalanced setting achieved only one-sided lossless expansion. Specifically, we show that the one-sided lossless expanders constructed by Kalev and Ta-Shma (RANDOM'22) -- that are based on multiplicity codes introduced by Kopparty, Saraf, and Yekhanin (STOC'11) -- are, in fact, two-sided lossless expanders. Moreover, we show that our result is tight, thus completely characterizing the graph of Kalev and Ta-Shma. Using our unbalanced bipartite expander, we easily obtain lossless (non-bipartite) expander graphs on vertices with polynomial degree and expanding sets of size .
Paper Structure (28 sections, 30 theorems, 19 equations)

This paper contains 28 sections, 30 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. Our results
  3. Lossless expanders in the balanced setting
  4. Concurrent works
  5. Proof Overview
  6. Two-sided lossless expander
  7. Tightness of right expansion
  8. Non-bipartite lossless expander
  9. Organization
  10. Preliminaries
  11. Notation
  12. Lossless expansion
  13. The KT graph
  14. A useful inequality
  15. Free group actions on graphs
  16. ...and 13 more sections

Key Result

Theorem 1

For infinitely many $N$ and all constant $0 < \delta \le 0.99$, there exists an explicit, biregular, two-sided $(K_{L}, \varepsilon_L = 0.01, K_{R}, \varepsilon_{R} = 0.01)$ lossless expander $\Gamma_{\shortrightarrow}{}: [N]\times [D_L]\to [M]$ where $D_L = \operatorname{poly}(\log N)$, $N^{1.01\de

Theorems & Definitions (84)

  • Definition 1.1: Two-sided lossless expander
  • Theorem 1: Informal version of \ref{['thm: main simplified theorem']}, bipartite two-sided lossless expander
  • Remark 1.2
  • Theorem 2: Informal version of \ref{['thm:normal lossless expander']}, non-bipartite lossless expander
  • Remark 1.3
  • Theorem 2.1: Informal version of \ref{['MainThm']}
  • Definition 2.2: The KT graph kalev_unbalanced_2022
  • Theorem 2.3: The KT graph losslessly expands from the right
  • Lemma 2.4: Right regularity
  • Lemma 2.5: Number of common left neighbors
  • ...and 74 more