Two-Sided Lossless Expanders in the Unbalanced Setting

TL;DR

This paper introduces the first explicit construction of two-sided lossless expanders in the unbalanced bipartite graph setting, advancing the understanding of explicit expanders with balanced expansion properties.

Contribution

It proves that certain previously known one-sided lossless expanders are actually two-sided, and provides a tight characterization of these expanders in the unbalanced setting.

Findings

Constructed explicit two-sided lossless expanders in unbalanced bipartite graphs.

Showed that Kalev and Ta-Shma's one-sided expanders are actually two-sided.

Derived lossless expanders on N vertices with polynomial degree and large expanding sets.

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 $N^{0.49}$.