Two-Sided Lossless Expanders in the Unbalanced Setting

2024-09-06

Electron. Colloquium Comput. Complex.

AI Summary

This work addresses the long-standing challenge of explicitly constructing *bilateral lossless expanders* for highly unbalanced bipartite graphs (where the left-side size $N$ vastly exceeds the right-side size $M$). Prior explicit constructions achieved losslessness only on one side. The authors first prove that the Kalev–Ta-Shma construction is inherently bilateral lossless, providing a tight characterization; they then devise a novel algebraic construction based on *multiplicity codes*, analyzed via combinatorial and graph-theoretic expansion techniques. The resulting expander achieves large left-side expansion—expanding sets of size up to $N^{0.49}$—with degree significantly sublinear in $N$. Moreover, it yields the first efficient *non-bipartite lossless expander*. The core breakthrough is the first explicit construction achieving *simultaneously* bilateral losslessness, high imbalance ($N gg M$), and lossless expansion—resolving a fundamental open problem in explicit expander theory.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Models for Web evolution

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 $N^{0.49}$.

Problem

• Construct two-sided lossless expanders
• Address unbalanced bipartite graphs
• Characterize Kalev-Ta-Shma graph

Innovation

• Explicit construction of expanders
• Two-sided lossless expansion
• Polynomial degree expander graphs