Rounding Large Independent Sets on Expanders
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3717823.3718137.pdf
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Author(s)
Bafna, Mitali
Hsieh, Jun-Ting
Kothari, Pravesh K.
Date Issued
June 15, 2025
Publisher
ACM|Proceedings of the 57th Annual ACM Symposium on Theory of Computing
Citation
Mitali Bafna, Jun-Ting Hsieh, and Pravesh K. Kothari. 2025. Rounding Large Independent Sets on Expanders. In Proceedings of the 57th Annual ACM Symposium on Theory of Computing (STOC '25). Association for Computing Machinery, New York, NY, USA, 631–642.
Version
Final published version
Abstract
We develop a new approach for approximating large independent sets when the input graph is a one-sided spectral expander - that is, the uniform random walk matrix of the graph has its second eigenvalue bounded away from 1. Consequently, we obtain a polynomial time algorithm to find linear-sized independent sets in one-sided expanders that are almost 3-colorable or are promised to contain an independent set of size (1/2−є)n. Our second result above can be refined to require only a weaker vertex expansion property with an efficient certificate. In a surprising contrast to our algorithmic result, we observe that the analogous task of finding a linear-sized independent set in almost 4-colorable one-sided expanders (even when the second eigenvalue is on(1)) is NP-hard, assuming the Unique Games Conjecture.
All prior algorithms that beat the worst-case guarantees for this problem rely on bottom eigenspace enumeration techniques (following the classical spectral methods of Alon and Kahale) and require two-sided expansion, meaning a bounded number of negative eigenvalues of magnitude Ω(1). Such techniques naturally extend to almost k-colorable graphs for any constant k, in contrast to analogous guarantees on one-sided expanders, which are Unique Games-hard to achieve for k ≥ 4.
Our rounding scheme builds on the method of simulating multiple samples from a pseudo-distribution introduced in Bafna et. al. for rounding Unique Games instances. The key to our analysis is a new clustering property of large independent sets in expanding graphs - every large independent set has a larger-than-expected intersection with some member of a small list - and its formalization in the low-degree sum-of-squares proof system.
Description
STOC ’25, Prague, Czechia
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1145/3717823.3718137
