Rounding Large Independent Sets on Expanders
<|reference_start|>Rounding Large Independent Sets on Expanders: 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 the 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 $(1/2-\epsilon)n$. Our second result above can be refined to require only a weaker vertex expansion property with an efficient certificate. Somewhat surprisingly, 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 $o_n(1)$) 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 $\Omega(1)$. 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 $k \geq 4$.   Our rounding builds on the method of simulating multiple samples from a pseudodistribution introduced by Barak 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.<|reference_end|>
@article{bafna2024rounding,
    title={Rounding Large Independent Sets on Expanders},
    author={Mitali Bafna, Jun-Ting Hsieh, Pravesh K. Kothari},
    journal={arXiv preprint arXiv:2405.10238},
    year={2024},
    archivePrefix={arXiv},
    eprint={2405.10238},
    primaryClass={cs.DS cs.CC}
}