Finding Colorings in One-Sided Expanders
Rares-Darius Buhai, Yiding Hua, David Steurer, Andor Vári-Kakas
TL;DR
This work tackles the tractability of coloring and independent-set problems on regular one-sided expanders by marrying spectral methods with a model-graph framework. It develops a comprehensive algorithmic pipeline that uses bottom-threshold spectral information, variance bounds, and a partition-recovery meta-theorem to either recover color classes or obtain large independent sets, with precise results when the color-class structure is represented by matrices $M$ with distinct versus repeated rows. The paper further establishes a sharp hardness dichotomy under the Unique Games Conjecture, showing that repeated rows in $M$ yield UG-hardness while distinct rows permit efficient algorithms, and it extends the framework to random planting and rounding techniques that achieve near-optimal colorings or independent sets under plausible spectral conditions. Taken together, these results delineate the boundary between tractable and intractable vertex-coloring problems on one-sided expanders and provide a versatile spectral toolkit for vertex-based graph problems in this regime.
Abstract
We establish new algorithmic guarantees with matching hardness results for coloring and independent set problems in one-sided expanders and related classes of graphs. For example, given a $3$-colorable regular one-sided expander, we compute in polynomial time either an independent set of relative size at least $1/2-o(1)$ or a proper $3$-coloring for all but an $o(1)$ fraction of the vertices, where $o(1)$ stands for a function that tends to $0$ with the second largest eigenvalue of the normalized adjacency matrix. This result improves on recent seminal work of Bafna, Hsieh, and Kothari (STOC 2025) developing an algorithm that efficiently finds independent sets of relative size at least $0.01$ in such graphs. We also obtain an efficient $1.6667$-factor approximation algorithm for VERTEX COVER in sufficiently strong regular one-sided expanders, improving over a previous $(2-ε)$-factor approximation in such graphs for an unspecified constant $ε>0$. We propose a new stratification of $k$-COLORING in terms of $k$-by-$k$ matrices akin to predicate sets for constraint satisfaction problems. We prove that whenever this matrix has repeated rows, the corresponding coloring problem is NP-hard for one-sided expanders under the Unique Games Conjecture. On the other hand, if this matrix has no repeated rows, our algorithms can solve the corresponding coloring problem on one-sided expanders in polynomial time. As starting point for our algorithmic results, we show a property of graph spectra that, to the best of our knowledge, has not been observed before: The number of negative eigenvalues smaller than $-τ$ is at most $O(1/τ^{4})$ times the number of eigenvalues larger than $τ^{2}/2$. While this result allows us to bound the number of eigenvalues bounded away from $0$ in one-sided spectral expanders, this property alone is insufficient for our algorithmic results.