Learning-augmented Maximum Independent Set
Vladimir Braverman, Prathamesh Dharangutte, Vihan Shah, Chen Wang
TL;DR
This work studies MIS on general graphs under a learning-augmented oracle that, for a fixed MIS $I^*$, answers vertex-membership queries with probability $1/2+\varepsilon$. In the persistent-noise model, the authors achieve a $\tilde{O}(\sqrt{\Delta})$-approximation in $O(m)$ time, by pruning vertices with many connections to $I^*$ using neighborhood oracle signals and then greedily constructing an MIS on the residual graph. In the non-persistent-noise setting, they obtain an $O(1)$-approximation using $O(n/\varepsilon^2)$ total oracle queries and $\tilde{O}(m)$ time, via a two-phase approach that combines a bandit-style elimination with a 2-approximate vertex cover to produce a large independent set. The results leverage Chernoff concentration and careful exploitation of MIS structure to convert unreliable predictions into provably strong graph algorithms, with practical implications for MIS in prediction-rich environments. Open questions include whether the persistent setting can surpass the $\tilde{O}(\sqrt{\Delta})$ barrier and how many oracle queries are truly necessary to recover substantial fractions of $I^*$ in the non-persistent regime.
Abstract
We study the Maximum Independent Set (MIS) problem on general graphs within the framework of learning-augmented algorithms. The MIS problem is known to be NP-hard and is also NP-hard to approximate to within a factor of $n^{1-δ}$ for any $δ>0$. We show that we can break this barrier in the presence of an oracle obtained through predictions from a machine learning model that answers vertex membership queries for a fixed MIS with probability $1/2+\varepsilon$. In the first setting we consider, the oracle can be queried once per vertex to know if a vertex belongs to a fixed MIS, and the oracle returns the correct answer with probability $1/2 + \varepsilon$. Under this setting, we show an algorithm that obtains an $\tilde{O}(\sqrtΔ/\varepsilon)$-approximation in $O(m)$ time where $Δ$ is the maximum degree of the graph. In the second setting, we allow multiple queries to the oracle for a vertex, each of which is correct with probability $1/2 + \varepsilon$. For this setting, we show an $O(1)$-approximation algorithm using $O(n/\varepsilon^2)$ total queries and $\tilde{O}(m)$ runtime.