Learning-augmented Maximum Independent Set

TL;DR

This paper introduces learning-augmented algorithms for the NP-hard Maximum Independent Set problem, leveraging probabilistic oracle predictions to achieve better approximation ratios than classical methods.

Contribution

It presents novel algorithms that utilize machine learning predictions to improve approximation ratios for MIS on general graphs, breaking traditional NP-hardness barriers.

Findings

First setting: $ ilde{O}(rac{ ext{sqrt}( ext{max degree})}{ ext{epsilon}})$-approximation in linear time.

Second setting: $O(1)$-approximation with $O(n/ ext{epsilon}^2)$ queries and near-linear runtime.

Algorithms outperform classical approaches by exploiting probabilistic oracle predictions.

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 $n^{1-\delta}$ for any $\delta>0$. 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 $\tilde{O}(\sqrt{\Delta}/\varepsilon)$-approximation in $O(m)$ time where $\Delta$ 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.