Learning-Augmented Streaming Algorithms for Approximating MAX-CUT

TL;DR

The paper addresses estimating the MAX-CUT value in graph streams under a noisy, ε-accurate predictor. It introduces a learning-augmented framework and a high/low-degree decomposition to surpass the classic 1/2-approximation barrier, achieving a $(\frac{1}{2} + Ω(ε^2))$-approximation with poly$(1/ε)$ space in insertion-only streams and poly$(1/ε, \log n)$ space in fully dynamic streams. The approach combines CountMin sketches, ℓ0-sampling or reservoir sampling, and greedy extensions to leverage predictions while preserving robustness; it also yields constant-query guarantees in random-order streams. This work demonstrates that predictions can substantially improve space-accuracy trade-offs in streaming MAX-CUT estimation and lays groundwork for prediction-augmented algorithms in sublinear-space graph analytics.

Abstract

We study learning-augmented streaming algorithms for estimating the value of MAX-CUT in a graph. In the classical streaming model, while a $1/2$-approximation for estimating the value of MAX-CUT can be trivially achieved with $O(1)$ words of space, Kapralov and Krachun [STOC'19] showed that this is essentially the best possible: for any $ε> 0$, any (randomized) single-pass streaming algorithm that achieves an approximation ratio of at least $1/2 + ε$ requires $Ω(n / 2^{\text{poly}(1/ε)})$ space. We show that it is possible to surpass the $1/2$-approximation barrier using just $O(1)$ words of space by leveraging a (machine learned) oracle. Specifically, we consider streaming algorithms that are equipped with an $ε$-accurate oracle that for each vertex in the graph, returns its correct label in $\{-1, +1\}$, corresponding to an optimal MAX-CUT solution in the graph, with some probability $1/2 + ε$, and the incorrect label otherwise. Within this framework, we present a single-pass algorithm that approximates the value of MAX-CUT to within a factor of $1/2 + Ω(ε^2)$ with probability at least $2/3$ for insertion-only streams, using only $\text{poly}(1/ε)$ words of space. We also extend our algorithm to fully dynamic streams while maintaining a space complexity of $\text{poly}(1/ε,\log n)$ words.

Theorems & Definitions (31)

• Theorem 1.1
• Theorem 2.1: CountMin Sketch cormode2005improved
• Definition 2.2: $\ell_0$-sampler jowhari2011tight
• Theorem 2.3: Theorem 2 in jowhari2011tight
• Theorem 3.1
• proof
• Theorem 4.1
• Lemma 4.2
• proof
• Lemma 4.3