Constant Factor Approximation for Balanced Cut in the PIE model

TL;DR

This paper introduces a new semi-random model for the Balanced Cut problem called the PIE model, and provides an approximation algorithm that achieves a constant factor approximation in this setting, even with arbitrary permutation-invariant random edges.

Contribution

The paper develops a novel semi-random model for Balanced Cut and presents an approximation algorithm with provable guarantees in this general setting.

Findings

Achieves a balanced cut with cost $O(|E_{random}|) + n ext{polylog}(n)$

Provides a constant factor approximation when $|E_{random}| = ext{Omega}(n ext{polylog}(n))$

Extends the applicability of approximation algorithms to more general semi-random models.

Abstract

We propose and study a new semi-random semi-adversarial model for Balanced Cut, a planted model with permutation-invariant random edges (PIE). Our model is much more general than planted models considered previously. Consider a set of vertices V partitioned into two clusters $L$ and $R$ of equal size. Let $G$ be an arbitrary graph on $V$ with no edges between $L$ and $R$. Let $E_{random}$ be a set of edges sampled from an arbitrary permutation-invariant distribution (a distribution that is invariant under permutation of vertices in $L$ and in $R$). Then we say that $G + E_{random}$ is a graph with permutation-invariant random edges. We present an approximation algorithm for the Balanced Cut problem that finds a balanced cut of cost $O(|E_{random}|) + n \text{polylog}(n)$ in this model. In the regime when $|E_{random}| = \Omega(n \text{polylog}(n))$, this is a constant factor approximation with respect to the cost of the planted cut.