Approximation Algorithm for Sparsest k-Partitioning

TL;DR

This paper presents a polynomial-time bi-criteria approximation algorithm for the sparsest k-partitioning problem, achieving near-optimal partitions with controlled expansion for each piece, extending the classical sparsest cut problem.

Contribution

It introduces a novel approximation algorithm for the generalized sparsest k-partitioning problem with guarantees on partition quality and size.

Findings

Provides a polynomial-time bi-criteria approximation algorithm.

Achieves each partition's expansion within a factor of $O_{ ext{ extit{ extbf{ extit{ ext{ε}}}}}}(\sqrt{ ext{ extit{ extbf{ extit{ ext{log}}}}} n ext{ extit{ extbf{ extit{ ext{log}}}}} k)$ of the optimal.

Outputs a $(1 - ext{ extit{ extbf{ extit{ ext{ε}}}}})k$-partition with controlled expansion.

Abstract

Given a graph $G$, the sparsest-cut problem asks to find the set of vertices $S$ which has the least expansion defined as $$\phi_G(S) := \frac{w(E(S,\bar{S}))}{\min \set{w(S), w(\bar{S})}}, $$ where $w$ is the total edge weight of a subset. Here we study the natural generalization of this problem: given an integer $k$, compute a $k$-partition $\set{P_1, \ldots, P_k}$ of the vertex set so as to minimize $$ \phi_k(\set{P_1, \ldots, P_k}) := \max_i \phi_G(P_i). $$ Our main result is a polynomial time bi-criteria approximation algorithm which outputs a $(1 - \e)k$-partition of the vertex set such that each piece has expansion at most $O_{\varepsilon}(\sqrt{\log n \log k})$ times $OPT$. We also study balanced versions of this problem.