Approximation and parameterized algorithms to find balanced connected partitions of graphs

TL;DR

This paper introduces approximation algorithms and fixed-parameter tractable methods for balanced connected graph partitioning, improving solution quality and efficiency for complex partitioning problems.

Contribution

It presents a simple pseudo-polynomial approximation algorithm and a polynomial approximation with scaling for balanced connected partitions, and a fixed-parameter tractable ILP-based algorithm.

Findings

A $rac{k}{2}$-approximation algorithm with $ ext{O}(W|V||E|)$ runtime.

A $(rac{k}{2} + ext{ε})$-approximation algorithm with $ ext{O}(|V|^3|E|/ ext{ε})$ runtime.

An ILP-based fixed-parameter tractable algorithm for unweighted cases.

Abstract

Partitioning a connected graph into $k$~vertex-disjoint connected subgraphs of similar (or given) orders is a classical problem that has been intensively investigated since late seventies. Given a connected graph $G=(V,E)$ and a weight function $w : V \to \mathbb{Q}_\geq$, a connected $k$-partition of $G$ is a partition of $V$ such that each class induces a connected subgraph. The balanced connected $k$-partition problem consists in finding a connected $k$-partition in which every class has roughly the same weight. To model this concept of balance, one may seek connected $k$-partitions that either maximize the weight of a lightest class $(\text{max-min BCP}_k)$ or minimize the weight of a heaviest class $(\text{min-max BCP}_k)$. Such problems are equivalent when $k=2$, but they are different when $k\geq 3$. In this work, we propose a simple pseudo-polynomial $\frac{k}{2}$-approximation algorithm for $\text{min-max BCP}_k$ which runs in time $\mathcal{O}(W|V||E|)$, where $W = \sum_{v \in V} w(v)$. Based on this algorithm and using a scaling technique, we design a (polynomial) $(\frac{k}{2} +\varepsilon)$-approximation for the same problem with running-time $\mathcal{O}(|V|^3|E|/\varepsilon)$, for any fixed $\varepsilon>0$. Additionally, we propose a fixed-parameter tractable algorithm based on integer linear programming for the unweighted $\text{max-min BCP}_k$ parameterized by the size of a vertex cover.