Balanced Crown Decomposition for Connectivity Constraints

TL;DR

This paper introduces the balanced crown decomposition, a new graph structure that improves algorithms for connectivity-constrained graph problems, including the first constant-factor approximation for the Balanced Connected Partition problem.

Contribution

The paper presents the balanced crown decomposition, enabling improved kernelization and approximation algorithms for connectivity-based graph partitioning problems.

Findings

First constant-factor approximation for BCP problem.

Derived a 3-approximation for key objectives in graph partitioning.

Enhanced algorithms for graph editing, packing, and partitioning tasks.

Abstract

We introduce the balanced crown decomposition that captures the structure imposed on graphs by their connected induced subgraphs of a given size. Such subgraphs are a popular modeling tool in various application areas, where the non-local nature of the connectivity condition usually results in very challenging algorithmic tasks. The balanced crown decomposition is a combination of a crown decomposition and a balanced partition which makes it applicable to graph editing as well as graph packing and partitioning problems. We illustrate this by deriving improved kernelization and approximation algorithms for a variety of such problems. In particular, through this structure, we obtain the first constant-factor approximation for the Balanced Connected Partition (BCP) problem, where the task is to partition a vertex-weighted graph into $k$ connected components of approximately equal weight. We derive a 3-approximation for the two most commonly used objectives of maximizing the weight of the lightest component or minimizing the weight of the heaviest component.