Graph Partitioning With Limited Moves

TL;DR

This paper introduces the $r$-move $k$-partitioning problem, a variant of multiway cut constrained by limited node moves, and provides a polynomial-time approximation algorithm along with complexity results.

Contribution

It formalizes the $r$-move $k$-partitioning problem, offers a 3(r+1)-approximation algorithm, proves W[1]-hardness, and develops an FPTAS for small $r$.

Findings

Polynomial-time 3(r+1) approximation algorithm.

Proves the problem is W[1]-hard.

Provides an FPTAS for small constant $r$.

Abstract

In many real world networks, there already exists a (not necessarily optimal) $k$-partitioning of the network. Oftentimes, one aims to find a $k$-partitioning with a smaller cut value for such networks by moving only a few nodes across partitions. The number of nodes that can be moved across partitions is often a constraint forced by budgetary limitations. Motivated by such real-world applications, we introduce and study the $r$-move $k$-partitioning~problem, a natural variant of the Multiway cut problem. Given a graph, a set of $k$ terminals and an initial partitioning of the graph, the $r$-move $k$-partitioning~problem aims to find a $k$-partitioning with the minimum-weighted cut among all the $k$-partitionings that can be obtained by moving at most $r$ non-terminal nodes to partitions different from their initial ones. Our main result is a polynomial time $3(r+1)$ approximation algorithm for this problem. We further show that this problem is $W[1]$-hard, and give an FPTAS for when $r$ is a small constant.