Graph Partitioning with a Move Budget
Mina Dalirrooyfard, Elaheh Fata, Majid Behbahani, Yuriy Nevmyvaka
Abstract
In many real world networks, there already exists a (not necessarily optimal) $k$-partitioning of the network. Oftentimes, for such networks, one aims to find a $k$-partitioning with a smaller cut value 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.
BibTeX
@InProceedings{pmlr-v238-dalirrooyfard24a,
title = {Graph Partitioning with a Move Budget},
author = {Dalirrooyfard, Mina and Fata, Elaheh and Behbahani, Majid and Nevmyvaka, Yuriy},
booktitle = {Proceedings of The 27th International Conference on Artificial Intelligence and Statistics},
pages = {568--576},
year = {2024},
editor = {Dasgupta, Sanjoy and Mandt, Stephan and Li, Yingzhen},
volume = {238},
series = {Proceedings of Machine Learning Research},
month = {02--04 May},
publisher = {PMLR},
pdf = {https://proceedings.mlr.press/v238/dalirrooyfard24a/dalirrooyfard24a.pdf},
url = {https://proceedings.mlr.press/v238/dalirrooyfard24a.html},
abstract = {In many real world networks, there already exists a (not necessarily optimal) $k$-partitioning of the network. Oftentimes, for such networks, one aims to find a $k$-partitioning with a smaller cut value 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.}
}