Parameterized Algorithms for Balanced Cluster Edge Modification Problems
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| author | Madathil, Jayakrishnan Meeks, Kitty |
| author_facet | Madathil, Jayakrishnan Meeks, Kitty |
| contents | We study {\sc Cluster Edge Modification} problems with constraints on the size of the clusters. A graph $G$ is a cluster graph if every connected component of $G$ is a clique. In a typical {\sc Cluster Edge Modification} problem such as the widely studied {\sc Cluster Editing}, we are given a graph $G$ and a non-negative integer $k$ as input, and we have to decide if we can turn $G$ into a cluster graph by way of at most $k$ edge modifications -- that is, by adding or deleting edges. In this paper, we study the parameterized complexity of such problems, but with an additional constraint: The size difference between any two connected components of the resulting cluster graph should not exceed a given threshold. Depending on which modifications are permissible -- only adding edges, only deleting edges, both adding and deleting edges -- we have three different computational problems. We show that all three problems, when parameterized by $k$, admit single-exponential time FPT algorithms and polynomial kernels. Our problems may be thought of as the size-constrained or balanced counterparts of the typical {\sc Cluster Edge Modification} problems, similar to the well-studied size-constrained or balanced counterparts of other clustering problems such as {\sc $k$-Means Clustering}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_03830 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Parameterized Algorithms for Balanced Cluster Edge Modification Problems Madathil, Jayakrishnan Meeks, Kitty Data Structures and Algorithms Discrete Mathematics We study {\sc Cluster Edge Modification} problems with constraints on the size of the clusters. A graph $G$ is a cluster graph if every connected component of $G$ is a clique. In a typical {\sc Cluster Edge Modification} problem such as the widely studied {\sc Cluster Editing}, we are given a graph $G$ and a non-negative integer $k$ as input, and we have to decide if we can turn $G$ into a cluster graph by way of at most $k$ edge modifications -- that is, by adding or deleting edges. In this paper, we study the parameterized complexity of such problems, but with an additional constraint: The size difference between any two connected components of the resulting cluster graph should not exceed a given threshold. Depending on which modifications are permissible -- only adding edges, only deleting edges, both adding and deleting edges -- we have three different computational problems. We show that all three problems, when parameterized by $k$, admit single-exponential time FPT algorithms and polynomial kernels. Our problems may be thought of as the size-constrained or balanced counterparts of the typical {\sc Cluster Edge Modification} problems, similar to the well-studied size-constrained or balanced counterparts of other clustering problems such as {\sc $k$-Means Clustering}. |
| title | Parameterized Algorithms for Balanced Cluster Edge Modification Problems |
| topic | Data Structures and Algorithms Discrete Mathematics |
| url | https://arxiv.org/abs/2403.03830 |