Algorithms and Complexity of Hedge Cluster Deletion Problems
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2025
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| _version_ | 1866909942706864128 |
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| author | Konstantinidis, Athanasios L. Papadopoulos, Charis Velissaris, Georgios |
| author_facet | Konstantinidis, Athanasios L. Papadopoulos, Charis Velissaris, Georgios |
| contents | A hedge graph is a graph whose edge set has been partitioned into groups called hedges. Here we consider a generalization of the well-known \textsc{Cluster Deletion} problem, named \textsc{Hedge Cluster Deletion}. The task is to compute the minimum number of hedges of a hedge graph so that their removal results in a graph that is isomorphic to a disjoint union of cliques.
We identify NP-completeness and polynomial-time solutions based on vertex-disjoint 3-vertex-paths as subgraphs. Regarding its approximability, we show that it is NP-hard to approximate \textsc{Hedge Cluster Deletion} within factor $2^{O(\log^{1-ε} r)}$ for any $ε>0$, where $r$ is the number of hedges in a given hedge graph. While \textsc{Hedge Cluster Deletion} is fixed-parameter tractable with respect to the solution size (i.e., the number of removal hedges), we prove that it does not admit a polynomial kernel, unless NP $\subseteq$ coNP/poly.
Moreover, we consider the hedge underlying structure. We give a polynomial-time algorithm with constant approximation ratio for \textsc{Hedge Cluster Deletion} whenever each triangle of the input graph is covered by at most two hedges. On the way to this result, an interesting ingredient that we solved efficiently is a variant of the \textsc{Vertex Cover} problem in which apart from the desired vertex set that covers the edge set, a given set of vertex-constraints should also be included in the solution. Moreover, as a possible workaround for the existence of efficient exact algorithms, we propose the hedge intersection graph which is the intersection graph spanned by the hedges. Towards this direction, we give a polynomial-time algorithm for \textsc{Hedge Cluster Deletion} whenever the hedge intersection graph is acyclic. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_10202 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Algorithms and Complexity of Hedge Cluster Deletion Problems Konstantinidis, Athanasios L. Papadopoulos, Charis Velissaris, Georgios Data Structures and Algorithms A hedge graph is a graph whose edge set has been partitioned into groups called hedges. Here we consider a generalization of the well-known \textsc{Cluster Deletion} problem, named \textsc{Hedge Cluster Deletion}. The task is to compute the minimum number of hedges of a hedge graph so that their removal results in a graph that is isomorphic to a disjoint union of cliques. We identify NP-completeness and polynomial-time solutions based on vertex-disjoint 3-vertex-paths as subgraphs. Regarding its approximability, we show that it is NP-hard to approximate \textsc{Hedge Cluster Deletion} within factor $2^{O(\log^{1-ε} r)}$ for any $ε>0$, where $r$ is the number of hedges in a given hedge graph. While \textsc{Hedge Cluster Deletion} is fixed-parameter tractable with respect to the solution size (i.e., the number of removal hedges), we prove that it does not admit a polynomial kernel, unless NP $\subseteq$ coNP/poly. Moreover, we consider the hedge underlying structure. We give a polynomial-time algorithm with constant approximation ratio for \textsc{Hedge Cluster Deletion} whenever each triangle of the input graph is covered by at most two hedges. On the way to this result, an interesting ingredient that we solved efficiently is a variant of the \textsc{Vertex Cover} problem in which apart from the desired vertex set that covers the edge set, a given set of vertex-constraints should also be included in the solution. Moreover, as a possible workaround for the existence of efficient exact algorithms, we propose the hedge intersection graph which is the intersection graph spanned by the hedges. Towards this direction, we give a polynomial-time algorithm for \textsc{Hedge Cluster Deletion} whenever the hedge intersection graph is acyclic. |
| title | Algorithms and Complexity of Hedge Cluster Deletion Problems |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2511.10202 |