Algorithms and Complexity of Hedge Cluster Deletion Problems

2025-11-13

AI Summary

This paper studies the Hedge Cluster Deletion problem: given a hedge graph—i.e., a graph whose edges are partitioned into hedges—find a minimum set of hedges to delete so that the remaining graph is a disjoint union of cliques. This constitutes the first extension of Cluster Deletion to hedge graphs. The authors establish an equivalence between Hedge Cluster Deletion and Min Horn Deletion, introduce a novel hedge intersection graph model, and formulate a vertex-constrained variant of Vertex Cover. Methodologically, they design efficient exact and approximation algorithms. Theoretical contributions include: (i) a polynomial-time algorithm for graphs with bounded 3-vertex-path number; (ii) constant-factor approximation algorithms for instances where the hedge intersection graph is acyclic or each triangle is covered by at most two hedges; and (iii) a hardness result showing that the problem admits no polynomial-time constant-factor approximation unless P = NP.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

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 extsc{Cluster Deletion} problem, named extsc{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 show that for graphs that contain bounded size of vertex-disjoint 3-vertex-paths as subgraphs, extsc{Hedge Cluster Deletion} can be solved in polynomial time. Regarding its approximability, we prove that the problem is tightly connected to the related complexity of the extsc{Min Horn Deletion} problem, a well-known boolean CSP problem. Our connection shows that it is NP-hard to approximate extsc{Hedge Cluster Deletion} within factor $2^{O(log^{1-epsilon} r)}$ for any $epsilon>0$, where $r$ is the number of hedges in a given hedge graph. Based on its classified (in)approximability and the difficulty imposed by the structure of almost all non-trivial graphs, we consider the hedge underlying structure. We give a polynomial-time algorithm with constant approximation ratio for extsc{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 extsc{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 extsc{Hedge Cluster Deletion} whenever the hedge intersection graph is acyclic.

Problem

• Computing minimum hedges removal for disjoint clique graphs
• Establishing complexity and approximability bounds for hedge deletion
• Developing polynomial algorithms for restricted hedge graph structures

Innovation

• Polynomial-time algorithm for bounded vertex-disjoint 3-vertex-paths
• Constant approximation ratio for triangles covered by two hedges
• Polynomial-time solution for acyclic hedge intersection graphs