Algorithms and Complexity of Hedge Cluster Deletion Problems

TL;DR

This paper studies the computational complexity and algorithms for Hedge Cluster Deletion, a generalization of Cluster Deletion involving hedge graphs, providing NP-hardness results, fixed-parameter tractability, and approximation algorithms.

Contribution

It introduces Hedge Cluster Deletion, establishes its NP-completeness, explores approximation and fixed-parameter algorithms, and proposes new approaches like hedge intersection graphs.

Findings

NP-complete and polynomial-time solvable cases identified.

Approximation hardness established with exponential factors.

Polynomial algorithms for special hedge structures provided.

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 \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 $2^{O(\log^{1-\epsilon} r)}$ for any $\epsilon >0$, 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.