Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Algorithms and Complexity of Hedge Cluster Deletion Problems

Athanasios L. Konstantinidis, Charis Papadopoulos, Georgios Velissaris

TL;DR

This work studies Hedge Cluster Deletion, a generalization of Cluster Deletion where edges are partitioned into hedges. It establishes a rich complexity landscape: NP-hardness in broad underlying structures, a dichotomy granting polynomial-time solvability when the maximal vertex-disjoint $P_3$ packing is bounded, and deep inapproximability results tied to Min Horn Deletion via A-reductions. It also provides a 2-approximation for bi-hedge graphs and a polynomial-time algorithm for the acyclic hedge-intersection-graph case, highlighting both difficulties and tractable special cases. Overall, the paper connects hedge-based edge deletions to well-studied CSP and kernelization barriers, offering both theoretical insights and practical approximation approaches.

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 $2^{O(\log^{1-ε} r)}$ 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.

Algorithms and Complexity of Hedge Cluster Deletion Problems

TL;DR

This work studies Hedge Cluster Deletion, a generalization of Cluster Deletion where edges are partitioned into hedges. It establishes a rich complexity landscape: NP-hardness in broad underlying structures, a dichotomy granting polynomial-time solvability when the maximal vertex-disjoint packing is bounded, and deep inapproximability results tied to Min Horn Deletion via A-reductions. It also provides a 2-approximation for bi-hedge graphs and a polynomial-time algorithm for the acyclic hedge-intersection-graph case, highlighting both difficulties and tractable special cases. Overall, the paper connects hedge-based edge deletions to well-studied CSP and kernelization barriers, offering both theoretical insights and practical approximation approaches.

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 for any , where 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 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.

Paper Structure

This paper contains 21 sections, 28 theorems, 2 equations, 4 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related work
  3. Overview of our results and techniques
  4. Hardness results
  5. Tractable results
  6. (In)approximability
  7. Incompressibility
  8. Constant-factor approximation algorithm
  9. Hedge intersection graph
  10. Preliminaries
  11. Basic definitions
  12. Approximation algorithms
  13. Kernelization lower bounds
  14. Hedge graphs
  15. Hardness result implied by the underlying structure
  16. ...and 6 more sections

Key Result

Lemma 2.5

Let $H$ be a hedge graph on $n$ vertices. There is an $O(n^3)$-time algorithm that constructs another hedge graph $H'$ with $O(n^3)$ vertices having the following properties:

Figures (4)

  • Figure 1: Illustration of the reduction given in \ref{['lem:paths']}. On the left side, we show an input graph $G$ for Vertex Cover and on the right side, we depict the corresponding hedge graph $H$, according to the construction.
  • Figure 2: Illustrating CNF formulas for MinOnes$(\mathcal{F}")$ and Propagational-$f_1$ Satisfiability, alongside the corresponding hedge graph given in \ref{['lemma:hcdtomones']} and \ref{['theo:nokernel']}, respectively. Note that the $K_3$ spanned solely by hedge ${x_6}$ does not correspond to any constraint in the MinOnes$(\mathcal{F}")$ formulation.
  • Figure 3: Illustrating the domination relation. On the left side, a hedge graph is shown by enumerating all $P_3$ and $K_3$ in the underlying graph. On the right side, the domination relation is depicted among the hedges by introducing a graph representation: for any two hedges $h$ and $h'$ (i) there is a directed edge from $h$ to $h'$ whenever $h \in D(h')$ and (ii) there is an edge $\{h,h'\}$ whenever there is a $P_3$ with edges spanned by $h$ and $h'$. Observe that if we remove the hedge $x_5$ then all hedges $x_1,x_2,x_3$ must be removed which constitute the hedges that dominate $x_5$, that is, $R(x_5)=\{x_1,x_2,x_3,x_5\}$. All hedges of $R(x_5)$ reach $x_5$ with a directed path in this graph representation.
  • Figure 4: This graph corresponds to the graph obtained from \ref{['fig:domination']} towards the construction given in \ref{['lem:HCDtoMVC']} for the Multi-Vertex Cover problem. Every vertex $v$ is equipped with a list $L(v)$; the unique vertex with $v \in L(v)$ is shown underlined. We also highlight the difference between a minimum vertex cover (represented by the black vertices) and a minimum multi-vertex cover (indicated by the shaded region).

Theorems & Definitions (56)

  • Definition 2.1: hedge-subgraph
  • Definition 2.2: cluster hedge-subgraph
  • proof
  • Lemma 2.5
  • proof
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • ...and 46 more