Hedge Cluster Deletion

• Hedge Cluster Deletion is the task of removing the fewest hedges—atomic groups of edges—to convert a hedge graph into a disjoint union of cliques.
• The problem generalizes the classical Cluster Deletion by treating edge groups atomically and is NP-complete with strong inapproximability bounds tied to CSP reductions.
• Recent work identifies tractable cases via bounded disjoint P3s, bi-hedge graphs with 2-approximations, and acyclic hedge intersections that allow for exact polynomial-time solutions.

Hedge Cluster Deletion is the problem of deleting the fewest hedges—where a hedge is a group of edges, forming a partition of the edge set—in a hedge graph so that the remaining graph becomes a cluster graph, i.e., a disjoint union of cliques. This generalizes classical Cluster Deletion (edge-deletion to a cluster graph) by introducing edge groups (hedges) as atomic deletion units. The problem unifies aspects of both combinatorial optimization and constraint satisfaction, with structural connections to CSPs such as Min Horn Deletion. Recent research precisely delineates the computational boundaries of Hedge Cluster Deletion, offering tight hardness and inapproximability results while also identifying polynomial-time tractable regimes determined by the structure of the underlying hedge partition and the input graph.

1. Formal Definitions and Problem Statement

A hedge graph is a triple $G = (V, E, \mathcal{H})$, where $V$ is a finite vertex set, $E \subseteq \binom{V}{2}$ is an undirected edge set, and $\mathcal{H} = \{ H_1, H_2, \dots, H_r \}$ partitions $E$ into pairwise disjoint hedges ($\bigcup_i H_i = E$, $H_i \cap H_j = \emptyset$ for $i \neq j$) (Konstantinidis et al., 13 Nov 2025). A cluster graph is a graph in which each connected component is a clique.

Hedge Cluster Deletion (decision version): Given a hedge graph $G$ and integer $k \ge 0$, decide whether there exists a set $S \subseteq \mathcal{H}$ of at most $k$ hedges such that deleting all edges in $\bigcup_{H \in S} H$ results in a cluster graph. The optimization variant seeks the minimal such $|S|$.

This generalizes Cluster Deletion, where each hedge is a singleton edge ($r = |E|$), and encompasses previous variants such as Highly Connected (Hedge) Cluster Deletion, where each cluster is required to be not a clique but a "highly connected" subgraph (minimum degree or edge connectivity exceeding half its order) (Bliznets et al., 2017).

2. Complexity and Inapproximability

Hedge Cluster Deletion is NP-complete in the general case—this is true even in highly restricted settings. Specifically, for every integer $\delta$, the problem is NP-complete even when the underlying graph is the disjoint union of $\delta$ copies of $P_3$ (3-vertex path), and the number of hedges $r \leq 3\delta$ (Konstantinidis et al., 13 Nov 2025). The key reduction is from Vertex Cover: for each edge of the input, create a $P_3$ split between two hedges, so that covering all $P_3$s corresponds precisely to covering all edges by hedges, i.e., vertices in the original graph.

A dichotomy emerges based on the largest number $\delta$ of vertex-disjoint $P_3$s in the graph:

• If $\delta = O(1)$, Hedge Cluster Deletion can be solved exactly in $n^{O(\delta)}$ time (where $n = |V|$).
• If $\delta$ is unbounded, the problem is NP-complete (Konstantinidis et al., 13 Nov 2025).

Inapproximability: Hedge Cluster Deletion is tightly connected to Min Horn Deletion (a canonical boolean CSP optimization problem) via $A$-reductions. As a consequence, the problem is NP-hard to approximate within factor $2^{O(\log^{1-\epsilon} r )}$ for any $\epsilon > 0$, unless $$\mathrm{NP} \subseteq \mathrm{DTIME}(2^{\log^{O(1)} n})$$, where $r = |\mathcal{H}|$ (Konstantinidis et al., 13 Nov 2025).

Graph Class	Tractability	Reference
General hedge graphs	NP-complete, hard to approx	(Konstantinidis et al., 13 Nov 2025)
Bounded disjoint $P_3$	Polytime for constant $\delta$	(Konstantinidis et al., 13 Nov 2025)
Bi-hedge graphs	Polytime 2-approximation	(Konstantinidis et al., 13 Nov 2025)
Acyclic hedge-intersection	Exact polytime solution	(Konstantinidis et al., 13 Nov 2025)

3. Approximation Algorithms and Tractable Cases

Despite general hardness, important subclasses admit efficient algorithms or constant-factor approximation.

3.1. Subclasses with Polynomial-Time or Approximable Solutions

Bounded Disjoint $P_3$s: If the largest collection of vertex-disjoint $P_3$s has constant size ($\delta$), enumerate all such placements and possible clusterings in $n^{O(\delta)}$ time (Konstantinidis et al., 13 Nov 2025).

Bi-hedge Graphs (Every triangle in ≤2 hedges): A polynomial-time 2-approximation is achievable. This is done by

• Deleting all hedges that span an “internal” $P_3$ (both edges of a $P_3$ in one hedge).
• Constructing an auxiliary graph with hedges as vertices and adding edges corresponding to inter-hedge $P_3$s; domination-closure lists $L(x)$ enable a reduction to multi-vertex cover, which in this case can be reduced to a vertex cover and 2-approximated (Konstantinidis et al., 13 Nov 2025).

Acyclic Hedge Intersection Graphs: If the intersection graph of hedges (vertices: hedges; edges: shared incident vertices between hedges) is a forest, then Hedge Cluster Deletion is exactly solvable in polynomial time. The solution involves deleting “critical” hedges, partitioning the intersection graph, constructing an auxiliary bipartite graph via domination closure, and solving minimum vertex cover on it (Konstantinidis et al., 13 Nov 2025). The full details and correctness are formalized in [(Konstantinidis et al., 13 Nov 2025), Lemmas 4.4–4.7].

3.2. Contrasts with Edge-Based Cluster Deletion

Standard Cluster Deletion (where each hedge is a single edge) is NP-complete but admits

• Parameterized algorithms: $O^*(1.404^k)$ FPT algorithm (Tsur, 2019).
• A 2-approximation via LP rounding and CC-pivot (Veldt et al., 2017).
• A deterministic 3-approximation using combinatorial pivoting schemes (Balmaseda et al., 24 Apr 2024).

Hedge Cluster Deletion resists such approximations due to its CSP-completeness; the best known approximation in the general case is super-polynomial in $r$ (Konstantinidis et al., 13 Nov 2025).

4. Algorithms and Structural Decompositions

4.1. Auxiliary Graph Constructions

In bi-hedge and acyclic intersection cases, auxiliary graphs or meta-graphs are constructed with hedges as vertices, and connections represent interactions via $P_3$s or triangles in the original graph. The domination-closure lists and multi-vertex cover property guarantee structured reductions to vertex cover problems, which are efficiently solvable when the underlying auxiliary graph is bipartite or admits certain nesting properties among domination lists (Konstantinidis et al., 13 Nov 2025).

4.2. Algorithmic Workflow for Acyclic Intersection

A precise procedure (Algorithm 1 in (Konstantinidis et al., 13 Nov 2025)) establishes:

• Build the intersection graph $\mathcal{F}$.
• Identify and delete hedges corresponding to critical subgraphs (internal $P_3$s, "mixed" triangle-edges).
• Iteratively partition the remaining instance and apply minimum vertex cover on the bipartite auxiliary graph representing domination-closures among hedges.

4.3. Complexity in Terms of Hedge Structure

The structure of the hedge partition fundamentally determines tractability:

• If the intersection graph is a forest, exact polynomial-time algorithms are possible.
• If each triangle is in at most two hedges, a 2-approximation exists.
• With unbounded intersection cycles or triangle complexity, hard CSP phenomena dominate, precluding efficient algorithms or PTASs (Konstantinidis et al., 13 Nov 2025).

5. Connections to Constraint Satisfaction and Broader Implications

Hedge Cluster Deletion is complete for Min Horn Deletion under $A$-reductions as formalized by Khanna et al., which has crucial consequences:

• Any algorithmic or approximability breakthrough for Hedge Cluster Deletion would extend to a wide class of CSPs (Konstantinidis et al., 13 Nov 2025).
• The intractability is robust to the hedge grouping: grouping edges into hedges neither simplifies nor worsens the core computational hardness.
• Hardness of approximation within polylogarithmic factors in $r$ is a direct corollary.

These connections embed Hedge Cluster Deletion firmly at the interface of graph modification problems and CSP approximation complexity. The classification of tractability and inapproximability uniquely hinges on the hedge structure—unifying graph-theoretic and CSP perspectives.

6. Summary of Key Results and Current Landscape

The table below summarizes the main algorithmic and hardness results for Hedge Cluster Deletion:

Instance property	Complexity/Algorithm	Proven Bound
Unrestricted hedges	NP-complete, Min Horn-complete	$2^{O(\log^{1-\epsilon} r)}$ inapproximable
Maximum disjoint $P_3$s constant	Exact polytime: $n^{O(\delta)}$	(Konstantinidis et al., 13 Nov 2025)
Bi-hedge graphs ($\leq$2 hedges/triangle)	Polynomial-time 2-approximation	(Konstantinidis et al., 13 Nov 2025)
Acyclic hedge intersection graph	Exact polytime by vertex cover	(Konstantinidis et al., 13 Nov 2025)

In summary, Hedge Cluster Deletion generalizes and sharpens the classical theory of cluster deletion by introducing hedge structures, resulting in strong dichotomies and inapproximability boundaries governed by the hedge-induced substructure. The problem's landscape is precisely mapped, with exact and approximate algorithms available on narrow subfamilies, and cryptographically hard-to-approximate elsewhere, coinciding with major CSP lower bounds (Konstantinidis et al., 13 Nov 2025).