Cluster Vertex Deletion on Chordal Graphs

Abstract: We present a polynomial-time algorithm for the cluster vertex deletion problem on chordal graphs, resolving an open question posed in different contexts by Cao et al. [Theoretical Computer Science, 2018], Aprile et al. [Mathematical Programming, 2023], Chakraborty et al. [Discrete Applied Mathematics, 2024], and Hsieh et al. [Algorithmica, 2024]. We use dynamic programming over clique trees and reduce the computation of the optimal subproblem value to the minimization of a submodular set function.

• The paper proves that cluster vertex deletion is polynomial-time solvable on chordal graphs, resolving an open question left by several earlier studies.
• The algorithm uses dynamic programming over clique trees and maximizes a supermodular constrained-optimum function through submodular minimization, achieving O(n^7) running time.
• The supermodularity proof relies on an acyclic interaction graph between clusters in two optimal solutions, offering a potentially reusable technique for related deletion problems on chordal graphs.

Background and problem statement

The cluster vertex deletion problem asks, given a vertex-weighted graph $G$, to remove a minimum-weight vertex set so that the remainder is a cluster graph (a disjoint union of cliques). Equivalently, since cluster graphs are exactly the $P_3$-free graphs, it is a hitting set problem for induced $P_3$'s. The problem is NP-hard in general (2604.20457), admits a 3-approximation via local ratio, improved to 2 by Aprile et al., and serves as a canonical testbed for parameterized algorithms, kernelization, and approximation techniques. The deletion weight is also used as a structural parameter ("distance to cluster"), notable for remaining meaningful on dense graphs.

The paper resolves a question posed independently by Cao et al. (TCS 2018), Aprile et al. (Mathematical Programming 2023), Chakraborty et al. (DAM 2024), and Hsieh et al. (Algorithmica 2024): is cluster vertex deletion polynomial-time solvable on chordal graphs? Prior work settled split graphs and interval graphs [cao-18], well-partitioned chordal graphs [chakraborty-24], but general chordal graphs remained open; Aprile et al. had even suggested hardness was possible there.

Main result

The central contribution is:

Theorem 1. There is a polynomial-time algorithm for cluster vertex deletion on chordal graphs.

The paper works with the equivalent maximization form: find a maximum-weight induced cluster subgraph $S$, denoted $\psi(G)$. The algorithm runs in $O(n^7)$ time, which the authors note can likely be improved with more careful analysis — an explicit concession that the exponent is not tight.

Algorithmic framework: dynamic programming over clique trees

Chordal graphs admit clique trees: trees whose nodes are maximal cliques such that for each vertex, the nodes containing it form a connected subtree [dirac-61]. The dynamic program roots the clique tree at an arbitrary node $R$ and defines subproblems on $G_K = G[\bigcup_{Q \in T_K} Q \setminus P(K)]$ — deliberately excluding vertices shared with the parent clique. This unusual definition ensures that if an optimal solution of $G_K$ avoids $K$, its value decomposes as $P_3$0 over children.

Two structural facts drive the recurrence. First, every cluster of an optimal solution of $P_3$1 lies inside some maximal clique $P_3$2 of the subtree $P_3$3 minus the parent's vertices (Proposition 3). Second, if the distinguished cluster $P_3$4 intersects $P_3$5, then every component of $P_3$6 equals a component of some $P_3$7 with $P_3$8 in $P_3$9 (Lemma 4), so $P_3$0 is readable from the table in $P_3$1 time by summing precomputed values over surviving subtree roots.

The remaining task reduces to finding a clique $P_3$2 intersecting $P_3$3 maximizing $P_3$4. Guessing the maximal clique $P_3$5 and a vertex $P_3$6, one must maximize over $P_3$7 the function

$P_3$8

A subtlety arises here: submodular minimization requires evaluating arbitrary sets, but the intersection of two subsets both meeting $P_3$9 may miss $S$0, making evaluation impossible from the table. Fixing $S$1 sidesteps this, since all evaluated sets contain $S$2.

Supermodularity: the structural core

The key ingredient is:

Theorem 2. For a fixed clique $S$3 of a weighted chordal graph $S$4, the function $S$5 — the weight of an optimal solution in which $S$6 appears as a cluster — is supermodular on $S$7: $S$8.

Since maximizing a supermodular function is equivalent to minimizing a submodular one, Jiang's oracle-based algorithm [jiang-22] applies, using $S$9 evaluations per $\psi(G)$0 pair.

The proof constructs two optimal solutions $\psi(G)$1 witnessing $\psi(G)$2 and builds an auxiliary bipartite graph $\psi(G)$3 whose nodes are clusters of $\psi(G)$4 and $\psi(G)$5, with edges representing adjacency in $\psi(G)$6. Chordality forces $\psi(G)$7 to be acyclic: a cycle would yield an induced cycle of length at least four in $\psi(G)$8 via a path threaded through the cycle's clusters. In particular, the edge $\psi(G)$9 is a bridge. Removing it splits $O(n^7)$0 into two components, from which two new solutions are assembled: $O(n^7)$1 containing $O(n^7)$2 as a cluster and $O(n^7)$3 containing $O(n^7)$4, with $O(n^7)$5. This repartitioning argument establishes the inequality directly.

Warm-up: interval graphs

Before the main algorithm, the paper gives a simpler $O(n^7)$6 dynamic program for interval graphs via their clique paths, based on the recurrence

$O(n^7)$7

which captures the rightmost cluster of an optimal solution. The chordal algorithm can be read as a tree-generalization of this path-based scheme.

Complexity accounting

For each node $O(n^7)$8 ($O(n^7)$9 choices), each $R$0 ($R$1), and each $R$2 ($R$3), Jiang's algorithm performs $R$4 oracle calls, each costing $R$5, giving $R$6 per node and $R$7 overall. The authors state plainly that this bound "can likely be improved."

Limitations and open questions

The paper leaves several points open. The $R$8 running time is acknowledged as improvable, and no lower bound or fine-grained analysis is given. Theorem 2 is proved only for chordal graphs; whether the supermodularity structure extends to related classes (e.g., $R$9-free graphs more broadly, where Aprile et al.'s approximation question originated) is not addressed. The complexity of cluster vertex deletion on other intermediate classes between well-partitioned chordal and chordal graphs is likewise untouched. Finally, the algorithm is purely existential in presentation; practical constants and implementation details are not explored.

Conclusion

This paper settles a long-standing open question by showing that cluster vertex deletion is polynomial-time solvable on chordal graphs, via dynamic programming over clique trees combined with supermodular maximization. The self-contained proof that the "constrained optimum" function $G_K = G[\bigcup_{Q \in T_K} Q \setminus P(K)]$0 is supermodular — resting on the fact that the interaction graph of two solutions' clusters must be acyclic — is presented in a general form likely to be reusable for other deletion problems on chordal graphs.