Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2505.00922

Cluster deletion in cographs, permutation graphs, and graphs with bounded clique number

Nicola Galesi·Tony Huynh·Arnaud Patey·Fariba Ranjbar
View original source ↗Read source document ↗Export metadata ↓

Abstract

The Cluster Deletion problem asks for a minimum-size edge set whose deletion turns a graph into a disjoint union of complete graphs. Equivalently, the Clique Partition problem asks for a partition of the vertex set into cliques that maximizes the number of edges within the parts. We give a simpler proof of a result of Gao, Hare, and Nastos (Discete Mathematics, 2013), that Cluster Deletion is polynomial-time solvable on cographs. In addition, we show that the natural linear programming formulation of Clique Partition is exact on cographs. We then show that Cluster Deletion is NP-complete on permutation graphs, which are a superclass of cographs. This answers an open question of Konstantinidis and Papadopoulos (Algorithmica, 2021). We also exhibit a permutation graph on nine vertices for which the linear programming formulation is not exact. Finally, for graphs with clique number at most $c$, we give a polynomial-time $2\binom{c}{2}/(\binom{c}{2}+1)$-approximation algorithm for Clique Partition. More generally, the algorithm runs in polynomial time on every graph class for which a maximum clique can be found in polynomial time. For each fixed $c\geq 3$, we also construct infinitely many examples attaining the stated approximation ratio. The same examples show that, for Cluster Deletion , the algorithm is a $2$-approximation and no better, for every fixed $c \geq 3$.

Explore related subjects

cs.DScs.DMmath.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Nicola Galesi, Tony Huynh, Arnaud Patey, Fariba Ranjbar. 2026-09-15. Cluster deletion in cographs, permutation graphs, and graphs with bounded clique number. https://arxiv.org/abs/2505.00922

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Beyond Kruskal: Polynomial-Time Tensor Decomposition under the Lovitz-Petrov Condition

Identifiability criteria certify that a given tensor decomposition is a unique rank decomposition. Kruskal's classical condition is one of the best-known deterministic criteria for identifiability. However, no polynomial-time decomposition algorithm is known under the Kruskal condition, and verifying the condition itself is NP-hard. Lovitz and Petrov introduced a strictly more general identifiability condition which, in contrast, is polynomial-time verifiable, but no polynomial-time decomposition algorithm was previously known under this condition. We give a polynomial-time algorithm for tensor decomposition under the Lovitz--Petrov condition. Moreover, combining our algorithm with polynomial-time verification of the Lovitz--Petrov condition yields an efficient end-to-end certification procedure: after computing a decomposition, one can deterministically certify in polynomial time that it is unique and therefore of minimum rank. This contrasts with an arbitrary tensor decomposition, which certifies only an upper bound on the tensor rank, while determining tensor rank is NP-hard in general.

cs.DS↗
arXiv2026

Poisson Exchange Beyond Submodularity: Effective Approximation Algorithms for Offline and Online Subset Selection over Matroids

Over the past decade, a growing body of research has shown that $γ$-weak submodularity broadly arises in numerous subset selection tasks, including feature selection, neural network pruning, and video summarization. Despite its prevalence, maximizing a $γ$-weakly submodular function subject to a general matroid constraint remains challenging. To date, the only known approximation guarantee is the conservative $(1+1/γ)^{-2}$ factor established by \citet{chen2018weakly}. To improve upon this result, this paper proposes a novel algorithm called \MGPE, which repeatedly performs maximum-gain local exchanges through careful control of a non-homogeneous Poisson clock, and proves that this \MGPE\ can attain an approximation ratio arbitrarily close to $ρ_γ=1-\left(γ/(2-γ)\right)^{ \frac{γ^2}{2(1-γ)} }$. In sharp contrast to the previous guarantee, our obtained factor $ρ_γ$ not only strictly improves upon $(1+1/γ)^{-2}$ for every $γ\in(0,1]$, but also can asymptotically approach the optimal $(1-1/e)$-approximation for submodular maximization as $γ\to1$. Furthermore, we surprisingly find that when the matroid constraint reduces to a cardinality or the objective satisfies the stronger notion of $α$-weak DR-submodularity, \MGPE\ can automatically recover the tight approximation ratios of $1-e^{-γ}$ and $1-e^{-α}$, respectively. Here, $α\in(0,1]$ denotes the DR ratio.

cs.DS↗
arXiv2026

Approximating Prize-Collecting TSP below 1.556

The prize-collecting traveling salesperson problem is a variant of the metric traveling salesperson problem in which vertices may be left unvisited by paying their associated penalties. The objective is to minimize the length of the tour plus the total penalty of the unvisited vertices. Blauth, Klein, and Nägele gave the previously best-known LP-relative $1.599$-approximation. We show that a simpler version of their algorithm, obtained by omitting the splitting-off preprocessing before the tree decomposition, has an LP-relative approximation ratio of $1.555761$. The improvement comes entirely from a new analysis of the parity-correction step: a simple analysis already gives $1.56$, and the stated factor follows from a numerical parameter search with exact verification.

cs.DS↗