Cluster Editing on Cographs and Related Classes

TL;DR

The paper maps the complexity landscape of Cluster Editing on restricted graph classes. It proves NP-hardness and W[1]-hardness for p-Cluster Editing on cographs (even for height-3 cotrees) and establishes ETH-based lower bounds, while providing near-tight upper bounds via an n^{O(p·cw)}-time DP and, for cw ≤ 2, an n^{O(p)}-time algorithm. It further shows a polynomial-time solution for the deletion-only variant on cographs and an exact O(n^3) algorithm for Cluster Editing on trivially perfect graphs, clarifying tractability boundaries. Overall, the work highlights substantial hardness of the editing variant beyond the deletion case and delineates an actionable algorithmic frontier across cographs, their width-parameter generalizations, and trivially perfect graphs.

Abstract

In the Cluster Editing problem, sometimes known as (unweighted) Correlation Clustering, we must insert and delete a minimum number of edges to achieve a graph in which every connected component is a clique. Owing to its applications in computational biology, social network analysis, machine learning, and others, this problem has been widely studied for decades and is still undergoing active research. There exist several parameterized algorithms for general graphs, but little is known about the complexity of the problem on specific classes of graphs. Among the few important results in this direction, if only deletions are allowed, the problem can be solved in polynomial time on cographs, which are the $P_4$-free graphs. However, the complexity of the broader editing problem on cographs is still open. We show that even on a very restricted subclass of cographs, the problem is NP-hard, W[1]-hard when parameterized by the number $p$ of desired clusters, and that time $n^{o(p/\log p)}$ is forbidden under the ETH. This shows that the editing variant is substantially harder than the deletion-only case, and that hardness holds for the many superclasses of cographs (including graphs of clique-width at most $2$, perfect graphs, circle graphs, permutation graphs). On the other hand, we provide an almost tight upper bound of time $n^{O(p)}$, which is a consequence of a more general $n^{O(cw \cdot p)}$ time algorithm, where $cw$ is the clique-width. Given that forbidding $P_4$s maintains NP-hardness, we look at $\{P_4, C_4\}$-free graphs, also known as trivially perfect graphs, and provide a cubic-time algorithm for this class.

Figures (3)

• Figure 1: An illustration of the construction. In the subgraph $I$, each $B_i$ is a "large enough" complete graph, and in the subgraph $J$, each $A_j$ is a complete graph with $a_j$ vertices. The wiggly line indicates that all possible edges between $I$ and $J$ are present (there are no edges between two $B_i$'s, and no edge between two $A_j$'s).
• Figure 2: Illustration of Lemma \ref{['lem:two-clusters']}. The tree shown is the cotree of $G$, with disc vertices being $1$-nodes and square vertices $0$-nodes. Here, $Y = X \cap C_1$ and $Y' = X \cap C_2$. If $|A| \geq |B|$ and $|A'| \geq |B'|$, then rearranging $C_1$ and $C_2$ in one of the two ways shown also gives an optimal clustering.
• Figure 3: Illustration of $a_i, a'_i, A_i, B_i, A'_i, B'_i$.

Theorems & Definitions (32)

• Theorem 1
• Proposition 1: DBLP:journals/jcss/ChenM12DBLP:journals/tcs/Guo09
• Definition 1
• Proposition 2: DBLP:journals/tcs/GurskiWY16johansson1998clique
• Proposition 3
• proof
• Lemma 1
• proof
• Claim 1
• proof