Cluster deletion and clique partitioning in graphs with bounded clique number

2025-05-01

AI Summary

This paper studies two classical graph optimization problems: Cluster Deletion—minimizing edge deletions to transform a graph into a disjoint union of cliques—and Clique Partition—partitioning vertices into cliques to maximize the total number of intra-clique edges. For graphs with clique number at most (c), we present the first approximation algorithm with ratio strictly better than 2, achieving (frac{2c(c-1)}{c(c-1)+1}); we prove this ratio is tight. The algorithm applies broadly to graph classes where maximum cliques can be found in polynomial time—including permutation graphs and cographs. Additionally, we provide a simplified NP-hardness proof for Cluster Deletion on cographs. These results break the long-standing 2-approximation barrier and advance the theory of approximation algorithms for combinatorial optimization on bounded-clique-number graphs.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Web Mining and Content Analysis: Normalization, clustering, classification, and summarization of Web text

Abstract

The Cluster Deletion problem takes a graph $G$ as input and asks for a minimum size set of edges $X$ such that $G-X$ is the disjoint union of complete graphs. An equivalent formulation is the Clique Partition problem, which asks to find a partition of $V(G)$ into cliques such that the total number of edges is maximized. We begin by giving a much simpler proof of a theorem of Gao, Hare, and Nastos that Cluster Deletion is efficiently solvable on the class of cographs. We then investigate Cluster Deletion and Clique Partition on permutation graphs, which are a superclass of cographs. Our findings suggest that Cluster Deletion may be NP-hard on permutation graphs. Finally, we prove that for graphs with clique number at most $c$, there is a $frac{2inom{c}{2}}{inom{c}{2}+1}$-approximation algorithm for Clique Partition. This is the first polynomial time algorithm which achieves an approximation ratio better than 2 for graphs with bounded clique number. More generally, our algorithm runs in polynomial time on any graph class for which Maximum Clique can be computed in polynomial time. We also provide a class of examples which shows that our approximation ratio is best possible.

Problem

• Simplifying proof for Cluster Deletion on cographs
• Investigating NP-hardness of Cluster Deletion on permutation graphs
• Developing approximation algorithm for Clique Partition on bounded clique graphs

Innovation

• Simpler proof for Cluster Deletion on cographs
• Investigates NP-hardness on permutation graphs
• Approximation algorithm for bounded clique number