An improved parameterized algorithm for the p-cluster vertex deletion problem

• Published in: Journal of combinatorial optimization Vol. 33; no. 2; pp. 373 - 388
• Main Authors: Wu, Bang Ye, Chen, Li-Hsuan
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.02.2017; Springer Nature B.V
• Subjects: Algorithms; Apexes; Clusters; Combinatorics; Complexity; Convex and Discrete Geometry; Deletion; Graph theory; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Operations Research/Decision Theory; Optimization; Theory of Computation; Upper bounds; Cluster graph; Parameterized algorithm; Graph modification; Exact algorithm
• ISSN: 1382-6905; 1573-2886
• DOI: 10.1007/s10878-015-9969-4

In the p - Cluster Vertex Deletion problem, we are given a graph G = ( V , E ) and two parameters k and p , and the goal is to determine if there exists a subset X of at most k vertices such that the removal of X results in a graph consisting of exactly p disjoint maximal cliques. Let r = p / k . In this paper, we design a branching algorithm with time complexity O ( α k + | V | | E | ) , where α depends on r and has a rough upper bound min { 1 . 618 1 + r , 2 } . With a more precise analysis, we show that α = 1.28 · 3 . 57 r for r ≤ 0.219 ; α = ( 1 - r ) r - 1 r - r for 0.219 < r < 1 / 2 ; and α = 2 for r ≥ 1 / 2 , respectively. Our algorithm also works with the same time complexity for the variant that the number of clusters is at most p . Our result improves the previous best time complexity O ∗ ( 1 . 84 p + k ) and implies that for fixed p the problem can be solved as efficiently as Vertex Cover .