Parameterized algorithms for min–max 2-cluster editing

• Published in: Journal of combinatorial optimization Vol. 34; no. 1; pp. 47 - 63
• Main Authors: Chen, Li-Hsuan, Wu, Bang Ye
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.07.2017; Springer Nature B.V
• Subjects: Algorithms; Clustering; Clusters; Combinatorics; Complexity; Convex and Discrete Geometry; Design parameters; Editing; Graph theory; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Operations Research/Decision Theory; Optimization; Parameterization; Satellites; Theory of Computation; Subexponential algorithm; Parameterized algorithm; Graph modification; Clustering; Randomized algorithm; Parameterized complexity
• ISSN: 1382-6905; 1573-2886
• DOI: 10.1007/s10878-016-0059-z

For a given graph and an integer t , the Min – Max 2-Clustering problem asks if there exists a modification of a given graph into two maximal disjoint cliques by inserting or deleting edges such that the number of the editing edges incident to each vertex is at most t . It has been shown that the problem can be solved in polynomial time for t < n / 4 , where n is the number of vertices. In this paper, we design parameterized algorithms for different ranges of t . Let k = t - n / 4 . We show that the problem is polynomial-time solvable when roughly k < n / 32 . When k ∈ o ( n ) , we design a randomized and a deterministic algorithm with sub-exponential time parameterized complexity, i.e., the problem is in SUBEPT. We also show that the problem can be solved in O ( 2 n / r · n 2 ) time for k < n / 12 and in O ( n 2 · 2 3 n / 4 + k ) time for n / 12 ≤ k < n / 4 , where r = 2 + ⌊ ( n / 4 - 3 k - 2 ) / ( 2 k + 1 ) ⌋ ≥ 2 .