Exact algorithms for problems related to the densest k-set problem

• Published in: Information processing letters Vol. 114; no. 9; pp. 510 - 513
• Main Authors: Chang, Maw-Shang, Chen, Li-Hsuan, Hung, Ling-Ju, Rossmanith, Peter, Wu, Guan-Han
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.09.2014; Elsevier Sequoia S.A
• Subjects: Algorithms; Cohesion; Data processing; Densest (sparsest) k-set; Design of algorithms; Exact algorithm; Graph theory; Graphs; Mathematical problems; Minimum (maximum) k-vertex cover; Partial vertex cover; Running; Set theory; Social networks; Studies; Subgroups; Vertex sets; Partial vertex cover; Minimum (maximum) k-vertex cover; Design of algorithms; Exact algorithm; Densest (sparsest) k-set
• ISSN: 0020-0190; 1872-6119
• DOI: 10.1016/j.ipl.2014.04.009

Many graph concepts such as cliques, k-clubs, and k-plexes are used to define cohesive subgroups in a social network. The concept of densest k-set is one of them. A densest k-set in an undirected graph G=(V,E) is a vertex set S⊆V of size k such that the number of edges in the subgraph of G induced by S is maximum among all subgraphs of G induced by vertex sets of size k. One can obtain a densest k-set of G in O(k2nk) time by exhaustive-search technique for an undirected graph of n vertices and a number k<n. However, if the value of k approaches n/2, the running time of the exhaustive-search algorithm is O⁎(2n). Whether there exists an O⁎(cn)-time algorithm with the fixed constant c<2 to find a densest k-set in an undirected graph of n vertices remains open in the literature. In this paper, we point out that the densest k-set problem and a class of problems related to the concept of densest k-sets can be solved in time O⁎(1.7315n). •An O⁎(1.7315n)-time exact algorithm for the densest k-set problem is proposed.•The algorithm can be applied to solve a series of problems related to the densest k-set problem.•All these related problems can also be solved in time O⁎(1.7315n).