Efficient algorithms with performance guarantees for some problems of finding several cliques in a complete undirected weighted graph
Edward Kh. Gimadi, Alexander V. Kel’manov, A. V. Pyatkin, Michael Khachay · Proceedings of the Steklov Institute of Mathematics · 2015
We consider the problem of finding a fixed number of vertex-disjoint cliques of given sizes in a complete undirected weighted graph so that the total weight of vertices and edges in the cliques would be minimal. We show that the problem is strongly NP-hard both in the general case and in two subclasses, which have important applications. An approximation algorithm for this problem is presented. We show that the algorithm finds a solution with a bounded approximation ratio for the considered subclasses of the problem, and the bound is attainable. In the case when the number of cliques to be found is fixed in advance (i.e., is a parameter), the time complexity of the algorithm is polynomial.