Clustering bipartite and chordal graphs: Complexity, sequential and parallel algorithms

We study a group of clustering problems on bipartite and chordal graphs. Our objective is to partition the vertices of a graph into a restricted number of sets so that a prespecified, diameter related, objective function is minimized. We unify a few problems using monotone diameter functions defined on sub-partitions of a graph. Among these problems are the following: partition vertices of a graph into a restricted number of subgraphs of bounded diameter, and partition vertices of a graph into a restricted number of subgraphs so the sum of the diameters of the subgraphs is bounded. We show that the first of the aforementioned problems is NP-complete on bipartite and chordal graphs, but has linear time sequential solutions on interval and bipartite permutation graphs. As well, we show that the unified problem has an NC parallel algorithm on interval graphs.