Complete partitions of graphs

A complete partition of a graph G is a partition of its vertex set in which any two distinct classes are connected by an edge. Let cp(G) denote the maximum number of classes in a complete partition of G. This measure was defined in 1969 by Gupta [G69], and is known to be NP-hard to compute for several classes of graphs. We obtain essentially tight lower and upper bounds on the approximability of this problem. We show that there is a randomized polynomial-time algorithm that given a graph G with n vertices, produces a complete partition of size Ω(cp(G)/ √ lg n). This algorithm can be derandomized. We show that the upper bound is essentially tight: there is a constant C > 1, such that if there is a randomized polynomial-time algorithm that for all large n, when given a graph G with n vertices produces a complete partition into at least C · cp(G)/ √ lg n classes, then NP ⊆ RTime(nO(lg lg n) ). The problem of finding a complete partition of a graph is thus the first natural...

• Type: Journal
• Year: 2007
• Where: COMBINATORICA
• Authors: Magnús M. Halldórsson, Guy Kortsarz, Jaikumar Radhakrishnan, Sivaramakrishnan Sivasubramanian
• Keywords: COMBINATORICA 2007, COMBINATORICA 2008, Complete Partition, Randomized Polynomial-time Algorithm, Upper Bound