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 , 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)/√lgn). 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)/√lgn classes, then NP ⊆ RTime(n O(lg lg n)). The problem of finding a complete partition of a graph is thus the first natural problem whose approximation threshold has been determined to be of the form Θ((lgn)c) for some constant c strictly between 0 and 1.
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The work reported here is a merger of the results reported in  and .
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Halldórsson, M.M., Kortsarz, G., Radhakrishnan, J. et al. Complete partitions of graphs. Combinatorica 27, 519 (2007). https://doi.org/10.1007/s00493-007-2169-9
Mathematics Subject Classification (2000)