### Abstract

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 [19], 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.

Original language | English (US) |
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Pages (from-to) | 519-550 |

Number of pages | 32 |

Journal | Combinatorica |

Volume | 27 |

Issue number | 5 |

DOIs | |

State | Published - Sep 1 2007 |

### All Science Journal Classification (ASJC) codes

- Computational Mathematics
- Discrete Mathematics and Combinatorics

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## Cite this

*Combinatorica*,

*27*(5), 519-550. https://doi.org/10.1007/s00493-007-2169-9