TY - JOUR

T1 - Complete partitions of graphs

AU - Halldórsson, Magnús M.

AU - Kortsarz, Guy

AU - Radhakrishnan, Jaikumar

AU - Sivasubramanian, Sivaramakrishnan

N1 - Copyright:
Copyright 2008 Elsevier B.V., All rights reserved.

PY - 2007/9

Y1 - 2007/9

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=44449116635&partnerID=8YFLogxK

U2 - 10.1007/s00493-007-2169-9

DO - 10.1007/s00493-007-2169-9

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AN - SCOPUS:44449116635

SN - 0209-9683

VL - 27

SP - 519

EP - 550

JO - Combinatorica

JF - Combinatorica

IS - 5

ER -