## 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*)/√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(*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*((lg*n*)^{c}) for some constant *c* strictly between 0 and 1.

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## References

- [1]
S. Arora and C. Lund: Hardness of approximations, in: D. Hochbaum, editor,

*Approximation Algorithms for NP-Hard Problems*, PWS Publishing Company, 1996. - [2]
S. Arora, C. Lund, R. Motwani, M. Sudan and M. Szegedy: Proof verification and hardness of approximation problems,

*J. ACM***45(3)**(1998), 501–555. - [3]
R. Balasubramanian, V. Raman and V. Yegnanarayanan: On the pseudoachromatic number of join of graphs,

*J. Computer Math.***80**(2003), 1131–1137. - [4]
H. L. Bodlaender: Achromatic number is NP-complete for cographs and interval graphs,

*Inform. Process. Lett.***31(3)**(1989), 135–138. - [5]
V. N. Bhave: On the pseudoachromatic number of a graph,

*Fund. Math.***102(3)**(1979), 159–164. - [6]
B. Bollobás, P. A. Catlin and P. Erdős: Hadwiger’s conjecture is true for almost all graphs,

*Europ. J. Combin.***1**(1980), 195–199. - [7]
B. Bollobás, B. Reed and A. Thomason: An extremal function for the achromatic number,

*Graph structure theory*, 161–165,*Contemp. Math.***147**, AMS 1993. - [8]
N. Cairnie and K. J. Edwards: Some results on the achromatic number,

*J. Graph Theory***26(3)**(1997), 129–136. - [9]
N. Cairnie and K. J. Edwards: The achromatic number of bounded degree trees,

*Discrete Math.***188**(1998), 87–97. - [10]
J. Chuzhoy, S. Guha, E. Halperin, S. Khanna, G. Kortsarz and J. Naor: Tight lower bounds for the asymmetric

*k*-center problem, in*Proc. 36th Annual ACM Symposium on Theory of Computing*(STOC), 21–27, 2004. - [11]
D. Dubhashi and D. Ranjan: Balls and bins: A study in negative dependence,

*Random Structures and Algorithms***13(2)**(1998), 99–124. - [12]
K. J. Edwards: The harmonious chromatic number and the achromatic number, in

*Surveys in Combinatorics 1997*, Cambridge University Press, pages 13–47, 1997. - [13]
K. J. Edwards: Achromatic number versus pseudoachromatic number: A counterexample to a conjecture of Hedetniemi,

*Discrete Math.***219**(2000), 271–274. - [14]
J. Edmonds and E. L. Johnson: Matching: A well solved class of integer linear programs, in

*Combinatorial Structures and Their Appl. (Proc. Calgary Internat. Conf.)*, pages 89–92, 1970. - [15]
K. J. Edwards and C. J. H. McDiarmid: The complexity of harmonious coloring for trees,

*Disc. Appl. Math.***57**(1995), 133–144. - [16]
U. Feige: A threshold of ln

*n*for approximating set cover,*J. ACM***45(2)**(1998), 634–652. - [17]
U. Feige, M. M. Halldórsson, G. Kortsarz and A. Srinivasan: Approximating the domatic number,

*SIAM J. Computing***32(1)**(2002), 172–195. - [18]
H. N. Gabow: An efficient reduction technique for degree constrained subgraph and bidirected network flow problems, in

*Proc. 15th Annual ACM Symposium on Theory of Computing*(STOC), pages 448–456, 1983. - [19]
R. P. Gupta: Bounds on the chromatic and achromatic numbers of complementary graphs, in

*Recent Progress in Combinatorics*Academic Press, New York (1969), pp. 229–235. - [20]
M. M. Halldórsson: Approximating the minimum maximal independence number,

*Inform. Process. Lett.***46(4)**(1993), 169–172. - [21]
M. M. Halldórsson: An approximation algorithm for complete partitions of regular graphs, Unpublished manuscript, http://www.hi.is/:_mmh/publications.html, February 2004.

- [22]
J. Håstad: Clique is hard to approximate within

*n*^{1−ε},*Acta Mathematica***182**(1999), 105–142. - [23]
J. Håstad: Some optimal inapproximability results,

*J. ACM***48**(2001), 798–859. - [24]
E. Halperin and R. Krauthgamer: Polylogarithmic inapproximability, in

*Proc. 35th Annual ACM Symposium on Theory of Computing*(STOC), 585–594, 2003. - [25]
S. Hedetniemi: Open problems in combinatorial optimization, Webpage at www.cs.clemson.edu/:_hedet/coloring.html, September 1998.

- [26]
C. Lund and M. Yannakakis: On the hardness of approximating minimization problems,

*J. ACM***41(5)**(1994), 960–981. - [27]
A. V. Kostochka: The minimum Hadwiger number for graphs with a given mean degree of vertices,

*Metody Diskret. Analiz.***38**(1982), 37–58 (in Russian). - [28]
G. Kortsarz: On the hardness of approximating spanners,

*Algorithmica***30(3)**(2001), 432–450. - [29]
G. Kortsarz and R. Krauthgamer: On the approximation of the achromatic number,

*SIAM Journal of Discrete Math.***14(3)**(2000), 408–422. - [30]
G. Kortsarz, J. Radhakrishnan and S. Sivasubramanian: Complete partitions of graphs, in

*Proc. 16th Annual SIAM-ACM Symp. Discrete Algorithms*(SODA), pages 860–869, 2005. - [31]
G. Kortsarz and S. Shende: Approximating the achromatic number problem on bipartite graphs, in

*Proceedings of 11th European Symposium on Algorithms*(Budapest, 2003),*Lecture Notes in Computer Science***2832**, Springer, Berlin (2003), pp. 385–396. - [32]
C. J. H. McDiarmid: Achromatic numbers of random graphs,

*Mathematical Proceedings of Cambridge Philosophical Society***92**(1982), 21–28. - [33]
C. McDiarmid: On the method of bounded differences,

*London Mathematical Society Lecture Notes Series***141**(1989), 148–188. - [34]
R. Raz: A parallel repetition theorem,

*SIAM J. Comput.***27(3)**(1998), 763–803. - [35]
N. Robertson, P. D. Seymour and R. Thomas: Hadwiger’s conjecture for

*K*_{6}-free graphs,*Combinatorica***13(3)**(1993), 279–361. - [36]
E. Sampathkumar and V. N. Bhave: Partition graphs and coloring numbers of a graph,

*Discorete Math.***16(1)**(1976), 57–60. - [37]
V. G. Vizing: On an estimate of the chromatic class of a

*p*-graph,*Diskret. Anal.***3**(1964), 25–30 (in Russian). - [38]
V. Yegnanarayanan: On pseudocoloring of graphs,

*Util. Math.***62**(2002), 199–216.

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## Additional information

The work reported here is a merger of the results reported in [30] and [21].

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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

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### Mathematics Subject Classification (2000)

- 68W25
- 68W20
- 68Q17
- 05C70