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Linear-Time Approximation Algorithms for the Max Cut Problem

Published online by Cambridge University Press:  12 September 2008

Nguyen van Ngoc  and
Zsolt Tuza
Nguyen van Ngoc
Affiliation:
Fl C20 Kim Giang, Dong Da – Ha Noi, Viet Nam
Zsolt Tuza
Affiliation:
Computer and Automation Institute, Hungarian Academy of Sciences, H-llll Budapest, Kende u. 13-17, Hungary
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Abstract

Let G be a connected graph with n vertices and m edges (multiple edges allowed), and let k ≥ 2 be an integer. There is an algorithm with (optimal) running time of O(m) that finds

(i) a bipartite subgraph of G with ≥ m/2 + (n − 1)/4 edges,

(ii) a bipartite subgraph of G with ≥ m/2 + 3(n−1)/8 edges if G is triangle-free,

(iii) a k-colourable subgraph of G with ≥ m − m/k + (n−1)/k + (k − 3)/2 edges if k ≥ 3 and G is not k-colorable.

Infinite families of graphs show that each of those lower bounds on the worst-case performance are best possible (for every algorithm). Moreover, even if short cycles are excluded, the general lower bound of m − m/k cannot be replaced by m − m/k + εm for any fixed ε > 0; and it is NP-complete to decide whether a graph with m edges contains a k-colorable subgraph with more than m − m/k + εm edges, for any k ≥ 2 and ε> 0, ε < 1/k.

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 2 , Issue 2 , June 1993 , pp. 201 - 210
Copyright
Copyright © Cambridge University Press 1993

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References

[1]Andersen, L. D., Grant, D. D. and Linial, N. (1983) Extremal k-colourable subgraphs. Ars Combinatoria 16 259–270.Google Scholar
[2]Caro, Y. and Tuza, Zs. (1991) Improved lower bounds on k-independence. J. Graph Theory 15 99–107.CrossRefGoogle Scholar
[3]Edwards, C. S. (1973) Some extremal properties of bipartite subgraphs. Canadian J. Math. 25 475–485.CrossRefGoogle Scholar
[4]Erdős, P. (1959) Graph theory and probability. Canadian J. Math. 11 34–38.CrossRefGoogle Scholar
[5]Erdős, P. (1964) On even subgraphs of graphs. Mat. Lapok 8 283–288 (in Hungarian).Google Scholar
[6]Erdős, P., Faudree, R., Pach, J. and Spencer, J. (1988) How to make a graph bipartite. J. Combinatorial Theory Ser. B 45 86–98.CrossRefGoogle Scholar
[7]Erdős, P. and Lovász, L. (1979) See §8, pp. 159–161. In: Erdős P., Problems and results in graph theory and combinatorial analysis. In: Bondy, J. A. and Murty, U. S. R. (eds), Graph Theory and Related Topics, Proc. conf. Waterloo (Canada) 1977, Academic Press, 153–163.Google Scholar
[8]Garey, M. R. and Johnson, D. S. (1979) Computers and Intractability – A Guide to the Theory of NP-Completeness, Freeman.Google Scholar
[9]Van Ngoc, Nguyen (1987) On Graph Colorings, Thesis, Hungarian Academy of Sciences, Budapest (in Hungarian).Google Scholar
[10]Poljak, S. and Turzik, D. (1982) A Polynomial algorithm for constructing a large bipartite subgraph, with an application to a satisfiability problem. Canadian J. Math. 34 519–524.CrossRefGoogle Scholar
[11]Poljak, S. and Turzik, D. (1986) A polynomial time heuristic for certain subgraph optimization problems with guaranteed worst case bound. Discrete Math. 58 99–104.CrossRefGoogle Scholar
[12]Poljak, S. and Tuza, Zs. (to appear) Bipartite subgraphs of triangle-free graphs. SIAM J. Discrete Math.Google Scholar
[13]Shearer, J. B. (1992) A note on bipartite subgraphs of triangle-free graphs. Random Structures & Algorithms 3 223–226.CrossRefGoogle Scholar
[14]Tuza, Zs. (1992) Graph coloring in linear time. J. Combinatorial Theory Ser B 55 236–243.CrossRefGoogle Scholar
[15]Tuza, Zs. (to appear) Theorem proving through depth-first search. Acta Univ. Carolinae–Math. Phys. 33.Google Scholar