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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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