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Journal of Combinatorial Optimization

, Volume 27, Issue 2, pp 315–327 | Cite as

Improved approximation algorithms for the max-bisection and the disjoint 2-catalog segmentation problems

  • Zi Xu
  • Donglei Du
  • Dachuan Xu
Article

Abstract

We consider the max-bisection problem and the disjoint 2-catalog segmentation problem, two well-known NP-hard combinatorial optimization problems. For the first problem, we apply the semidefinite programming (SDP) relaxation and the RPR2 technique of Feige and Langberg (J. Algorithms 60:1–23, 2006) to obtain a performance curve as a function of the ratio of the optimal SDP value over the total weight through finer analysis under the assumption of convexity of the RPR2 function. This ratio is shown to be in the range of [0.5,1]. The performance curve implies better approximation performance when this ratio is away from 0.92, corresponding to the lowest point on this curve with the currently best approximation ratio of 0.7031 due to Feige and Langberg (J. Algorithms 60:1–23, 2006). For the second problem, similar technique results in an approximation ratio of 0.7469, improving the previously best known result 0.7317 due to Wu et al. (J. Ind. Manag. Optim. 8:117–126, 2012).

Keywords

Max-bisection problem 2-Catalog segmentation problem Approximation algorithm Semidefinite programming RPR2 rounding 

Notes

Acknowledgements

The authors are grateful for the anonymous reviewers to give some helpful comments to improve this paper.

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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of MathematicsShanghai UniversityShanghaiP.R. China
  2. 2.Faculty of Business AdministrationUniversity of New BrunswickFrederictonCanada
  3. 3.Department of Applied MathematicsBeijing University of TechnologyBeijingP.R. China

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