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

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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).

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Acknowledgements

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

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Authors and Affiliations

  1. Department of Mathematics, Shanghai University, 99 Shangda Road, Baoshan District, Shanghai, 200444, P.R. China

    Zi Xu

  2. Faculty of Business Administration, University of New Brunswick, Fredericton, NB, Canada, E3B 5A3

    Donglei Du

  3. Department of Applied Mathematics, Beijing University of Technology, 100 Pingleyuan, Chaoyang District, Beijing, 100124, P.R. China

    Dachuan Xu

Authors
  1. Zi Xu
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  2. Donglei Du
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  3. Dachuan Xu
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Corresponding author

Correspondence to Dachuan Xu.

Additional information

The first author is supported by China NSF under the grant 11101261 and Key Disciplines of Shanghai Municipality (S30104). The research of the second author is supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) grant 283103. The third author’s research is supported by NSF of China (No. 11071268), Beijing Natural Science Foundation (No. 1102001), and PHR(IHLB).

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Xu, Z., Du, D. & Xu, D. Improved approximation algorithms for the max-bisection and the disjoint 2-catalog segmentation problems. J Comb Optim 27, 315–327 (2014). https://doi.org/10.1007/s10878-012-9526-3

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