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

Authors:
Zi Xu;Donglei Du;Dachuan Xu
Affiliations:
Department of Mathematics, Shanghai University, Shanghai, P.R. China 200444;Faculty of Business Administration, University of New Brunswick, Fredericton, Canada E3B 5A3;Department of Applied Mathematics, Beijing University of Technology, Beijing, P.R. China 100124
Venue:
Journal of Combinatorial Optimization
Year:
2014

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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).