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Approximating the 2-catalog segmentation problem using semidefinite programming relaxations
Xu, DC; Ye, YY; Zhang, JW
2003-12-01
发表期刊OPTIMIZATION METHODS & SOFTWARE
ISSN1055-6788
卷号18期号:6页码:705-719
摘要We consider the 2-Catalog Segmentation problem (2-CatSP) introduced by Kleinberg et al. [J. Kleinberg, C. Papadimitriou and P. Raghavan (1998). Segmentation problems. In Proceedings of the 30th Symposium on Theory of Computation , pp. 473-482.], where we are given a ground set I of n items, a family {S-1, S-2,...,S-m } of subsets of I and an integer 1 less than or equal to k less than or equal to n . The objective is to find subsets A(1), A(2) subset of I such that \A(1)\ = \A(2)\ = k and Sigma(i=1)(m) max {\S-i boolean AND A(1)\, \S-i boolean AND A(2)\} is maximized. It is known that a simple and elegant greedy algorithm has a performance guarantee 1/2. Furthermore, using a semidefinite programming (SDP) relaxation Doids et al. [Y. Doids, V. Guruswami and S. Khanna (1999). The 2-catalog segmentation problem. In Proceedings of SODA , pp. 378-380.] showed that 2-CatSP can be approximated by a factor of 0.56 when k = n/2. Motivated by these results, we develop improved approximation algorithms for 2-CatSP on a range of k in this paper. The performance guarantee of our algorithm is 1/2 for general k , and is strictly greater than 1/2 when k greater than or equal to n /3. In particular, we obtain a ratio of 0.67 for 2-CatSP when k = n/2. Unlike the relaxation used by Doids et al. , our extended and direct SDP relaxation deals with general k , which enables us to obtain better approximation for 2-CatSP. Another contribution of this paper is a new variation of the random hyperplane rounding technique, which allows us to explore the structure of 2-CatSP. This rounding technique might be of independent interest. It can be also used to obtain improved approximation for several other graph partitioning problems considered in Feige and Langberg [U. Fiege and M. Langberg (2002). Approximation algorithms for maximization problems arising in graph partitioning. Journal of Algorithm .], Ye and Zhang [Y. Ye and J. Zhang (2003). Approximation for dense-n/2-subgraph and the complement of min-bisection. Journal of Global Optimization , 25 , 55-73.], and Halperin and Zwick [E. Halperin and U. Zwick (2001). A unified framework for obtaining improved approximation algorithms for maximum graph bisection problems. In IPCO , Lecture Notes in Computer Science. Springer, Berlin.].
关键词catalog segmentation problem semidefinite programming approximation algorithm
DOI10.1080/10556780310001634082
语种英语
WOS研究方向Computer Science ; Operations Research & Management Science ; Mathematics
WOS类目Computer Science, Software Engineering ; Operations Research & Management Science ; Mathematics, Applied
WOS记录号WOS:000187296300007
出版者TAYLOR & FRANCIS LTD
引用统计
文献类型期刊论文
条目标识符http://ir.amss.ac.cn/handle/2S8OKBNM/19131
专题中国科学院数学与系统科学研究院
通讯作者Ye, YY
作者单位1.Stanford Univ, Dept Management Sci & Engn, Stanford, CA 94305 USA
2.Chinese Acad Sci, Acad Math & Syst Sci, Inst Appl Math, Beijing 100080, Peoples R China
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Xu, DC,Ye, YY,Zhang, JW. Approximating the 2-catalog segmentation problem using semidefinite programming relaxations[J]. OPTIMIZATION METHODS & SOFTWARE,2003,18(6):705-719.
APA Xu, DC,Ye, YY,&Zhang, JW.(2003).Approximating the 2-catalog segmentation problem using semidefinite programming relaxations.OPTIMIZATION METHODS & SOFTWARE,18(6),705-719.
MLA Xu, DC,et al."Approximating the 2-catalog segmentation problem using semidefinite programming relaxations".OPTIMIZATION METHODS & SOFTWARE 18.6(2003):705-719.
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