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 ...
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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 complemen
the Council wishes to draw the attention of members to the internationalconference on Large Electric Systems (C.I.G.R.E.) which will hold its sixteenth session in Paris from the 30th May to the 9th June 1956.
the Council wishes to draw the attention of members to the internationalconference on Large Electric Systems (C.I.G.R.E.) which will hold its sixteenth session in Paris from the 30th May to the 9th June 1956.
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