Approximating the 2-catalog segmentation problem using semidefinite programming relaxations

Abstract

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 ≤ k ≤ n. The objective is to find subsets A 1, A 2 ⊂ I such that |A 1| = |A 2| = k and ∑ i=1 m max{|S i ∩ A 1|, |S i ∩ 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 ≥ 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.]. *E-mail: xudc@lsec.cc.ac.cn ‡E-mail: jiazhang@stanford.edu