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Improved approximating $2$-CatSP for $\sigma\geq 0.50$ with an unbalanced rounding matrix

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  • $2$-CatSP is a fundamental NP-Complete optimization problem, which is formulated as given a ground set $U$ of $n$ items, $m$ subsets $S_1,S_2,\cdots,$ $S_m$ of $U$ and an integer $1\leq t\leq n$, find two subsets $A_1$ and $A_2$ of $U$ such that $|A_1|=|A_2|\leq t$ and $\sum_{i=1}^{m}\max\left(|S_i\cap A_1|,|S_i\cap A_2|\right)$ is maximized. In this paper, we reconsider randomized approximation algorithms for $2$-CatSP without and with triangle inequalities in terms of a new positive semidefinite matrix reflecting more information on unbalanced properties. The performance ratios of our algorithm are much better than the current best ones of Xu et al in (Optim. Method. Softw., 18(6): 705-719, 2003) for general cases under unbalanced parameters $\sigma=t/n\geq 0.50$ by our rigorous analysis. In particular, we get 0.6700 and 0.7250 without and with triangle inequalities for the most important subcase $\sigma=0.50$, improving previously best ratios 0.6430 and 0.6736 in corresponding environment. Indeed recently Wu et al (J. Ind. Manag. Optim. 8: 117-126, 2012) and Xu et al in (J. Comb. Optim., 27: 315-327, 2014) also considered approximate strategies for its disjoint subcase $A_1\cap A_2=\emptyset$ as $\sigma=0.50$ with ratios 0.7317 and 0.7469, which can be reduced to max bisection problems of graphs.
    Mathematics Subject Classification: Primary: 90C25, 90C27; Secondary: 68Q25, 68W25.


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