关于最小测试集的线性规划松弛近似

目前最小测试集的最佳近似比是贪心算法的2 ln n＋o（1）.这个近似比能否改进是一个公开的问题.本文讨论了最小测试集的基于线性规划松弛的近似比证明方法的能力问题.我们证明最小测试集的整性间隙至少为0.72 ln n,而且最小测试集整性间隙的系数可以与最小集合覆盖的整性间隙的系数一样大.另外,我们说明加权最小测试集的贪心算法的近似比不能通过对偶拟合方法改进超过一个常数.

On LP-relaxation Approximation of Minimum Test Set

The up-to-date best approximation ratio of minimum test set is 21nn+o(1)of the greedy algorithm. It is an open problem if this approximation ratio can be improved. This paper discusses the ability of the methods for proof of the approximation ratios based on the LP-relaxation. We prove the integrality gap of minimum test set is at least 0.72 In n and the coefficient of the integrality gap of minimum test set can be as large as that of minimum set cover. Moreo- ver, we show one cannot improve the approximation ratio of the greedy algorithm for weighted minimum test set by more than a constant.