We consider the offline sorting buffer problem. The input is a sequence of items of different types. All items must be processed one by one by a server. The server is equipped with a random-access buffer of limited capacity which can be used to rearrange items. The problem is to design a scheduling strategy that decides upon the order in which items from the buffer are sent to the server. Each type change incurs unit cost, and thus, the cost minimizing objective is to minimize the total number of type changes for serving the entire sequence. This problem is motivated by various applications in manufacturing processes and computer science, and it has attracted significant attention in the last few years. The main focus has been on online competitive algorithms. Surprisingly little is known on the basic offline problem. In this paper, we show that the sorting buffer problem with uniform cost is NP-hard and, thus, close one of the most fundamental questions for the offline problem. On the positive side, we give an O(1)-approximation algorithm when the scheduler is given a buffer only slightly larger than double the original size. We also give a dynamic programming algorithm for the special case of buffer size two that solves the problem exactly in linear time, improving on the standard DP which runs in cubic time.
School of Computing University of North Florida（计算学院北弗罗里达大学）；Computer Science Department Florida Polytechnic University（计算机科学系弗罗里达技术大学）；WIDiCoReL Research Lab（WIDiCoReL研究实验室）
本文通过将 Block Sorting 多项式归约，利用 cage 和 hinge 等局部 gadget 建立精确 SbSS 调度与完美块调度的双射等价关系，证明了 Sorting by Strip Swaps 是 NP-hard 的。
A new 1.375-approximation algorithm for Sorting By Transpositions
在基因组重排中，突变事件转座交换了一条染色体中两个相邻的基因块。转置距离问题（TDP）的目的是找到将一个染色体转换为另一个染色体所需的最小转置次数，两者都表示为排列。TDP可以归结为换位排序（SBT）问题。SBT是$\mathcal{NP}$-难的，Elias和Hartman提出了具有$1.375$比的最佳近似算法.他们的算法采用了简化，一种用于将输入置换$π$转换为简单置换$\hatπ$的技术，可能更容易处理。置换$\hatπ$是通过以在$\hatπ$上保持$π$的转置距离的下界的方式将新符号插入到$π$中而获得的。简化保证保持下界，而不是换位距离。在本文中，我们首先表明，算法的埃利亚斯和哈特曼（EH算法）可能需要一个额外的转置以上的近似比$1.375$，这取决于如何输入置换被简化。其次，利用代数方法，提出了一个新的换位距离上界和一个新的$1.375$-近似算法，解决了SBT跳跃简化问题，并保证了$1.375$对所有$S_n$的近似比.我们实现了我们的算法和EH。关于EH算法的实现，需要解决两个问题。我们针对大小为$n$、$2\leq n \leq 12$的所有排列测试了这两种算法。结果表明，对于大于$7$的置换，EH算法的逼近比超过了$1.375$.最后，我们研究了这两种实现的最大长度$500$较长的排列的性能。
