Đang chuẩn bị nút TẢI XUỐNG, xin hãy chờ
Tải xuống
Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: The complexity of constructing gerechte designs. | The complexity of constructing gerechte designs E. R. Vaughan School of Mathematical Sciences Queen Mary University of London UK e.vaughan@qmul.ac.uk Submitted Nov 20 2007 Accepted Jan 20 2009 Published Jan 30 2009 Mathematics Subject Classihcation 05B15 68Q17 Abstract Gerechte designs are a specialisation of latin squares. A gerechte design is an n X n array containing the symbols 1 . ng together with a partition of the cells of the array into n regions of n cells each. The entries in the cells are required to be such that each row column and region contains each symbol exactly once. We show that the problem of deciding if a gerechte design exists for a given partition of the cells is NP-complete. It follows that there is no polynomial time algorithm for Ending gerechte designs with specihed partitions unless P NP. 1 Introduction Gerechte designs were introduced by W. U. Behrens in 1956 2 as a specialisation of latin squares. A gerechte skeleton of order n is an n X n array whose n2 cells are partitioned into n regions containing n cells each. A gerechte design of order n consists of a gerechte skeleton of order n together with an assignment of a symbol from the set 1 . ng to each cell such that each symbol occurs once in each row once in each column and once in each region of the array. We call such an assignment of symbols to cells a completion. Not all gerechte skeletons have completions. For example suppose we have a skeleton in which one of the regions consists of all the cells in a row except for the cell in column n and a cell in column n in a different row. Clearly a skeleton that contains such a region cannot have a completion. Bailey Cameron and Connelly in their article on sudoku 1 asked what is the computational complexity of determining if a given gerechte skeleton has a completion This question was also presented at the problem session of the CGCS Luminy conference in May 2007 3 . We prove the following Theorem 1. The problem of determining if a .
