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Simultaneous consecutive ones submatrix and editing problems: Classical complexity and fixed-parameter tractable results
Theoretical Computer Science ( IF 0.9 ) Pub Date : 2019-07-04 , DOI: 10.1016/j.tcs.2019.05.043
Rani M. R , Subashini R , Mohith Jagalmohanan

A binary matrix M has the consecutive ones property (C1P) for rows (resp. columns) if there exists a permutation of its columns (resp. rows) that arranges the ones consecutively in all the rows (resp. columns). If M has the C1P for rows and the C1P for columns, then M is said to have the simultaneous consecutive ones property (SC1P). In this article, we consider the classical complexity and fixed-parameter tractability of a few variants of decision problems related to the SC1P. Given a binary matrix M and a positive integer d, we focus on problems that decide whether there exists a set of rows, columns, and rows as well as columns, respectively, of size at most d in M, whose deletion results in a matrix with the SC1P. We also consider problems that decide whether there exists a set of 0-entries, 1-entries and 0-entries as well as 1-entries, respectively, of size at most d in M, whose flipping results in a matrix with the SC1P. In this paper, we show that all the above mentioned problems are NP-complete. We could also prove that all these problems are fixed-parameter tractable with respect to solution size as the parameter, except for two variants (flipping 1-entries and flipping 0/1-entries). We also give improved FPT algorithms for certain problems on restricted binary matrices.



中文翻译:

同时连续的子矩阵和编辑问题:经典复杂度和固定参数易处理的结果

如果存在二进制列矩阵(列)的排列(在所有行(列)中连续排列),则二进制矩阵M的行(列)具有连续的(C1P)属性。如果M具有用于行的C1P和具有列的C1P,则称M具有同时连续的1属性(SC1P)。在本文中,我们考虑了与SC1P相关的决策问题的一些变体的经典复杂性和固定参数易处理性。给定一个二进制矩阵M和一个正整数d,我们关注于确定是否存在一组行,列,行和列的集合,分别在M中最大为d的问题。,其删除会导致与SC1P形成矩阵。我们还考虑了一些问题,这些问题决定了是否存在一组0个条目,1个条目和0个条目以及1个条目,它们的大小最大为dM中,其翻转会导致与SC1P形成矩阵。在本文中,我们证明上述所有问题都是NP完全的。我们还可以证明,除了两个变量(翻转1项和翻转0/1项)以外,所有这些问题对于解决方案大小作为参数都是固定参数易处理的。对于受限二进制矩阵上的某些问题,我们还提供了改进的FPT算法。

更新日期:2019-07-04
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