带半径约束的矩阵完备化参数化算法
Parameterized Algorithms for Matrix Completion With Radius Constraints
摘要：
考虑矩阵的缺失项，我们研究NP-难矩阵完成问题，由此产生的完成矩阵应具有有限的（本地）半径。在纯半径版本中，这意味着目标是填充条目，使得存在到所有矩阵行的汉明距离尽可能小的“中心串”。在字符串学中，这个问题也被称为带有通配符的最接近字符串。在局部半径版本中，请求的中心字符串必须是完整矩阵的行之一。Hermelin和Rozenberg [CPM 2014，TCS 2016]对带有通配符的最近字符串进行了参数化复杂性研究。我们回答了他们的一个公开问题，修复了他们工作中关于固定参数易处理性结果的错误，并改进了一些运行时间上限。对于局部半径的情况下，我们揭示了一个计算复杂性二分法。在一般情况下，我们的研究结果表明，虽然是NP难的，以及，这种变体往往允许更快的（固定参数）算法。
英文摘要：
Considering matrices with missing entries, we study NP-hard matrix completion problems where the resulting completed matrix shall have limited (local) radius. In the pure radius version, this means that the goal is to fill in the entries such that there exists a 'center string' which has Hamming distance to all matrix rows as small as possible. In stringology, this problem is also known as Closest String with Wildcards. In the local radius version, the requested center string must be one of the rows of the completed matrix. Hermelin and Rozenberg [CPM 2014, TCS 2016] performed parameterized complexity studies for Closest String with Wildcards. We answer one of their open questions, fix a bug concerning a fixed-parameter tractability result in their work, and improve some upper running time bounds. For the local radius case, we reveal a computational complexity dichotomy. In general, our results indicate that, although being NP-hard as well, this variant often allows for faster (fixed-parameter) algorithms.
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