加权序列的最长公共子序列
Longest Common Subsequence on Weighted Sequences
摘要：
本文研究加权序列的最长公共子序列问题。加权序列是经典字符串的扩展，其中在每个位置，字母表中的每个字母都可能以一定的概率出现。先前的结果提供了PTAS，并注意到除非P=NP，否则不可能有FPTAS。在本文中，我们基本上关闭上界和下界之间的差距，提高两者。首先，我们给出了有界字母表的EPTAS（这是最自然的情况），并证明了除非FPT=W，否则不存在无界字母表的EPTAS [1]。此外，根据指数时间假设，我们提供了一个下限，这表明没有显着更好的PTAS可以存在无限的字母。作为一个侧记，我们证明，它是足够的工作，只有一个阈值的一般变体的问题。
英文摘要：
We consider the general problem of the Longest Common Subsequence (LCS) on weighted sequences. Weighted sequences are an extension of classical strings, where in each position every letter of the alphabet may occur with some probability. Previous results presented a PTAS and noticed that no FPTAS is possible unless P=NP. In this paper we essentially close the gap between upper and lower bounds by improving both. First of all, we provide an EPTAS for bounded alphabets (which is the most natural case), and prove that there does not exist any EPTAS for unbounded alphabets unless FPT=W[1]. Furthermore, under the Exponential Time Hypothesis, we provide a lower bound which shows that no significantly better PTAS can exist for unbounded alphabets. As a side note, we prove that it is sufficient to work with only one threshold in the general variant of the problem.
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