Longest Common Subsequence on Weighted Sequences
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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. In this paper we provide faster algorithms and prove a series of hardness results for more general variants of the problem. In particular, we provide an NP-Completeness result on the general variant of the problem instead of the log-probability version used in earlier papers, already for alphabets of size 2. Furthermore, we design an EPTAS for bounded alphabets, which is also an improved, compared to previous results, PTAS for unbounded alphabets. These are in a sense optimal, since it is known that there is no FPTAS for bounded alphabets, while we prove that there is no EPTAS for unbounded alphabets. Finally, we provide a matching conditional (under the Exponential Time Hypothesis) lower bound for any PTAS. 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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