# Longest Common Subsequence on Weighted Sequences

## TL;DR

This paper advances the understanding of the Longest Common Subsequence problem on weighted sequences by providing efficient approximation schemes for bounded alphabets and establishing complexity bounds for unbounded alphabets.

## Contribution

It introduces an EPTAS for bounded alphabets and proves hardness results for unbounded alphabets, closing the gap between upper and lower bounds.

## Key findings

- EPTAS achieved for bounded alphabets
- No EPTAS exists for unbounded alphabets unless FPT=W[1]
- Lower bounds under ETH restrict PTAS improvements for unbounded alphabets

## Abstract

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.