Exact Algorithms for the Bounded Repetition Longest Common Subsequence Problem

Abstract

In this paper, we study exact, exponential-time algorithms for a variant of the classic Longest Common Subsequence problem called the r-Repetition Longest Common Subsequence problem (or r-RLCS, for short): Given two sequences X and Y over an alphabet S, find a longest common subsequence of X and Y such that each symbol appears at most r times in the obtained subsequence. Without loss of generality, we will assume that $|X| \le |Y|$ from here on. The special case of 1-RLCS, also known as the Repetition-Free Longest Common Subsequence problem (RFLCS), has been studied previously; e.g., in [1], Adi et al. presented an (exponential-time) integer linear programming-based exact algorithm for 1-RLCS. However, they did not analyze its time complexity, and to the best of our knowledge, there are no previous results on the running times of any exact algorithms for this problem. In this paper, we first propose a simple algorithm for 1-RLCS based on the strategy used in [1] and show explicitly that its running time is bounded by $O(1.44225^{|X|}|X||Y|)$. Next, we provide a DP-based algorithm for r-RLCS and prove that its running time is $O((r+1)^{|X|/(r+1)}|X||Y|)$ for any $r \ge 1$. In particular, our new algorithm runs in $O(1.41422^{|X|}|X||Y|)$ time for 1-RLCS, which is faster than the previous one.