Exact algorithms for the repetition-bounded longest common subsequence problem

In this paper, we study exact, exponential-time algorithms for a variant of the classic LONGEST COMMON SUBSEQUENCE problem called the REPETITION-BOUNDED LONGEST COMMON SUBSEQUENCE problem (or RBLCS, for short): Let an alphabet S be a finite set of symbols and an occurrence constraint C_occ be a function C_occ:S→N, assigning an upper bound on the number of occurrences of each symbol in S. Given two sequences X and Y over the alphabet S and an occurrence constraint C_occ, the goal of RBLCS is to find a longest common subsequence of X and Y such that each symbol s∈S appears at most C_occ(s) times in the obtained subsequence. The special case where C_occ(s)=1 for every symbol s∈S is known as the REPETITION-FREE LONGEST COMMON SUBSEQUENCE problem (RFLCS) and has been studied previously; e.g., in [1], Adi et al. presented a simple (exponential-time) exact algorithm for RFLCS. However, they did not analyze its time complexity in detail, and to the best of our knowledge, there are no previous results on the running times of any exact algorithms for this problem. Without loss of generality, we will assume that |X|≤|Y| and |X|=n. In this paper, we first propose a simpler algorithm for RFLCS based on the strategy used in [1] and show explicitly that its running time is O(1.44225ⁿ). Next, we provide a dynamic programming (DP) based algorithm for RBLCS and prove that its running time is O(1.44225ⁿ) for any occurrence constraint C_occ, and even less in certain special cases. In particular, for RFLCS, our DP-based algorithm runs in O(1.41422ⁿ) time, which is faster than the previous one. Furthermore, we prove NP-hardness and APX-hardness results for RBLCS on restricted instances.