Constrained pairwise and center-star sequences alignment problems

Sequence alignment is a fundamental problem in computational biology, which is also important in theoretical computer science. In this paper, we consider the problem of aligning a set of sequences subject to a given constrained sequence. Given two sequences A= a₁a₂… a_n and B= b₁b₂… b_n with a given distance function and a constrained sequence C= c₁c₂… c_k, our goal is to find the optimal sequence alignment of A and B w.r.t. the constraint C. We investigate several variants of this problem. If C= c^k, i.e., all characters in C are same, the optimal constrained pairwise sequence alignment can be solved in O(min { kn², (t- k) n²}) time, where t is the minimum number of occurrences of character c in A and B. If in the final alignment, the alignment score between any two consecutive constrained characters is upper bounded by some value, which is called GB-CPSA, we give a dynamic programming with the time complexity O(kn⁴/ log n). For the constrained center-star sequence alignment (CCSA), we prove that it is NP-hard to achieve the optimal alignment even over the binary alphabet. Furthermore, we show a negative result for CCSA, i.e., there is no polynomial-time algorithm to approximate the CCSA within any constant ratio.