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 $$A=a_1a_2\ldots a_n$$ A = a 1 a 2 … a n and $$B=b_1b_2\ldots b_n$$ B = b 1 b 2 … b n with a given distance function and a constrained sequence $$C=c_1c_2\ldots c_k$$ C = c 1 c 2 … 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 $$C=c^k$$ C = c k , i.e., all characters in C are same, the optimal constrained pairwise sequence alignment can be solved in $$O(\min \{kn^2,(t-k)n^2\})$$ O ( min { k n 2 , ( t - k ) n 2 } ) 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 $$O(kn^4/\log n)$$ O ( k n 4 / 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.