Constrained pairwise and center-star sequences alignment problems

Abstract

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 $A=a_1a_2\ldots a_n$ and $B=b_1b_2\ldots b_n$ with a given distance function and a constrained sequence $C=c_1c_2\ldots c_k$, 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 $O(\min \{kn^2,(t-k)n^2\})$ 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 $O(kn^4/\log n)$. 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.