# Constrained pairwise and center-star sequences alignment problems

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## 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.

## Keywords

Sequence alignment Dynamic programming Complexity## Notes

### Acknowledgments

The authors thank the anonymous referees for their helpful comments to improve the presentation of this paper. This work was supported by NSFC (61433012, U1435215, 11171086), HK RGC Grant (HKU 7114/13E, HKU 7164/12E, HKU 7111/12E), HKU small project funding 201309176064, Natural Science Foundation of Hebei A2013201218, Chinese Academy of Sciences research Grant (No. KGZD-EW-103-5(9)), Fundamental Research Foundation of Northwestern Polytechnical University in China (Grant No. JC201164), Fundamental Research Funds for the Central Universities (Grant No. 3102015ZY081), and China Postdoctoral Science Foundation (Grant No. 2012M521803).

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