Hardness and approximation of the asynchronous border minimization problem

An exponential time algorithm has been proposed for the problem but it is unknown whether the problem is NP-hard or not. In this paper, we give a comprehensive study of different variations of BMP by presenting NP-hardness proofs and approximation algorithms. We show that BMP, P-BMP, and 1D-BMP are all NP-hard and 1D-BMP is polynomial time solvable. The interesting implications include (i) the BMP is NP-hard regardless of the dimension (1D or 2D) of the array; (ii) the array dimension differentiates the complexity of the P-BMP; and (iii) for 1D array, whether placement is given differentiates the complexity of the BMP. Another contribution of the paper is devising approximation algorithms, and in particular, we present a randomized approximation algorithm for BMP with approximation ratio O(n1∕4log2n), where n is the total number of sequences.