Hardness and Approximation of the Asynchronous Border Minimization Problem [chapter]

Alexandru Popa, Prudence W. H. Wong, Fencol C. C. Yung
2012 Lecture Notes in Computer Science  
We study a combinatorial problem arising from the microarrays synthesis. The objective of the BMP is to place a set of sequences in the array and to find an embedding of these sequences into a common supersequence such that the sum of the "border length" is minimized. A variant of the problem, called P-BMP, is that the placement is given and the concern is simply to find the embedding. Approximation algorithms have been proposed for the problem [21] but it is unknown whether the problem is
more » ... rd or not. In this paper, we give a comprehensive study of different variations of BMP by presenting NP-hardness proofs and improved approximation algorithms. We show that P-BMP, 1D-BMP, and BMP are all NP-hard. In contrast with the result in [21] that 1D-P-BMP is polynomial time solvable, the interesting implications include (i) the array dimension (1D or 2D) differentiates the complexity of P-BMP; (ii) for 1D array, whether placement is given differentiates the complexity of BMP; (iii) BMP is NP-hard regardless of the dimension of the array. Another contribution of the paper is improving the approximation for BMP from O(n 1/2 log 2 n) to O(n 1/4 log 2 n), where n is the total number of sequences.
doi:10.1007/978-3-642-29952-0_20 fatcat:akrmp4fo3zhpjnbjdboide7uua