Approximating Border Length for DNA Microarray Synthesis

Abstract

Abstract. We study the border minimization problem (BMP), which arises in microarray synthesis to place and embed probes in the array. The synthesis is based on a light-directed chemical process in which unintended illumination may contaminate the quality of the experiments. Border length is a measure of the amount of unintended illumination and the objective of BMP is to find a placement and embedding of probes such that the border length is minimized. The problem is believed to be NP-hard. In this paper we show that BMP admits an O( √ n log 2 n)-approximation, where n is the number of probes to be synthesized. In the case where the placement is given in advance, we show that the problem is O(log 2 n)-approximable. We also study a related problem called agreement maximization problem (AMP). In contrast to BMP, we show that AMP admits a constant approximation even when placement is not given in advance.

Hardness and Approximation of the Asynchronous Border Minimization Problem

“…Contrasting with the previous result in [22] that 1D-P-BMP is polynomial time solvable, our hardness results show that (i) the dimension 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.…”

“…WMSA and MRCT. As shown in [22], P-BMP can be reduced to the weighted multiple sequence alignment problem (WMSA), which in turn can be reduced to the minimum routing cost tree problem (MRCT). In the WMSA problem [2,9,14,27], we are given k sequences S = {S 1 , S 2 , · · · , S k }.…”

“…It is stated in [22] that Bartal's algorithm [1] finds a routing spanning tree by embedding a metric space into a distribution of trees with expected distortion O(log 2 n). MRCT is O(log 2 n)approximable [1].…”

“…• For P-BMP, we show that the Shortest Common Supersequence problem [26] can be reduced to P-BMP, implying that P-BMP is NP-hard. This means that the dimension differentiates the complexity of P-BMP as we have seen in [22] that 1D-P-BMP is polynomial time solvable.…”

“…• For 1D-BMP (placement not given), we give a reduction from the Hamiltonian Path problem [11] to 1D-BMP, implying the NP-hardness of 1D-BMP. This means that when the array is one dimension, whether placement is given differentiates the complexity of BMP (as 1D-P-BMP is polynomial time solvable [22]).…”