Hardness and Approximation of The Asynchronous Border Minimization Problem

TL;DR

This paper investigates the computational complexity and approximation algorithms for the Border Minimization Problem in microarray synthesis, proving NP-hardness for various cases and improving approximation bounds.

Contribution

It establishes NP-hardness for multiple BMP variants and enhances the approximation algorithm from O(n^{1/2} log^2 n) to O(n^{1/4} log^2 n).

Findings

BMP is NP-hard in 1D and 2D arrays.

1D-P-BMP is polynomial time solvable.

Improved approximation algorithm for BMP.

Abstract

We study a combinatorial problem arising from microarrays synthesis. The synthesis is done by a light-directed chemical process. The objective is to minimize unintended illumination that may contaminate the quality of experiments. Unintended illumination is measured by a notion called border length and the problem is called Border Minimization Problem (BMP). The objective of the BMP is to place a set of probe sequences in the array and find an embedding (deposition of nucleotides/residues to the array cells) such that the sum of 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 previously proposed for the problem but it is unknown whether the problem is NP-hard or not. In this paper, we give a thorough study of different variations of BMP by giving NP-hardness proofs and improved approximation algorithms. We show that P-BMP, 1D-BMP, and BMP are all NP-hard. Contrast with the previous result 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.