Border Length Minimization Problem on a Square Array

Abstract

Abstract

Protein/peptide microarrays are rapidly gaining momentum in the diagnosis of cancer. High-density and high-throughput peptide arrays are being extensively used to detect tumor biomarkers, examine kinase activity, identify antibodies having low serum titers, and locate antibody signatures. Improving the yield of microarray fabrication involves solving a hard combinatorial optimization problem called the border length minimization problem (BLMP). An important question that remained open for the past 7 years is if the BLMP is tractable or not. We settle this open problem by proving that the BLMP is \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${ \cal NP}$$\end{document} -hard. We also present a hierarchical refinement algorithm that can refine any heuristic solution for the BLMP and prove that the TSP+1-threading heuristic is an O(N)-approximation.