On the Border Length Minimization Problem (BLMP) on a Square Array

TL;DR

This paper proves that the Border Length Minimization Problem (BLMP) is NP-hard and introduces a hierarchical refinement algorithm to improve solutions, with the heuristic TSP+1-threading shown to be an O(N)-approximation.

Contribution

The paper establishes the NP-hardness of the BLMP and provides a hierarchical refinement algorithm along with an approximation bound for a heuristic.

Findings

BLMP is NP-hard.

Hierarchical refinement improves solutions.

TSP+1-threading heuristic is an O(N)-approximation.

Abstract

Protein/Peptide microarrays are rapidly gaining momentum in the diagnosis of cancer. High-density and highthroughput 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 seven years is if the BLMP is tractable or not. We settle this open problem by proving that the BLMP is NP-hard. We also present a hierarchical refinement algorithm which can refine any heuristic solution for the BLMP problem. We also prove that the TSP+1-threading heuristic is an O(N)- approximation. The hierarchical refinement solver is available as an opensource code at http://launchpad.net/blm-solve.