On Approximability of Block Sorting

TL;DR

This paper proves that Block Sorting, a problem relevant to OCR and biology, is APX-Hard, establishing its computational difficulty and providing new bounds through a reduction from Max-3SAT and a novel parametrized problem.

Contribution

It establishes the APX-Hardness of Block Sorting via a linear reduction from Max-3SAT and introduces a new lower bound through the k-Block Merging problem.

Findings

Block Sorting is Max-SNP-Hard (APX-Hard).

A linear reduction from Max-3SAT to Block Sorting was constructed.

A new lower bound for Block Sorting was established using k-Block Merging.

Abstract

Block Sorting is a well studied problem, motivated by its applications in Optical Character Recognition (OCR), and Computational Biology. Block Sorting has been shown to be NP-Hard, and two separate polynomial time 2-approximation algorithms have been designed for the problem. But questions like whether a better approximation algorithm can be designed, and whether the problem is APX-Hard have been open for quite a while now. In this work we answer the latter question by proving Block Sorting to be Max-SNP-Hard (APX-Hard). The APX-Hardness result is based on a linear reduction of Max-3SAT to Block Sorting. We also provide a new lower bound for the problem via a new parametrized problem k-Block Merging.