On Approximability of Block Sorting

Abstract. Block Sorting is a well studied problem, motivated by its applications in Optical Character Recognition (OCR), and Computa-tional Biology. Block Sorting has been shown to be NP-Hard, and two separate polynomial time 2-approximation algorithms have been de-signed for the problem. But questions like whether a better approxima-tion 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.