Block Sorting Is APX-Hard

Abstract

Block Sorting is an NP-hard combinatorial optimization problem motivated by applications in Computational Biology and Optical Character Recognition (OCR). It has been approximated in P time within a factor of 2 using two different techniques and the complexity of better approximations has been open for close to a decade now. In this work we prove that Block Sorting does not admit a PTAS unless P = NP i.e. it is APX-Hard. The hardness result is based on new properties, that we identify, of the existing NP-hardness reduction from E3-SAT to Block Sorting. In an attempt to obtain an improved approximation for Block Sorting, we consider a generalization of the well-studied Block Merging, called $k$-Block Merging which is defined for each $k \ge 1$, and the $1$- Block Merging problem is the same as the Block Merging problem. We show that the optimum $k$-Block Merging is an $1+ \frac{1}{k}$-approximation to the optimum block sorting. We then show that for each $k \ge 2$, we prove $k$-Block Merging to be NP-Hard, thus proving a dichotomy result associated with block sorting.