Block Sorting Is APX-Hard

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9079)


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.


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© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of Computer Science and EngineeringIndian Institute of Technology MadrasChennaiIndia
  2. 2.School of ComputingUniversity of North FloridaJacksonvilleUSA

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