Short Block-Move–CPP is NP-Complete

The Closest Object Problem aims to find one object in the center of all others. It was studied for strings with respect to the Hamming distance, where the Hamming Closest String Problem was settled to be NP-complete. The Closest Permutation Problem (CPP) was also studied, since permutations are the natural restrictions of general strings, and we have settled that the Block interchange–CPP and the Breakpoint–CPP are NPcomplete. We consider a restricted form of block-interchange, called short block-move, defined by exchanging two contiguous blocks of elements of total length at most 3, for which the computational complexity of the distance problem is still open. We provide sufficient conditions to determine the short block-move distance by showing that the optimal sorting sequence of short block-moves of a given permutation can be obtained by sorting each connected component separately on the permutation graph, and we prove that Short Block Move–CPP is NP-complete. 2000 AMS Subject Classification: 68Q17, 68P10, 05A05.