The computational complexity of calculating partition functions of optimal medians with Hamming distance

We study the complexity of computing the partition function of medians for binary strings with Hamming distance using various weight functions. When the weight function is the factorial function, this partition function has application in bioinformatics, counting the most parsimonious scenarios on a star tree under the Single Cut-or-Join model for genome rearrangement. Although this model is computationally simple, we show that it is #P-complete to compute the partition function. Our results are also extended to binary trees as we show that it is #P-complete to calculate the most parsimonious scenarios on an arbitrary binary tree under the Single Cut-or-Join model. These results also apply to substitution models for many biological sequences. (C) 2018 Elsevier Inc. All rights reserved.