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

In this paper, we show that calculating the partition function of optimal medians of binary strings with Hamming distance is #P-complete for several weight functions. The case when the weight function is the factorial function has application in bioinformatics. In that case, the partition function counts the most parsimonious evolutionary scenarios on a star tree under several models in bioinformatics. The results are extended to binary trees and we show that it is also #P-complete to calculate the most parsimonious evolutionary scenarios on an arbitrary binary tree under the substitution model of biological sequences and under the Single Cut-or-Join model for genome rearrangements. 1. The Partition Function For n,m ∈ Z, fix a multiset of binary strings B = {ν1, ν2, . . . , νm} where νi ∈ {0, 1} for each i ∈ [m]. An optimal median μ is a binary string also in {0, 1} which minimizes ∑m i=1 H(νi, μ) where H(νi, μ) is the Hamming distance between νi and μ. Define M(B) to be the set of all optimal medians for the multiset B. Take f to be a non-negative, real-valued function. In this paper, we examine the partition function