A Kernelization Algorithm for Finding a Perfect Phylogeny From Mixed Tumor Samples

The split-row problem (SR), introduced by Hajirasouliha and Raphael [WABI 2014], models an effective method for reconstructing a perfect phylogeny from mixed tumor samples. In this problem, an m × n binary matrix M is given. A split-row operation on M is defined as replacing a row r by k > 1 rows whose bitwise OR is equal to r. The cost of the operation is the number of additional rows induced, that is, k − 1. The objective is to find a sequence of operations that transforms M into a matrix corresponding to a perfect phylogeny and the total cost is minimized. Hujdurović et al. [TCBB 2018] proved the NP-hardness of SR. Let ϵ(M) denote the minimum total cost. In this paper, we show that SR admits a polynomial size kernel, which has at most 3ϵ(M) rows and 4ϵ(M) − 1 columns. Our kernelization algorithm requires $O( {\text{max}( {{{m}^{0.373}}{{n}^2},\ m{{n}^{1.373}}} )} )$ time. When ϵ(M) is small, it can be used as a preprocessing procedure to speed up all previous exact algorithms for SR.