PARAMETERIZED COMPLEXITY FOR FINDING A PERFECT PHYLOGENY FROM MIXED TUMOR SAMPLES.

Motivated by an application in cancer genomics, Hajirasouliha and Raphael [Proceedings of the 14th International Workshop on Algorithms in Bioinformatics, 2014, pp. 354-367] proposed the split-row problem (SR). In this problem, an m\times n binary matrix M is given. A split-row operation on M is defined as replacing a row r by k > 1 rows r(1), r(2), . . ., r(k) whose bitwise OR is equal to r. The cost of the operation is the number of additional rows induced, that is, k 1. The goal is to find a sequence of split-row operations that transforms M into a matrix corresponding to a perfect phylogeny and the total cost is minimized. Recently, Hujdurovi\'c et al. [ACM Trans. Algorithms, 14 (2018), 26] proved the APX-hardness of SR and presented efficient exact and approximation algorithms. The parameterized study of SR was left as a direction for future work. Let\varepsilon (M) denote the minimum total cost. This paper gives an O*(2min(n,2x(M))-time exact algorithm for SR. This result indicates that SR is fixed-parameter tractable when parameterized by E (M). In addition, in the worst case, our algorithm requires O* (2n) time, significantly improving the previous upper bound of O\ast (nn). Hujdurovi\'c et al.'s exact algorithm can be modified to solve a variant of SR, called the distinct split-row problem (DSR). Our algorithm can be adapted to this variant as well. In addition, our algorithms can be extended to solve SR and DSR with the following additional constraint: only the rows in a given subset are allowed to be split. [ABSTRACT FROM AUTHOR]