This paper categorizes the parameterized complexity of the algorithmic problems Perfect Phylogeny and Triangulating Colored Graphs when parameterized by the number of genes and colors, respectively. We show that they are complete for the parameterized complexity class XALP using a reduction from Tree-chained Multicolor Independent Set and a proof of membership. We introduce the problem Triangulating Multicolored Graphs as a stepping stone and prove XALP-completeness for this problem as well. We also show that, assuming the Exponential Time Hypothesis, there exists no algorithm that solves any of these problems in time $f(k) n^{o(k)}$, where $n$ is the input size, $k$ the parameter, and $f$ any computable function.

翻译：本文研究了算法问题“完美系统发育”和“彩色图三角化”在以基因数和颜色数分别作为参数时的参数化复杂性。我们通过树链多色独立集问题的归约以及成员证明，展示了这些问题对于参数化复杂性类XALP是完备的。我们引入了“多色图三角化”问题作为过渡，并证明了该问题也是XALP完备的。此外，我们证明，在指数时间假设成立的前提下，不存在任何算法能在时间$f(k) n^{o(k)}$内解决这些问题，其中$n$是输入规模，$k$是参数，$f$是任意可计算函数。