This paper categorizes the parameterized complexity of the algorithmic problems Perfect Phylogeny and Triangulating Colored Graphs. 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：本文对算法问题“完美谱系”和“着色图三角化”的参数化复杂度进行了分类。我们通过从“树链多色独立集”的归约以及成员资格的证明，表明这些问题在参数化复杂度类XALP中是完备的。我们引入了“多色图三角化”问题作为中间步骤，并证明了该问题也是XALP完备的。我们还证明，假设指数时间假说成立，对于这些问题，不存在时间复杂度为$f(k) n^{o(k)}$的算法，其中$n$是输入规模，$k$是参数，$f$是任意可计算函数。