A heapable sequence is a sequence of numbers that can be arranged in a "min-heap data structure". Finding a longest heapable subsequence of a given sequence was proposed by Byers, Heeringa, Mitzenmacher, and Zervas (ANALCO 2011) as a generalization of the well-studied longest increasing subsequence problem and its complexity still remains open. An equivalent formulation of the longest heapable subsequence problem is that of finding a maximum-sized binary tree in a given permutation directed acyclic graph (permutation DAG). In this work, we study parameterized algorithms for both longest heapable subsequence as well as maximum-sized binary tree. We show the following results: 1. The longest heapable subsequence problem can be solved in $k^{O(\log{k})}n$ time, where $k$ is the number of distinct values in the input sequence. We introduce the "alphabet size" as a new parameter in the study of computational problems in permutation DAGs. Our result on longest heapable subsequence implies that the maximum-sized binary tree problem in a given permutation DAG is fixed-parameter tractable when parameterized by the alphabet size. 2. We show that the alphabet size with respect to a fixed topological ordering can be computed in polynomial time, admits a min-max relation, and has a polyhedral description. 3. We design a fixed-parameter algorithm with run-time $w^{O(w)}n$ for the maximum-sized binary tree problem in undirected graphs when parameterized by treewidth $w$. Our results make progress towards understanding the complexity of the longest heapable subsequence and maximum-sized binary tree in permutation DAGs from the perspective of parameterized algorithms. We believe that the parameter alphabet size that we introduce is likely to be useful in the context of optimization problems defined over permutation DAGs.

翻译：可喜序列是一个数字序列序列, 可以在“ 微增量数据结构” 中排列。 在此工作中, Byers、 Heeringa、 Mitzenmacher 和 Zervas ( ANALCO 2011) 提议对一个给定序列中最长时间的堆肥子序列进行亚序列分析, 其复杂性仍然开放。 一个最长的堆肥子序列序列的配方是, 在一个给定的离子变压数据结构中找到一个最大尺寸的二进制树形图。 在此工作中, 我们研究一个给定序列中最长时间的堆肥亚序列和最大尺寸的二进程树序列的代算法。 最长时间的堆肥次序列问题可以在 $%O (\log{k}}} nd时间中解决, 其中美元是输入序列中不同值的数值。 我们引入了“ 碱化” 以 Weal drowbet 大小作为新的参数, 在对计算问题的研究中, 最长时间的变数序列序列中, 最晚的平数将一个结果显示为我们最晚的平数的平数的平程的变变的变变变变变的变的变。