We introduce and investigate the approximability of the maximum binary tree problem (MBT) in directed and undirected graphs. The goal in MBT is to find a maximum-sized binary tree in a given graph. MBT is a natural variant of the well-studied longest path problem, since both can be viewed as finding a maximum-sized tree of bounded degree in a given graph. The connection to longest path motivates the study of MBT in directed acyclic graphs (DAGs), since the longest path problem is solvable efficiently in DAGs. In contrast, we show that MBT in DAGs is hard: it has no efficient \(\exp (-O(\log n/ \log \log n))\)-approximation under the exponential time hypothesis, where n is the number of vertices in the input graph. In undirected graphs, we show that MBT has no efficient \(\exp (-O(\log ^{0.63}{n}))\)-approximation under the exponential time hypothesis. Our inapproximability results rely on self-improving reductions and structural properties of binary trees. We also show constant-factor inapproximability assuming \({\mathbf {P}}\ne \mathbf {NP}\). In addition to inapproximability results, we present algorithmic results along two different flavors: (1) We design a randomized algorithm to verify if a given directed graph on n vertices contains a binary tree of size k in \(2^k \mathsf {poly}(n)\) time. (2) Motivated by the longest heapable subsequence problem, introduced by Byers, Heeringa, Mitzenmacher, and Zervas, ANALCO 2011, which is equivalent to MBT in permutation DAGs, we design efficient algorithms for MBT in bipartite permutation graphs.

我们介绍并研究了有向和无向图中最大二叉树问题(MBT)的逼近性。MBT 的目标是在给定图中找到最大尺寸的二叉树。MBT 是经过充分研究的最长路径问题的自然变体，因为两者都可以被视为在给定图中找到最大大小的有界树。与最长路径的联系激发了对有向无环图 (DAG) 中 MBT 的研究，因为最长路径问题在 DAG 中可以有效解决。相比之下，我们表明 DAG 中的 MBT 很难：在指数时间假设下，它没有有效的\(\exp (-O(\log n/ \log \log n))\) -近似值，其中n是输入图中的顶点数。在无向图中，我们表明 MBT在指数时间假设下没有有效的\(\exp (-O(\log ^{0.63}{n}))\) -近似。我们的不可逼近性结果依赖于二叉树的自我改进减少和结构特性。我们还展示了假设\({\mathbf {P}}\ne \mathbf {NP}\) 的常数因子不可近似性。除了不可逼近性结果之外，我们还展示了两种不同风格的算法结果：（1）我们设计了一个随机算法来验证给定的n个顶点上的有向图是否包含一个大小为k的二叉树in \(2^k \mathsf {poly }(n)\)时间。(2) 受由 Byers、Heeringa、Mitzenmacher 和 Zervas 在ANALCO 2011 中引入的最长可堆子序列问题的启发，该问题等效于置换 DAG 中的MBT ，我们为二部置换图中的 MBT 设计了高效的算法。