  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 in fact hard: it has no efficient
$\exp(-O(\log n/ \log \log n))$-approximation algorithm 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 $\text{P}\neq \text{NP}$.
  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
$2^k \text{poly}(n)$ 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.
