  Analysis of chemical graphs is a major research topic in computational
molecular biology due to its potential applications to drug design. One
approach is inverse quantitative structure activity/property relationship
(inverse QSAR/QSPR) analysis, which is to infer chemical structures from given
chemical activities/properties. Recently, a framework has been proposed for
inverse QSAR/QSPR using artificial neural networks (ANN) and mixed integer
linear programming (MILP). This method consists of a prediction phase and an
inverse prediction phase. In the first phase, a feature vector $f(G)$ of a
chemical graph $G$ is introduced and a prediction function $\psi$ on a chemical
property $\pi$ is constructed with an ANN. In the second phase, given a target
value $y^*$ of property $\pi$, a feature vector $x^*$ is inferred by solving an
MILP formulated from the trained ANN so that $\psi(x^*)$ is close to $y^*$ and
then a set of chemical structures $G^*$ such that $f(G^*)= x^*$ is enumerated
by a graph search algorithm. The framework has been applied to the case of
chemical compounds with cycle index up to 2. The computational results
conducted on instances with $n$ non-hydrogen atoms show that a feature vector
$x^*$ can be inferred for up to around $n=40$ whereas graphs $G^*$ can be
enumerated for up to $n=15$. When applied to the case of chemical acyclic
graphs, the maximum computable diameter of $G^*$ was around up to around 8. We
introduce a new characterization of graph structure, "branch-height," based on
which an MILP formulation and a graph search algorithm are designed for
chemical acyclic graphs. The results of computational experiments using
properties such as octanol/water partition coefficient, boiling point and heat
of combustion suggest that the proposed method can infer chemical acyclic
graphs $G^*$ with $n=50$ and diameter 30.
