  Given a graph $G=(V,E)$ with a weight $w_v$ associated with each vertex $v\in
V$, the maximum weighted induced forest problem consists of encountering a
maximum weighted subset $V'\subseteq V$ of the vertices such that $V'$ induces
a forest. This NP-hard problem is known to be equivalent to the minimum
weighted feedback vertex set problem, which has applicability in a variety of
domains. The closely related maximum weighted induced tree problem, on the
other hand, requires that the subset $V'\subseteq V$ induces a tree. We propose
two new integer programming formulations with an exponential number of
constraints and branch-and-cut procedures for the maximum weighted induced
forest problem. Furthermore, we show how formulations for the problem can be
very easily adapted to obtain maximum weighted induced trees. Computational
experiments using benchmark instances are performed comparing several
formulations, including the newly proposed approaches and those available in
the literature, when implemented in a standard commercial MIP (mixed integer
programming) solver. More specifically, five formulations are compared, two
compact (i.e., with a polynomial number of variables and constraints) ones and
the three others with an exponential number of constraints. The results
indicate that one of the newly proposed formulations, denoted tree with cycle
elimination (TCYC), outperforms those available in the literature when it comes
to the average times for proving optimality for the small instances, especially
the more challenging ones. Additionally, this formulation can achieve much
lower average times for solving the larger random instances that can be
optimally solved. The results also illustrate the impact of offering high
quality initial feasible solutions in the performance of the formulations.
