Maximum weighted induced forests and trees: New formulations and a computational comparative review

Given a graph $G=(V,E)$ with a weight $w_v$ associated with each vertex $v\in V$, the maximum weighted induced forest problem (MWIF) 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 (MWIT), 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 MWIF. Computational experiments using benchmark instances are performed comparing several formulations, including the newly proposed approaches and those available in the literature, when solved by a standard commercial 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 experiments show that a new formulation for the problem based on directed cutset inequalities for eliminating cycles (DCUT) offers stronger linear relaxation bounds earlier in the search process. The results also indicate that the other new formulation, 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. Furthermore, we show how the formulations for MWIF can be easily extended for MWIT. Such extension allowed us to compare the optimal solution values of the two problems.