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

Given a graph with weight associated each vertex , the maximum weighted induced forest problem (MWIF) consists of encountering subset vertices such that induces forest. This NP-hard is known to be equivalent minimum feedback set problem, which has applicability in variety domains. The closely related tree (MWIT), on other hand, requires tree. We propose two new integer programming formulations an exponential number constraints and branch-and-cut procedures for MWIF. Computational experiments using benchmark instances are performed comparing several formulations, including newly proposed approaches those available literature, when solved by standard commercial mixed solver. More specifically, five compared, compact ones (i.e., polynomial variables constraints) three others constraints. show formulation based directed cutset inequalities eliminating cycles (DCUT) offers stronger linear relaxation bounds earlier search process. results also indicate formulation, denoted cycle elimination (TCYC), outperforms literature it comes average times proving optimality up 81 vertices. Additionally, this can achieve much lower solving larger random optimally solved. Furthermore, we how MWIF easily extended MWIT. Such extension allowed us analyze difference between optimal solution values problems considering different classes graphs.