An Experimental Study of ILP Formulations for the Longest Induced Path Problem

Given a graph $G=(V,E)$, the longest induced path problem asks for a maximum cardinality node subset $W\subseteq V$ such that the graph induced by $W$ is a path. It is a long established problem with applications, e.g., in network analysis. We propose novel integer linear programming (ILP) formulations for the problem and discuss efficient implementations thereof. Comparing them with known formulations from literature, we prove that they are beneficial in theory, yielding stronger relaxations. Moreover, our experiments show their practical superiority

Mixed-Integer Approaches to Constrained Optimum Communication Spanning Tree Problem

Several novel mixed-integer linear and bilinear formulations are proposed for the optimum communication spanning tree problem. They implement the distance-based approach: graph distances are directly modeled by continuous, integral, or binary variables, and interconnection between distance variables is established using the recursive Bellman-type conditions or using matrix equations from algebraic graph theory. These non-linear relations are used either directly giving rise to the bilinear formulations, or, through the big-M reformulation, resulting in the linear programs. A branch-and-bound framework of Gurobi 9.0 optimization software is employed to compare performance of the novel formulations on the example of an optimum requirement spanning tree problem with additional vertex degree constraints. Several real-world requirements matrices from transportation industry are used to generate a number of examples of different size, and computational experiments show the superiority of the two novel linear distance-based formulations over the the traditional multicommodity flow model.Comment: 30 pages, 9 figures, 4 table

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