Compact formulations and an iterated local search-based matheuristic for the minimum weighted feedback vertex set problem

Abstract

Given a weighted graph G=(V,E), the minimum weighted feedback vertex set problem consists in obtaining a minimum weight subset F⊆V of the vertex set whose removal makes the graph acyclic. Differently from other approaches in the literature, in this work we tackle this problem via the maximum weighted induced forest problem. First, we propose two new compact mixed integer programming (MIP) formulations, using a polynomial number of variables and constraints. Next, we develop a matheuristic that hybridizes a multi-start iterated local search heuristic with a MIP-based local search procedure. Extensive computational experiments carried out on a set of benchmark instances show that the newly proposed MILS+-mtz matheuristic is extremely effective and outperforms or at least match the best heuristics available in the literature in terms of solution quality for 79 out of 99 (79.80%) large instance groups, with 55 (55.56%) of them being strictly better than the previously best known. Furthermore, the compact formulations were able to optimally solve 474 out of 810 test instances in less than 3600 seconds of running time.