Solving the feedback vertex set problem on undirected graphs

Abstract

Feedback problems consist of removing a minimal number ofvertices of a directed or undirected graph in order to make it acyclic. The problem is known to be NPcomplete. In this paper we consider the variant on undirected graphs. The polyhedral structure of the Feedback V ertex Set polytope is studied. We prove that this polytope is full dimensional and show that some inequalities are facet de ning. We describe a new large class of valid constraints, the subset inequalities. A branch-and-cut algorithm for the exact solution of the problem is then outlined, and separation algorithms for the inequalities studied in the paper are proposed. A Local Search heuristic is described next. Finally we create a library of 1400 random generated instances with the geometric structure suggested by the applications, and we computationally compare the two algorithmic approaches on our library.

A memetic algorithm for the weighted feedback vertex set problem

“…In particular, we want to highlight how the effectiveness and performance of the algorithms are affected by problem size, by density of the graph, by the weight ranges, and by the classes of graphs. We compare our MA algorithm with the two tabu search ITS and XTS .…”

“…For the cases that are not known to be polynomially solvable, there have been intensive efforts on approximation algorithms whereas very few heuristics are proposed in the literature for the WFVS. To the best of our knowledge, for the FVS problem a GRASP procedure and a simulated annealing algorithm are introduced whereas two metaheuristics XTS and ITS are proposed for the WFVS. The tabu search XTS is based on the “eXploring Tabu Search” schema .…”

A Tabu Search Heuristic Based on k-Diamonds for the Weighted Feedback Vertex Set Problem

“…Moreover, we show how this new class of graphs can be used to define a neighborhood structure (namely, the k-diamond Neighborhood) of a given feasible solution and, successively, we show how to solve the problem on general graphs by means of a tabu search technique using the k-diamond neighborhood. Such a class of neighborhood was already introduced in [4], where, however, the computational complexity of finding an optimum WFVP on a k-diamond graph was left open and a heuristic approach was used to solve the problem. We solve such an open problem (by giving a linear time algorithm) and also show the effectiveness of the chosen neighborhood in improving a given initial feasible solution when explored by means of our exploration strategy.…”

On the Minimum Feedback Vertex Set Problem: Exact and Enumeration Algorithms

“…It is also one of the classical NP-complete problems from Karp's list [22]. Thus not surprisingly, for several decades, many different algorithmic approaches were tried on this problem including approximation algorithms [1,2,12,23], linear programming [9], local search [4], polyhedral combinatorics [7,20], probabilistic algorithms [26], and parameterized complexity [10,11,21].…”