Applying local search to the feedback vertex set problem

Optimal Segmentation of Directed Graph and the Minimum Number of Feedback Arcs

“…The basic idea of the BPD algorithm is straightforward: the vertices i of an input digraph G are fixed to different layers h according to their marginal distributions q h i to simplify the optimization problem [26]. In our implementation, we first iterate the BP equation (8) in combination with the condition (17) a few number t 0 of times to drive the probability distributions close to a fixed point (t 0 = 100 in our code). Then we fill the D different layers up to their capacity as determined by Eq.…”

“…We rank in descending order the remaining un-assigned vertices i according to their marginal probability value q H i , and then assign the first n 0 (e.g., n 0 = 10 −3 N ) vertices to layer H (in the special situation of n 0 > n(H), only the first n(H) vertices are assigned to layer H). To finish the r-th decimation step, we then iterate the BP equation (8) in combination with Eq. (17) a few number t 1 of times (t 1 = 10) and re-evaluate the marginal probability distributions of the remaining un-assigned vertices according to Eq.…”

“…Accordingly, the self-consistent equation for the occupation cost C(h) is changed to Table 2 and Table 3. The total layer number D is set to be D = 2, 4, 8,16,32,64 (from bottom to top) for the six curves of each panel. The inverse temperature is fixed to be β = 10 in all these simulations.…”

“…In a recent work [4], Zhao and one of the present authors proposed a spin glass model for this cycle-constrained optimization problem and derived a mean field theory for estimating the minimum number of feedback arcs and for constructing near-minimum feedback arc sets. The physics-inspired message-passing algorithm performs slightly worse in comparison with a simulated annealing (SA) algorithm adapted from [8].…”

A memetic algorithm for the weighted feedback vertex set problem

“…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 .…”