A Memetic Algorithm for the Weighted Feedback Vertex Set
Problem
Francesco Carrabs
∗
, Carmine Cerrone and Raffaele Cerulli
Department of Mathematics, University of Salerno, Italy.
fcarrabs@unisa.it, ccerrone@unisa.it, raffaele@unisa.it
Abstract
Given an undirected and vertex weighted graph G = (V, E, w), the Weighted Feedback Vertex Set
Problem consists of finding the subset F ⊆ V of vertices, with minimum weight, whose removal results
in an acyclic graph. Finding the minimum feedback vertex set in a graph is an important combinato-
rial problem that has a variety of real applications. In this paper we introduce a memetic algorithm for
this problem. We propose an efficient greedy procedure that quickly generates chromosomes with specific
characteristics and a wise application of a recent local search procedures based on k-diamonds. Computa-
tional results show that the proposed algorithm outperforms the effectiveness of two other metaheuristics
recently proposed in the literature for this problem.
Keywords: Memetic Algorithm, Feedback Vertex Set, Loop Cutset, k-diamond.
1. Introduction
Given an undirected graph G = (V, E), a feedback vertex set (fvs) F ⊆ V of G is a subset of vertices whose
removal results in an acyclic graph (a forest). The feedback vertex set problem (FVS) consists of finding a
fvs of G with minimum cardinality. If G is a vertex weighted graph, then we have the weighted version of
problem (WFVS) that consists of finding the fvs of G with minimum weight.
This problem finds applications in many fields. For example, in the context of operating systems, it
models the problem to prevent and/or remove deadlocks [26] generated by cyclical processes requests of
already locked resources. A similar problem arises in the context of combinatorial circuit design where
the circuits are represented by graphs in which a cycle could generate a “race condition”, that is, some
∗
Corresponding author.
1
circuit elements could receive new inputs before being stabilized. Another application concerns the study
of “monopolies” in synchronous distributed systems [22, 23] where the connection networks are represented
by grid and toroidal graphs. Other applications include program verification [24] and constraint satisfaction
problems [2].
The feedback vertex set problem has been extensively studied (see [10] for a recent survey), and it is
among the first problems shown to be NP-complete [14]. However, it results solvable in polynomial time on
particular class of graphs like: k-diamond graphs [6, 7], permutation graphs [16, 25], reducible flow graphs
[18], interval graphs [19], co-comparability graphs and convex bipartite graphs [13, 17].
For the cases that are not known to be polynomially solvable there have been intensive efforts on approx-
imation algorithms [1, 2, 3, 9, 15] 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 [21] and a simulated annealing
algorithm [12] are introduced whereas two metaheuristics XTS [4] and ITS [7] are proposed for the WFVS.
The tabu search XTS is based on the “eXploring Tabu Search” schema [8]. Thanks to this schema, XTS
finds solutions very close to the optimal one in few seconds. In this metaheuristic the neighborhoods are
represented by k-diamond graphs generated by moving a single vertex from the fvs to the forest. These
neighborhoods are explored using an approximation algorithm [1]. The computational complexity of finding
an optimal fvs on a k-diamond graph was left as an open question. This open question is solved in [5, 6]
where a linear time procedure is proposed. The authors embedded this procedure into an iterative tabu
search ITS for the WFVS problem [7].
Looking at neighborhoods used by XTS and ITS algorithms, we can observe that they essentially perform
vertex exchanges between the fvs and the residual graph induced by removing the fvs from the graph. Due
to the WFVS problem characteristics, the generation of new neighborhoods, different from simple vertices
exchange, does not appear to be a simple task. In this contest, a genetic approach, in which several fvs
are created by mixing the vertices of the existing ones, appears to be particularly suitable. For this reason
we propose a memetic algorithm whose main aim is to be more effective than the other metaheuristics for
WFVS. The accomplishment of our aim is certified by computational results carried out on benchmark
instances. These results show that our algorithm, on the small instances, often finds the optimal solution.
Moreover, on the large instances, our algorithm finds better solutions than ITS and XTS algorithms.
The remainder of the paper is organized as follows. Section 2 introduces the definitions and notations
that are used throughout the paper. Section 3.1.1 contains the description of our greedy algorithm for the
creation of chromosomes. The memetic algorithm is described in Section 4. Finally, the computational
results are presented in Section 5 and some concluding remarks are given in Section 6.
2
2. Definitions and Notations
Let G = (V, E, w) be an undirected and vertex weighted graph, where V is the set of n vertices, E is the
set of m edges, and, w(v) is a positive weight associated with each vertex v ∈ V . Given a subset X ⊆ V of
vertices, we define
¯
X = V \X and W (X) as the sum of the weights of its elements, i.e. W (X) =
P
v∈X
w(v).
We denote by G[X] the subgraph of G induced by the set of vertices X ⊆ V . Formally, G[X] = (X, E
[X]
, w)
where E
[X]
= {(u, v) ∈ E : u, v ∈ X}. Let δ
X
(v) and d
X
(v) be the set of vertices adjacent to v and the
degree of the vertex v in G[X], respectively. When X = V , we simply denote these data by δ(v) and d(v),
respectively. Moreover, let rnd(a, b) be a function that returns a random integer value within the interval
[a, b].
A tree is an acyclic and connected graph while a forest is an acyclic graph in which any connected
component is a tree. Given a set of vertices X ⊆ V , the residual graph of G, generated by X, is the
subgraph G[
¯
X]. The set X is a feedback vertex set (fvs) of G if the residual graph G[
¯
X] is a forest. For
the graph G depicted in Figure 1a, the set of vertices X = {6, 16} is a fvs of G because G[
¯
X] is a forest
(Figure 1b).
From now on, we denote by F (G) and by F
∗
(G) any fvs and a minimum weight fvs (optimal solution)
of G, respectively. When no confusion may arise, we simply denote these sets by F and F
∗
, respectively. A
vertex v ∈ F (G) is redundant if F (G) \{v} is again a fvs of G. F (G) is a minimal fvs if it does not contain
redundant vertices, i.e. there can not be a fvs F
0
(G) such that |F
0
(G)| < |F (G)| and F
0
(G) ⊂ F (G). Any
vertex v ∈ V is said to be appended if it is not included in any cycle of G. In Figure 1c the vertices 2, 9 and
14 are appended. We say that a graph is reduced if it does not contain any appended vertex (Figure 1d).
Finally let us define the k-diamond graphs [7].
Definition 1. A weighted k-diamond D
R,z
= (V
R
, E
R
, w), where k ≥ 1, R = {r
1
, r
2
, . . . , r
k
} and z ∈ V
R
, is
an undirected and vertex weighted graph where (i) each vertex v ∈ V
R
is included in a simple path between
exactly one of the k apices r
i
∈ R and the vertex z and (ii) D
R,z
[V
R
\ {z}] is a forest with k connected
components.
In Figure 1d the 2-diamond graph D
{5,7},6
is depicted, where r
1
= 5, r
2
= 7 and z = 6. As we will see
in section 3.5, our local search procedure builds k-diamonds graphs as subgraphs of G and it uses them to
improve the solutions. For instance, the k-diamond graph D
{5,7},6
shown in Figure 1d can be obtained from
G by removing the vertex 16 and the appended vertices generated by its removal.
To simplify the notation, in the rest of the paper we denote a diamond D
R,z
just as D
z
, since for our
aims the upper apices are negligible.
3
752
112 13 14
6
10 15
75
112 13
6
10 15
(c) (d)
2 − diamond
752
112
9
13 14
6
10 15
(a)
G
16
9
752
112
9
13 14
10 15
(b)
G[
¯
X]
Figure 1: (a) A generic graph G. (b) The residual graph G[
¯
X] where X = {6, 16}. Since G[
¯
X] is a forest, X is a fvs of G. (c)
The subgraph of G obtained reinserting the vertex 6 in the residual graph G[
¯
X]. This graph contains three appended vertices:
2,9,14. (d) Removing these vertices we obtain a 2-diamonds graph.
3. A memetic approach: basic components
Memetic algorithms belong to the class of evolutionary algorithms that use local search within a classical
genetic algorithm framework to intensify the search phase. The algorithm starts with a set of solutions for the
WFVS problem (represented by chromosomes), called population, and it uses the solutions in this population
to generate a new population. In particular, at each iteration, two chromosomes (parents) are selected from
the population and the crossover operator is invoked on these parents to generate a new chromosome (child)
on which the mutation operator is applied. Finally, a local search, based on k-diamonds, is applied on the
child before it replaces one of the two parents into the population. This is motivated by the hope that the
new population will be better than the previous one. Solutions which are selected to form new solutions
are selected mainly according to their fitness: the more suitable they are the more chances they have to be
4
1
2
5
7
6
12
8
21
1732
24
4
3
4
1 5 7
F
1
R
(F
1
) = 50
5 3
F
2
R
(F
2
) = 45
3 6 2 4
F
3
G
fitness
redundant vertex
Figure 2: Three fvs of graph G. F
1
and F
2
are chromosomes because they are minimal fvs of G. Their fitness is obtained by
adding the weight of their vertices (there are no penalties in this example). F
3
is not a chromosome because the vertex 3 is
redundant.
selected for the crossover operation. This is repeated until a stop criterion is fulfilled. For a complete and
detailed description of the memetic algorithms and their characteristics the reader can refer to [20].
The elements that compose our memetic algorithm are described below in details.
3.1 Chromosomes and fitness
We define the chromosome as a set F ⊆ V of vertices (each vertex in F can be viewed as a gene) such that
G[
¯
F ] is a forest and there are no redundant vertices in F . In other words, a chromosome is a minimal fvs of
the graph. The fitness f (F ) of chromosome F is defined as the sum of weight w(v) and (possible) penalty
ρ(v) of each vertex v in F . Formally: f(F ) =
P
v∈F
{w(v) + ρ(v)}.
In Figure 2 a graph G and three fvs of graph are shown. F
1
and F
2
are chromosomes of G and their
fitness value is shown on the right. In this example, the fitness value is equal to the sum of the weights of
the vertices because there are no penalties associated to the vertices. The last fvs F
3
is not a chromosome
because, by removing the vertex 3 from F
3
, we obtain a smaller fvs of G. As we will see later, the sets
of vertices generated by crossover and mutation operators can contain redundant vertices. To avoid this,
every time a new fvs is created or modified by these operators, a redundancy control is carried out by our
algorithm.
3.1.1 Chromosomes generation: the Snd procedure
In this section we introduce the procedure we used to generate chromosomes, namely the Snd procedure.
Given a graph G = (V, E, w), Snd computes a fvs F of G according to the pseudocode shown in Algorithm 1.
At the beginning the set of vertices X, of the residual graph, is equal to V while F is an empty set (line 1).
5
![Figure 1: (a) A generic graph G. (b) The residual graph G[X̄] where X = {6, 16}. Since G[X̄] is a forest, X is a fvs of G. (c) The subgraph of G obtained reinserting the vertex 6 in the residual graph G[X̄]. This graph contains three appended vertices: 2,9,14. (d) Removing these vertices we obtain a 2-diamonds graph.](/figures/figure-1-a-a-generic-graph-g-b-the-residual-graph-g-x-where-2xohmn93.webp)