A PRUNING BASED ANT COLONY ALGORITHM FOR
MINIMUM VERTEX COVER PROBLEM
Ali D. Mehrabi
§
, Saeed Mehrabi
and Abbas Mehrabi
§
Department of Mathematical and Computer Sciences, Yazd University, Yazd, Iran
Department of Computer Science, Shahid Bahonar University of Kerman, Kerman, Iran
Department of Computer Engineering, Islamic Azad University, South Tehran Branch, Tehran, Iran
Keywords: Minimum vertex cover problem, Meta-heuristic approaches, Ant colony optimization algorithms.
Abstract: Given an undirected, unweighted graph G = (V , E) the minimum vertex cover (MVC) problem is a subset
of V whose cardinality is minimum subject to the premise that the selected vertices cover all edges in the
graph. In this paper, we propose a meta-heuristic based on Ant Colony Optimization (ACO) approach to
find approximate solutions to the minimum vertex cover problem. By introducing a visible set based on
pruning paradigm for ants, in each step of their traversal, they are not forced to consider all of the remaining
vertices to select the next one for continuing the traversal, resulting very high improvement in both time and
convergence rate of the algorithm. We compare our algorithm with two existing algorithms which are based
on Genetic Algorithms (GAs) as well as its testing on a variety of benchmarks. Computational experiments
evince that the ACO algorithm demonstrates much effectiveness and consistency for solving the minimum
vertex cover problem.
1 INTRODUCTION
1.1 Minimum Vertex Cover Problem
Given an undirected, unweighted graph G = (V , E),
the minimum vertex cover problem is to find a
subset of V such that for each edge in E, at least one
of its two end vertices is in the subset and that its
cardinality is minimum. Formally, given an
undirected, unweighted graph G = (V , E), the
minimum vertex cover problem seeks to find a
subset
VV '
satisfying
Eve
Vv
=
)(
'
, where
)(ve
denotes the edges incident on vertex v, such
that
'V
is minimized.
In (Karp, 1972) the decision version of the
minimum vertex cover problem had been shown as
NP-complete. After then, the vertex cover problem
is one of the core NP-complete problems that have
been frequently used for delivering to NP-hardness
(Gary and Johnson, 1979). In practice, the minimum
vertex cover problem can be used to model many
real world situations in the areas of circuit design,
telecommunications, network flow and so on. For
example, whenever one wants to monitor the
operation of a large network by monitoring as few
nodes as possible, the importance of the MVC
problem comes into the rule (Papadimitriou and
Steiglitz, 1982).
Due to the computational intractability of the
problem, many researchers have instead focused
their attention on the design of approximation
algorithms for delivering quality solutions in a
reasonable time. An intuitive greedy approach for
solving the problem is to successively select the
vertex with the largest degree until all of the edges
are covered by the vertices in V'. This
straightforward heuristic is not a good one as
demonstrated by (Papadimitriou & Steiglitz 1982, p.
407). They considered regular graphs, each of which
consists of three levels. The first two levels have the
same number of vertices while the third level has
two vertices less than the number of vertices found
on the previous two levels. The regular graph for
k=3 can be found in (Papadimitriou and Steiglitz,
1982). As (Papadimitriou, 1994) shows, this greedy
algorithm never produces a solution which is more
than
)ln(n times the optimum, where n is the
number of vertices. However, the best
281
D. Mehrabi A., Mehrabi S. and Mehrabi A. (2009).
A PRUNING BASED ANT COLONY ALGORITHM FOR MINIMUM VERTEX COVER PROBLEM.
In Proceedings of the International Joint Conference on Computational Intelligence, pages 281-286
DOI: 10.5220/0002313802810286
c
SciTePress
approximation algorithm known for the minimum
vertex cover problem has been reported in (Khuri
and Back, 1994), in which at each iteration, we
randomly choose an edge, say
end points to current vertex cover being constructed
deleting them from the graph. The algorithm reveals
a performance guarantee of 2. As a part of our
experimental results, we compare our proposed
algorithm with two existing algorithms based on
GAs on this type of graphs. Finally (Dinur and
Safra, 2001) show that it is impossible to attain
approximate solutions to the minimum vertex cover
problem within any factor smaller than 1.36067,
unless
.NPP = After this, the meta-heuristic
approaches for solving the problem come into the
role.
In solving the minimum vertex cover problem,
we also have a solution for another graph problem:
The maximum independent set (MIS) problem (exact
definition of this problem can be found in (West,
2001), for example). The close relationship between
these problems is shown by the following Lemma
(see e.g. (Papadimitriou and Steiglitz, 1972)):
Lemma 1. For any graph G = (V , E) and V' V, the
following statements are equivalent:
V' is the minimum vertex cover in G.
VV' is the maximum independent set of G.
Consequently, one can obtain a solution of the
maximum independent set problem by taking the
complement of solution to the minimum vertex
cover problem (Hifi, 1997). However, many meta-
heuristic approaches for solving the MIS have been
used, so far. Specifically, we have used a hybrid GA
for its solving which results in very near to optimum
solutions for a variety of large scale MIS problem
benchmarks (Mehrabi and et al., 2009).
1.2 Ant Colony Optimization
In the optimization literature, meta-heuristic also
foster a viable alternative for delivering quality
approximate solutions (Mehrabi and Mehrabi, 2009).
Like genetic algorithms, simulated annealing and
tabu search, the Ant Colony Optimization (ACO) is
a meta-heuristic using natural metaphor to solve
complex combinatorial optimization problems such
as the traveling salesman problem, graph coloring
problem (Costa and Hertz, 1997) and so on. The
general framework of the ACO algorithms is
presented in Fig. 1. Basically, the problem under
study is transformed into a weighted graph. Then,
the ACO algorithm iteratively distributes a set of
artificial ants onto the graph to construct the paths
corresponding to potential optimal solutions.
The optimization mechanism of the ACO is
based on two important features: The state transition
rule and the pheromone updating rule. The first one,
which is a probabilistic operation, is applied when
an ant is choosing the next vertex to visit. The
second one dynamically changes the preference
degree for the edges that have been traversed
through. As the literature shows, apply these two
simple parts of an ACO can solve many complex
optimization problems. This paper also explains
their role in algorithm by next section.
2 THE ACO ALGORITHM
As we mentioned, the structure of MVC problem is
different from those of problems which have been
solved by ACO in literature. In fact, the solution to
MVC is an unordered subset of vertices obtained by
each ant, while the most ACO-based solvers so far
get the solution as an ordered or unordered subset of
edges (Dorigo and Gambardella, 1997). So it
becomes more challenging and desirable to
transform the MVC problem characteristics into an
appropriate graph representation. Also, the
implementation of a standard ACO phases,
providing an efficient local heuristic for the state
transition rule and the design of pheromone updating
rule becomes an important issue. We devoted this
section for presenting our implementations in detail.
Figure 1: The outline of an ACO algorithm.
2.1 Graph Representation
Suppose that G = (V , E) denote the underlying
graph for MVC problem and the solution to this
IJCCI 2009 - International Joint Conference on Computational Intelligence
282
instance is an unordered vertex subset VV ' .
Each ant should traverse some path across the edges
of graph to cover exactly and only the vertices in
'V , however this path may really do not exist. For
overcoming this problem, we constitute a complete
graph G
c
= (V , E
c
) including the vertex set V of G
such that every pair of vertices are connected by an
edge in E
c
. To aware some ant, say ant k, of
distinguishing between original and added edges in
E
c
, we define a binary connectivity function
{}
1,0:
c
EC for each edge ),( ji as:
=
EEji
Eji
jiC
c
),(0
),(1
),(
(1)
For a better understanding aims, we explained
our implementations on a graph with 5 vertices
which have shown in Fig. 2(a). Also, applying Eq.
(1) to it is given in Fig. 2(b). We know that this
special instance has more than one solution such as
{}
ECA ,, ,
{}
ECB ,, and so on, and so for most of
the solutions we can not find a corresponding path in
graph that covers exactly and only the solution
vertices. However, by our graph transformation the
solution paths, which can obtained in any order of
vertices, can be reached.
2.2 Connectivity Updating Rule
The value of an edge, connectivity value, in E is
updated when one of its edge vertices was visited by
some ant, say ant k, in the following way:
n
jiC
1
),( =
, if edge Eji ),( and either
vertex i or j is visited by ant k.
(2)
in which
n is the number of graph vertices, V .
Fig. 3 shows the graph of previous example in which
the connectivity values of edges
),( BA , ),( CA
and
),( EA
have been updated to 51 when vertex
A
is visited. The update settings according to Eq.
(2) have a twofold benefit. First, the desirability of
each vertex, as the next one to visit, can be evaluated
dynamically. To be exact, we define:
=
c
Ejr
k
j
jrCD
),(
),(
(3)
as the preference number for vertex j. So, the larger
preference values the higher desirability of vertex j
for ant k. Second, since a solution to MVC is a
subset of vertices the ants may need to pass some
steps in order to complete their own tours. However,
if
1<
k
j
D for all j, the ant k has completed its own
tour, a good stopping criteria. Note that, before
starting the next cycle, the connectivity values
should be reset using Eq. (1) to restore the original
graph information.
2.3 State Transition Rule
One of the main parts of an ACO-based solver,
which results the improvement in both optimality
and efficiency of the algorithm, is the state transition
rule. So we have devised an efficient pruning-based
heuristic for this phase of our ACO for MVC
problem.
(a)
(b)
Figure 2: An example. (a) The original graph instance. (b)
Illustration of our graph transformation.
Unlike the most of ACO-based solvers, that the
preference information is deposited on edges, we
(due to underlying problem characteristics)
deposited the preference information on vertices.
Our state transition rule which describes the
probability of selecting vertex j, as next vertex, for
ant k by:
=
k
Ar
rk
r
jk
j
k
j
P
βα
βα
ητ
ητ
(4)
A PRUNING BASED ANT COLONY ALGORITHM FOR MINIMUM VERTEX COVER PROBLEM
283
where
k
A is the set of accessible vertices for ant k.
also,
j
α
τ
and
β
η
jk
represent the global pheromone
updating factor and local desirability scale for vertex
j, respectively. As the pruning part of Eq. (4), the ant
k is not forced to consider all of the vertices to select
the best one, vertex j. Instead, we define
k
V as the
visible set for ant k. In fact, we provide:
=
k
Ar
k
jrCjV 1),(
(5)
as the checklist for ant k and prune the other
vertices. We will show soon that this simple
heuristic of Eq. (5) results very high efficiency in
both time and solution optimality for MVC problem.
2.4 Pheromone Updating Rule
The ACO relies on the synergy among a population
of ant agents. Here, we use the global and local
pheromone updating rules as follows. First, at the
end of each cycle, the pheromone left on the vertices
of the currently best solution is reinforced. Suppose
that V'
c
is the currently best solution. For each vertex
Figure 3: Applying connectivity updating rule to our
running example.
c
Vi ' we will update its pheromone according to
global updating by:
iii
τ
ρ
τ
ρ
τ
Δ
+
= )1(
(6)
where
c
i
V '
1
=Δ
τ
(7)
and
)1,0(
ρ
is a parameter which simulates the
evaporation rate of the pheromone intensity and
enables the ants to forget the bad decisions
previously done.
Second, we apply a local pheromone updating
rule to explore the solution space as far as possible
and permit new ants to visit the unvisited vertices
with a higher probability which leads to diversity of
the solutions obtained. This is accomplished by the
following rule:
0
')'1(
τ
ρ
τ
ρ
τ
+
=
ii
(8)
where
)1,0('
ρ
pheromone previously laid on vertex i and
0
τ
is the
same as the initial value of pheromone laid on each
vertex before starting the algorithm. Reduction in
the pheromone intensity of the vertex i is obvious
from Eq. (8), as expected.
2.5 Stopping Criterion
The stopping criterion of an ACO could be a
maximum number of iterations, a constant CPU time
limit, or any other fixed criteria which leads to best
improvement of the algorithm. In this paper, we use
an alternative, a given number of iterations in which
no improvement on the solution is obtained.
3 EXPERIMENTAL RESULTS
In this section, we present the experimental results
obtained by MVC-AC for solving the problem.
The algorithm has been implemented by Java
programming language on windows platform and
Intel Pentium(R) 4 CPU 2.40 GHz processor. The
parameters used in our implementation are
α
=0.7
and
β
=0.3. The parameter setting for
ρ
has an
important role in the ACO algorithms. For some
fixed values of
and
β
, we run our algorithm
with different values of
ρ
. According to
experiences, MVC-AC with
ρ
=0.03 exhibits the
best performance for getting the optimum solution.
After that, we first, run our algorithm on regular
graphs of (Papadimitriou and Steiglitz, 1972), with
k=32 and k=66, namely "ps100" and "ps202",
respectively. We got the optimum solutions in 100
runs of MVC-AC, consistently. Our comparisons
with GENEsYs, a genetic algorithm software
package, from (Khuri and Back, 1994) and HGA of
Kotecha and Gambhava, 2003) on these graphs are
reported in Tables 1 and 2.
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284
Table 1: Comparison results for "ps100" graph.
Table 2: Comparison results for "ps200" graph.
Then, we tested algorithm on more challenging
instances of random graphs which described in
(Khuri and Back, 1994). We got very surprising
results in comparison with Khuri and Back's results
(Khuri and Back, 1994) and vercov heuristic's
reported there. The results of 100 runs of MVC-AC
on "mvcp100-02", "mvcp100-03", "mvcp200-01"
and "mvcp200-02", which are more challenging,
summarized in Tables 3 and 4. According to
comparisons, MVC-AC treats very consistent and
effective for solving the minimum vertex cover
problem.
4 CONCLUSIONS
One of the most challenging problems of the graph
theory is the NP-complete minimum vertex cover
problem. In this paper, we introduced a simple but
efficient Ant Colony Optimization algorithm, called
MVC-AC, for solving this problem. Most of our
ACO components incorporate with the standard
ACO algorithms. According to ACO literature, we
speed up the ants traversal by considering a heuristic
into the state transition rule of our ACO. Also, by
introducing a new pruning based approach, the
visible set for each ant, we restricted the ant search
space only to vertices in its visible set, resulting
substantial improvement for both time and
convergence rate of the algorithm.
For experience, we compared our algorithm with
some efficient existing algorithms based on
evolutionary algorithms, such as GENEsYs and
HGA. Also a variety of benchmarks is used to test
MVC-AC. As the experimental results show, MVC-
AC not only outperforms the algorithms above, but
it also treats very efficient and consistent with for
solving the minimum vertex cover problem.
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