Operations Research Letters 37 (2009) 181–186
Contents lists available at ScienceDirect
Operations Research Letters
journal homepage: www.elsevier.com/locate/orl
An edge-reduction algorithm for the vertex cover problem
Qiaoming Han
a
, Abraham P. Punnen
b,∗
, Yinyu Ye
c
a
School of Mathematics and Statistics, Zhejiang University of Finance and Economics, Hangzhou 310018, PR China
b
Department of Mathematics, Simon Fraser University, 14th Floor Central City Tower, 13450 102nd Ave., Surrey, BC V3T5X3, Canada
c
Department of Management Science and Engineering, School of Engineering, Stanford University, Stanford, CA 94305, USA
article info
Article history:
Received 19 April 2008
Accepted 20 January 2009
Available online 3 February 2009
Keywords:
Vertex cover problem
Approximation algorithm
LP-relaxation
abstract
An approximation algorithm for the vertex cover problem is proposed with performance ratio
3
2
on special
graphs. On an arbitrary graph, the algorithm guarantees a vertex cover S
1
such that |S
1
|≤
3
2
|S
∗
|+ξ where
S
∗
is an optimal cover and ξ is an error bound identified.
© 2009 Elsevier B.V. All rights reserved.
1. Introduction
Let G = (V , E ) be an undirected graph on the vertex set V =
{1, 2,..., n}.A vertex cover of G is a subset S of V such that each
edge of G has at least one endpoint in S . The vertex cover problem
(VCP) is to compute a vertex cover of smallest cardinality in G. VCP
is a classical NP-hard problem.
It is well known that an optimal vertex cover of a graph
can be approximated within a factor of 2 in polynomial time by
taking all the vertices of a maximal (not necessarily maximum)
matching in the graph or rounding the LP relaxation solution
of an integer programming formulation [1,15]. There has been
considerable work (see e.g. survey paper [2]) on the problem over
the past 30 years on finding a polynomial-time approximation
algorithm with an improved performance guarantee. It is known
that computing a δ-approximate solution in polynomial time for
the VCP is NP-Hard for any δ ≤ 10
√
5 − 21 ≃ 1.36 [3]. In fact,
no polynomial-time (2 − ǫ)-approximation algorithm is known
for the VCP for any constant ǫ > 0. Under the assumption of
unique game conjecture [4–6] is valid, many researchers believe
that a polynomial time 2 − ǫ approximation algorithm is not
possible for the VCP for any constant ǫ > 0. The current best
known bound on the performance ratio of a polynomial time
approximation algorithm for VCP is 2 − Θ(
1
√
log n
) [7]. Halperin [8]
showed that an approximation ratio of 2 −
2 log log ∆
log ∆
can be obtained
with the semidefinite programming (SDP) relaxation of VCP where
∗
Corresponding author.
E-mail addresses: qmhan@nju.edu.cn (Q. Han), apunnen@sfu.ca (A.P. Punnen),
yinyu-ye@stanford.edu (Y. Ye).
∆ is the maximum degree of G. Other SDP-relaxations of the VCP
were studied in [9,10]. On four colorable graphs, a
3
2
-approximate
solution can be identified in polynomial time [11]. Recently
Asgeirsson and Stein [12,13] reported extensive experimental
results using a heuristic algorithm which obtained no worse than
3
2
-approximate solutions for all the test problems they considered.
A natural integer programming formulation of VCP can be
described as follows:
(VC )
min
n
i=1
x
i
s.t . x
i
+ x
j
≥ 1, (i, j) ∈ E ,
x
i
∈{0, 1}, i = 1, 2,..., n.
(1)
Let ¯ x = ( ¯ x
1
, ¯ x
2
,..., ¯ x
n
) be an optimal solution to (1). Then R =
{i |¯ x
i
= 1} is an optimal vertex cover of the graph G. The linear
programming relaxation of the above integer program, denoted by
LP, is given by relaxing the integrality constraints to x
i
≥ 0, i =
1, 2,..., n.
Any vertex cover must contain at least s + 1 vertices of an odd
cycle of length 2s + 1. This motivates the following extended linear
programming (ELP) relaxation of the VCP:
(ELP )
min
n
i=1
x
i
s.t . x
i
+ x
j
≥ 1, (i, j) ∈ E ,
i∈ω
k
x
i
≥ s
k
+ 1, ω
k
∈ Ω,
x
i
≥ 0, i = 1, 2,..., n,
(2)
where Ω denotes the set of all odd-cycles of G and ω
k
∈ Ω
contains 2s
k
+ 1 vertices for some integer s
k
. Note that even
0167-6377/$ – see front matter © 2009 Elsevier B.V. All rights reserved.
doi:10.1016/j.orl.2009.01.010