J Glob Optim (2014) 59:663–671DOI 10.1007/s10898-013-0116-4

Minimum vertex cover in ball graphs through local search

Zhao Zhang · Weili Wu · Lidan Fan · Ding-Zhu Du

Received: 24 August 2012 / Accepted: 4 October 2013 / Published online: 13 October 2013© Springer Science+Business Media New York 2013

Abstract Using local search method, this paper provides a polynomial time approximationscheme for the minimum vertex cover problem on d-dimensional ball graphs where d ≥ 3.The key to the proof is a new separator theorem for ball graphs in higher dimensional space.

Keywords Vertex cover · Ball graph · Local search · Separator theorem

1 Introduction

For a graph G, a vertex cover is a vertex subset U ⊆ V (G) such that every edge has atleast one end in U . The minimum vertex cover problem (MVC) is to find a vertex cover ofthe minimum cardinality. MVC is a classic optimization problem which was one of Karp’s21 NP-complete problems, and is used in many books as the first example to introduceapproximation algorithm (see for example [4,22,24]).

It is well known that MVC is NP-hard even for planar graphs with no vertex degreeexceeding 4 [10]. Papadimitriou et al. [23] proved that MVC is APX-hard. Later on, Dinurand Safra [6] proved that MVC cannot be approximated within a factor of 1.3606 unlessP=NP. Despite these negative results, a factor-2 approximation algorithm has long beenknown [22], simply taking the ends of a maximal matching. More involved methods leadto a slightly better approximation. An approximation factor of 2 − (log log n)/(2 log n) wasindependently obtained by Bar-Yehuda and Even [2] and Monien and Speckenmeyer [19],which was imporoved by Karakostas [14] to 2 − O(1/ log

√n).

Z. Zhang (B)College of Mathematics and System Sciences, Xinjiang University Urumqi,Xinjiang 830046, People’s Republic of Chinae-mail: hxhzz@sina.com

W. Wu · L. Fan · D.-Z. DuDepartment of Computer Science, University of Texas at Dallas, Richardson,TX 75080, USA

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For graphs with some geometry, better approximation can be achieved. In an intersectiongraph G, every vertex corresponds to a geometric region on the plane. Two vertices areadjacent in G if and only if their regions have a nonempty intersection. In particular, if everyvertex corresponds to a disk, then the intersection graph is a disk graph (DG). If furthermorethe disks have the same radius, then the graph is a unit disk graph (UDG).

It was shown in [17] that in UDG, a lot of NP-hard problems can be approximated withina good factor, such as the maximum independent set problem (MIS), the minimum vertexcover problem (MVC), the minimum dominating set problem (MDS), etc. An algorithm is apolynomial time approximation scheme (PTAS) if it achieves approximation factor (1 + ε)

for any real number ε > 0 and runs in polynomial time (depending on ε). By the partitionand shifting strategy which was first proposed by Baker [1] and Hochbaum and Maass [12],PTAS can be obtained for the above problems in UDG [13].

Achieving good approximation in DG is much harder than in UDG. In [7], Erlebach etal. gave a PTAS for the weighted MIS problem and the weighted MVC problem in DG,using dynamic programming on a multi-layer partition combined with the shifting strategy.This method is quite complicated. However, as pointed out by Erlebach and van Leeuwenin [8], this method is not able to deal with the MDS problem. A break through was madein 2009, when Chan and Har-Peled [3] and Mustafa and Ray [20] independently observedthat a simple local search method can produce PTAS for some NP-hard problems in somegeometric graphs including DG. Chan and Har-Peled [3] used local search method to give aPTAS for MIS on admissible graphs, where a set of regions is admissible if for any two regionsr1 and r2, both r1\r2 and r2\r1 are connected and do not contain holes. Mustafa and Ray[20] used local search method to give the first PTAS for the Minimum Hitting Set problem[20] for half-spaces in R3 and r -admissible regions in R2 (a set of admissible regions isr -admissible if the boundaries of any two regions intersect at most r times). Continueing thestory, Gibson and Pirwani [11] used local search method to give the first PTAS for MDS inDG. The main idea behind these results is to apply a separator theorem for planar graphs [9]to a particular planar graph which relates a globally optimal solution with a locally optimalsolution. Motivated by these inspiring achievements, this paper intends to explore the powerof local search method in higher dimensional space.

In this paper, we study the MVC problem in ball graphs. For a d-dimensional ball graphG, each vertex corresponds to a d-dimensional ball, two vertices are adjacent in G if andonly if the two balls corresponding to them have non-empty intersection. The set of balls iscalled a geometric representation of G. In particular, a 2-dimensional ball graph is exactly adisk graph.

The contribution of this paper is two-fold. First, we show that a similar idea to Frederick-son’s separator theorem [9] can be applied to ball graphs in which

every point in Rd is contained in at most k balls, (1)

where k is a constant. Like Frederickson’s result [9] which is a careful generalization of awell-known separator theorem for planar graphs due to Lipton and Tarjan [16], our result isa careful generalization of some previous separator theorems due to Miller et al. [18] andDijdjev and Gilbert [5]. Then, applying the new separator theorem to an appropriately definedball graph satisfying condition (1) with k = 2, we show that local search method yields aPTAS for the MVC problem on ball graphs.

The paper is organized as follows. Section 2 introduces some separator theorems andprove a new one for ball graphs satisfying condition (1). Section 3 proves that a local searchprovides a PTAS for MVC in ball graphs. Section 4 concludes the paper and points out someopen problems.

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2 Separator in unit ball graph

Let G be a graph on vertex set V , and w be a weight function on V . For a vertex subsetA ⊆ V , the weight of A is w(A) = ∑

v∈A w(v).Divide and conquer is an important method in parallel computing. The idea is to divide

a problem into subproblems, solve them separately, and then assemble these local solutionsinto a global solution to the original problem. In order that the loss is small during suchan assembling, one has to ensure that the boundary is thin. Furthermore, it is desirable thatthe sizes of the subproblems are comparatively balanced. These are the focus of separatortheorem. A class of graphs satisfies an f (n)-separator theorem if there are constants α < 1and β > 0 such that every n-vertex graph in this class has a vertex subset, called a separator,of at most β f (n) vertices, the removal of which separates the graph into two parts, each parthaving weight at most αw(V ). Here f (n) is a higher order infinitesimal of n. A graph isplanar if there is an embedding of the graph on the plane such that its edges intersect only attheir ends. The following theorem is one of the earliest separator theorem.

Theorem 2.1 [16] Let G be an n-vertex planar graph having non-negative vertex weightfunction w such that w(V ) ≤ 1. Then the vertex set of G can be partitioned into three setsAI , AO , AB such that

(i) no edge joins a vertex in AI to a vertex in AO ;(ii) w(AI ), w(AO) ≤ 2/3;

(iii) |AB | ≤ 2√

2√

n.

Condition (i i) says that the two separated parts AI and AO have comparatively balancedweights, and Condition (i i i) says that the separator AB is thin.

By assigning each vertex with a weight 1/n, the well-known√

n-separator theorem forplanar graph is obtained from Theorem 2.1, saying that the vertex set of any n-vertex pla-nar graph can be partitioned into three sets AI , AO , AB satisfying conditions (i), (i i i) and|AI |, |AO | ≤ 2n/3.

The cardinality version of this theorem is generalized to d-dimensional ball graphs byMiller et al. [18]. One can also refer to the book [21] for a simpler proof.

Theorem 2.2 [18] Let B = {B1, . . . , Bn} be a geometric representation of a d-dimensionalball graph G. Suppose every point in R

d belongs to at most k members of B. Then the vertexset of G can be partitioned into three subsets AI , AO , AB, such that

(i) no edge joins a vertex in AI to a vertex in AO ;(ii) |AI |, |AO | ≤ d+1

d+2 n;

(iii) |AB | ≤ cdk1/dn1−1/d , where, cd is a constant depending only on d.

By Koebe’s result [15] which says that any planar graph can be represented by the inter-section graph of a packing of disks, it can be seen that Theorem 2.2 is a generalization of the√

n-separator theorem for planar graphs in higher dimensional spaces.It should be noted that this generalization is for the cardinality case. However, a weighted

case can be obtained by the following result proved by Djidjev and Gilbert [5].

Theorem 2.3 [5] Let G be an n-vertex graph all of whose subgraphs satisfy a cardinalitynλ-separator theorem with 0 < α, λ < 1 and β > 0. Suppose w is a vertex weight functionwith w(V ) = 1. Then for any tI > 0 and tO > 0 with tI + tO = 1, vertices of G can bepartitioned into three sets AI , AO , AB such that

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(i) no edge joins a vertex in AI to a vertex in AO ;(ii) w(AI ) ≤ tI and w(AO) ≤ tO ;

(iii) |AB | ≤ δnλ, where δ is a constant depending only on α, β and λ.

In fact, Djidjev and Gilbert proved more. They showed that if there are k vertex weightfunctions, a separator exists such that AI and AO are balanced with respect to all these kweight functions.

Notice that if a separator theorem is true for a vertex induced subgraph, then it is alsotrue for an edge induced subgraph. Theorem 2.2 says that any d-dimensional ball graphmeeting the requirement of condition (1) satisfies a cardinality n1−1/d -separator theoremwith α = (d + 1)/(d + 2) and β = cdk1/d . Notice that any vertex induced subgraph of aball graph is still a ball graph. Hence a ball graph satisfies the conditions of Theorem 2.3,and thus by taking tI = to = 1/2, one obtains the weighted version of Theorem 2.2.

Corollary 2.4 Let B = {B1, . . . , Bn} be a geometric representation of a d-dimensional ballgraph G, and let w be a vertex weight function such that w(V ) = 1. Suppose every point inR

d belongs to at most k members of B. Then the vertex set of G can be partitioned into threesubsets AI , AO , AB, such that

(i) no edge joins a vertex in AI to a vertex in AO ;(ii) w(AI ), w(AO) ≤ 1/2;

(iii) |AB | ≤ cn1−1/d , where c is a constant depending only on d and k.

We also need to use the cardinality version of the above result, which is obtained bychanging condition (i i) to |AI |, |AO | ≤ n/2.

For the purpose of this paper, we need to split a graph into many parts such that each parthas a thin boundary. This is the subject of the following theorem, which is the main result inthis section. A similar result for planar graph was given by Frederickson in [9]. For a vertexset U ⊆ V (G), let NG(U ) be the set of vertices in V (G)\U which are adjacent to somevertex of U in G. When the graph G is clear in the context, subscript G is omitted.

Theorem 2.5 Let B = {B1, . . . , Bn} be a geometric representation of a d-dimensional ballgraph G. Suppose every point in R

d belongs to at most k members of B. Then for any real

number r >( 2c

21/d−1

)d, where c is the constant in Corollary 2.4, the vertex set of G can be

partitioned into A ∪ V1 ∪ . . . ∪ Vt satisfying the following conditions:

(i) t = O(n/r);(ii) there is no edge between any Vi and Vj for i �= j ;

(iii) |Vi | ≤ r for each i = 1, . . . , t ;(iv) |N (Vi ) ∩ A| ≤ r1−1/(2d) for each i = 1, . . . , t ;(v) |A| = O(n/ 2d

√r).

Proof If r ≥ n, taking A = ∅ and t = 1 satisfies the theorem. Hence suppose r ≤ n in thefollowing. The theorem is proved by two phases of splitting.

In the first phase, apply the cardinality version of Corollary 2.4 to graph G. The resultingsubgraphs of G induced by AI ∪ AB and AO ∪ AB are called A-components of G. Recursivelyapplying the cardinality version of Corollary 2.4 to each A-component whose size is greaterthan r until every A-component has at most r vertices. A vertex is a boundary vertex if itbelongs to at least two A-components. We are to show that the number of A-components is�(n/r) and the total number of boundary vertices (counting multiplicity) is O(n/ 2d

√r).

Denote by BV the set of boundary vertices. For each boundary vertex v, let m(v) be thenumber of A-components that contain v, and let b(n, r) = ∑

v∈BV (m(v)− 1). Applying the

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cardinality version of Corollary 2.4 to G results in two A-components, each having size atmost n/2 + cn1−1/d . Hence b(n, r) satisfies the following recursive relation: b(n, r) = 0 forr ≥ n and

b(n, r) ≤ cn1−1/d + 2b(n/2 + cn1−1/d , r). (2)

We shall show that there exist two constants a1, a2 such that

b(n, r) ≤ a1n

2d√

r− a2n1−1/d . (3)

In fact, such constants a1 and a2 can be chosen with the following restriction:

1

21/d − 1

(

c + 2a1c2d√

r

)

≤ a2 ≤ a1r1/(2d) − c. (4)

Before proving inequality (3), we remark that in order that inequality (4) is compatible, itsleft term must be smaller than its right term, equivalently, we must have

21/dc ≤ r−1/(2d)((21/d − 1)r1/d − 2c

)a1. (5)

By the assumption that r >( 2c

21/d−1

)d , constant a1 satisfying inequality (5) exists, and thenone can choose constant a2 satisfying inequality (4) accordingly.

Now, we prove inequality (3) by induction on n. For the basis step, n = r . By the rightinequality of (4), we have

a1r

2d√

r− a2r1−1/d = r1−1/d(a1r1/(2d) − a2) ≥ cr1−1/d > 0.

Combining this with b(r, r) = 0, the basis step is proved. Next, suppose n > r . If n/2 +cn1−1/d < r (such n is possible when r > (2c)d ), then b(n/2 + cn1−1/d , r) = 0. Byinequality (2), b(n, r) ≤ cn1−1/d . Since a2 ≤ a1r1/(2d) − c [(see inequality (4)] and r ≤ n,

we have a2 ≤ a1r1/d

r1/(2d) −c ≤ a1n1/d

r1/(2d) −c. Then, it can be seen that cn1−1/d is no greater than

the righthand side of inequality (3). If n/2 + cn1−1/d ≥ r , applying induction hypothesis toinequality (2), we have

b(n, r) ≤ cn1−1/d + a1n + 2cn1−1/d

2d√

r− 2a2(n/2 + cn1−1/d)1−1/d

≤ a1n

2d√

r−

(

21/da2 − c − 2a1c2d√

r

)

n1−1/d

≤ a1n

2d√

r− a2n1−1/d ,

where the second inequality is obtained by leaving cn1−1/d in the last term, and the thirdinequality is true by the left inequality of (4). The induction step is completed.

Clearly, every final A-component R′ has |R′| ≤ r . Suppose R′ is obtained by splitting R,which is an A-component in the previous step, and the other A-component resulted from thissplitting is R′′. Then |R| > r since it needs to be split, and |R′|+|R′′| = |R|+|AB(R)|, whereAB(R) is the separator of R. By |R′′| ≤ |R|/2 + |AB(R)|, we see that |R′| ≥ |R|/2 > r/2.Hence |R′| = �(r). Combing this observation with the fact that the total number of vertices(counting multiplicity of boundary vertices) is n + b(n, r) = n + O(n/ 2d

√r), we see that the

number of A-components is �(n/r).

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Fig. 1 An illustration for theproof that N (C\A) ⊆ C ∩ A

Suppose there are ni A-components containing i boundary vertices at the end of the firststage. Then the number of boundary vertices (counting multiplicity) after the first stage,denoted by bv1, is

∑v∈BV m(v) = ∑

i≥1 ini . Since m(v) ≥ 2 for any v ∈ BV , we see that2b(n, r) = 2

∑v∈BV (m(v) − 1) ≥ ∑

v∈BV m(v). Hence

bv1 =∑

i≥1

ini ≤ 2b(n, r) = O(n/ 2d√

r). (6)

After the first phase, if there exists an A-component which contains i boundary vertices withi > r1−1/(2d), then continue to split it into smaller parts by applying the weighted versionof Corollary 2.4, where the weight on each boundary vertex is 1/ i and the weight on eachnon-boundary vertex is zero. Under such a weight assignment, the requirement that eachpart contains at most r1−1/(2d) boundary vertices is equivalent to the requirement that eachpart has weight no more than r1−1/(2d)/ i . Since the total weight is 1 and the boundary isthin, it can be seen that O(i/r1−1/(2d)) splits are sufficient, producing O(i/r1−1/(2d)) newparts. Since each split creates O(r/ 2d

√r) = O(r1−1/(2d)) new boundary vertices, splitting

an A-component which contains i boundary vertices at the end of the first stage creates atmost O(r1−1/(2d) · i/r1−1/(2d)) = O(i) new boundary vertices. Hence the number of newboundary vertices created in the second stage is bv2 = O(

∑i ini ), where the sum is over

all i with i > r1−1/(2d). Combining this with (6), the total number of boundary vertices isbv1 + bv2 = O(n/ 2d

√r). Furthermore, the number of new parts resulted from the second

stage is O(∑

i>r1−1/(2d) ini/r1−1/(2d)) = O(n/r) (again by [(6)]. Taking into account thoseA-components which remain intact in the second stage, the number of final parts is O(n/r).

Denote by A the set of boundary vertices during the process. We claim that every A-component C has the following property: N (C\A) ⊆ C ∩ A. This can be proved by inductionon the number of splits. For the first split, this is obviously true. Let j ≥ 2, and consideran A-component C which is split in the j-th iteration into two A-components C1 and C2

(see Fig.1). In this figure, A1, A2, A3 are boundary vertices before the j-th split, and A4

contains those new boundary vertices created in the j-th split. By the induction hypothesis,N (C\A) ⊆ A1 ∪ A2 ∪ A3. By the operation of split, there is no edge between C1\(A2 ∪ A4)

and C2\(A2 ∪ A4). So, N (C1\A) ⊆ A1 ∪ A2 ∪ A4 = C1 ∩ A and N (C2\A) ⊆ A3 ∪ A2 ∪A4 = C2 ∩ A. The claim is proved.

Suppose after the second phase, there are t A-components A1, . . . , At . Let Vi = Ai\A.Then, conditions (i), (i i i), and (v) are satisfied. By the above claim, condition (i i) is satisfied,and N (Vi )∩ A ⊆ Ai\Vi . Because Ai\Vi is the set of boundary vertices contained in Ai whichhas cardinality at most r1−1/(2d) by the termination criterion in the second phase, condition(iv) is satisfied. �

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3 PTAS for MVC in ball graphs

For a positive integer b, a vertex cover U is said to be b-local optimal if for any vertex subsetX ⊆ U with |X | ≤ b, there is no vertex subset Y ⊆ V (G)\U with |Y | < |X | such that(U\X)∪ Y is still a vertex cover of G. The next theorem shows that a b-local optimal vertexcover can be found in polynomial time.

Theorem 3.1 For a fixed positive integer b, finding a b-local optimal vertex cover can bedone in time O(mnb+1), where n and m are the number of vertices and the number of edges,respectively.

Proof Notice that if X can be replaced by a set Y to produce a smaller vertex cover, thenX must be an independent set and Y must contain all vertices in N (X)\U . Hence, one canstart from the trivial vertex cover U = V (G). Then for every independent subset X ⊆ Uwith |X | ≤ b, verify whether Y = N (X)\U has smaller size than X . There are at most∑b

i=1

(ni

)

= O(nb

)subsets to serve as X , and for a fixed set X , set Y can be found in time

O(m). Every time one finds a possible replacement, the size of the vertex cover is reducedby at least one. Hence the above procedure is executed at most n times. It follows that thetotal time complexity is O(mnb+1). �

The next theorem analyzes the approximation ratio of a b-local optimal vertex cover toan optimal one.

Theorem 3.2 Let b be an integer with b >( 2c

21/d−1

)d, where c is the constant in Corollary

2.4. Then, a b-local optimal vertex cover can approximate the optimal solution within a factorof 1 + O(1/

2d√

b).

Proof In the proof, we shall not distinguish a vertex set and the set of balls representingthem, and we shall not distinguish a ball graph and its geometric representation.

Let U∗ be an optimal vertex cover and U be a b-local optimal vertex cover. DenoteQ = U∗ ∩ U, R = U∗\U , and S = U\U∗. Since the complement of a vertex cover is anindependent set, we see that both R and S are independent sets. Let H be the subgraph of Ginduced by vertex set R ∪ S. Then H is also a ball graph. Furthermore, by the independenceof R and S, each point in R

3 belongs to at most one ball from R and at most one ball fromS. Hence H satisfies the condition of Theorem 2.5 with k = 2. Taking r = b, H has apartition as described in Theorem 2.5. For the ease of deduction, we suppose t ≤ c1nH /rand |A| ≤ c2nH / 2d

√r for two constants c1 and c2, where nH = |V (H)| = |S| + |R|.

For i = 1, . . . , t , denote Ri = R ∩ Vi and Si = S ∩ Vi . We claim that (U\Si ) ∪ NH (Si )

is a vertex cover of G. In fact, since Si is an independent set, any edge uv which is coveredby Si has only one end in Si , say u ∈ Si and v �∈ Si . Since U∗ also covers edge uv butu ∈ Si ⊆ U\U∗, we have v ∈ U∗. If v ∈ U∗ ∩ U , then v ∈ U\S ⊆ U\Si . Otherwise,v ∈ U∗\U = R, and thus v ∈ NH (Si ). In any case, any edge covered by Si is also coveredby (U\Si ) ∪ NH (Si ).

Since |Si | ≤ |Vi | ≤ r = b, by b-local optimality of U , we have |NH (Si )| ≥ |Si |. Then,

|S| ≤ |A| +t∑

i=1

|Si |

≤ |A| +t∑

i=1

|NH (Si )|

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670 J Glob Optim (2014) 59:663–671

≤ |A| +t∑

i=1

(|Ri | + |NH (Si ) ∩ A|)

≤ c2|S| + |R|

2d√

r+ |R| + c1

|S| + |R|r

· r1−1/(2d)

= c1 + c22d√

r(|S| + |R|) + |R|.

It follows that

|S| ≤(

1 + O

(c1 + c2

2d√

r

))

|R| =(

1 + O

(1

2d√

b

))

|R|,

and thus

|U | = |Q| + |S| ≤ |Q| + |R| + O

(1

2d√

b

)

|R| ≤(

1 + O

(1

2d√

b

))

|U∗|.

The proof of the theorem is completeed. �

Combining Theorems 3.1 and 3.2, it can be seen that for any real number ε > 0, takingb suitably large, one can find a vertex cover in time O(mnO(1/ε2d )) which approximates theoptimal solution within a factor of 1 + ε.

4 Conclusion

In this paper, we first proved a new separator theorem for unit ball graphs in d-dimensionalspace with the requirement that every point in R

d belongs to at most a constant number ofballs. Applying this new separator theorem, we showed that local search method yields aPTAS for the MVC problem on unit ball graphs in higher dimensional space. However, ourmethod does not work for the weighted case. The reason is as follows: Theorem 2.5 says thatthe number of vertices in the separator is small. However, in the weighted case, the weightof the separator might be large.

As far as we know, local search method is the only method which has succeeded inproviding a PTAS for the MDS problem on disk graphs [11]. Unfortunately, our method failsto provide a PTAS for the MDS problem on unit ball graphs in d-dimensional space withd ≥ 3. In [11], for the MDS problem on a 2-dimensional disk graph, separator theorem isused not on the disk graph itself, but on an auxiliary planar graph. In fact, it is used on ageneralized Delaunay triangulation (called the dual of the weighted Voronoi diagram in [11]),which is a plane graph. This is to guarantee a so-called locality condition which is crucialto the theoretical analysis. Although Delaunay triangulation has a generalization in higherdimensional space, whether there exists a separator theorem for such a graph is unknown.The question whether there exists a PTAS for the MDS problem on unit ball graphs in higherdimensional space is still open.

Acknowledgments The work is supported by NSFC (61222201) and SRFDP (20126501110001) XinjiangTalent Project (2013711011), and by National Science Foundation of USA under grants CNS0831579 andCCF0728851.

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