HardnessandApproximationofNetwork Vulnerabilitymythai/files/vulchap.pdf · or performance guarantees to large-scale networks. To this end, this chapter introduces two new optimization

Hardness and Approximation of NetworkVulnerability

My T. Thai, Thang N. Dinh and Yilin Shen

Contents

1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16322 Hardness Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1634

2.1 NP-Completeness of Critical Edge Disruptor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16342.2 NP-Completeness of Critical Vertex Disruptor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16392.3 NP-Completeness of ˇ-Edge Disruptor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16452.4 Hardness of ˇ-Vertex Disruptor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1648

3 Approximation Algorithm for ˇ-Edge Disruptor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16483.1 Balanced Tree Decomposition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16493.2 Pseudo-approximation Algorithm and Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1650

4 Approximation Algorithm for ˇ-Vertex Disruptor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16544.1 Algorithm Description. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16554.2 Theoretical Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1657

5 A Second Look. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16585.1 O.

plog n/ Pseudo-approximation for ˇ-Edge Disruptor. . . . . . . . . . . . . . . . . . . . . . . . . . . 1659

5.2 O.p

log n/ Pseudo-approximation for ˇ-Vertex Disruptor. . . . . . . . . . . . . . . . . . . . . . . . . . 16626 Literature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16647 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1665Recommended Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1665

AbstractAssessing network vulnerability is a central research topic to understandnetworks structures, thus providing an efficient way to protect them from attacksand other disruptive events. Existing vulnerability assessments mainly focus oninvestigating the inhomogeneous properties of graph elements, node degree, forexample; however, these measures and the corresponding heuristic solutionscannot either provide an accurate evaluation over general network topologies

M.T. Thai (�) • T.N. Dinh • Y. ShenDepartment of Computer Science, Information Science and Engineering, University of Florida,Gainesville, FL, USAe-mail: [email protected]; [email protected]; [email protected]

P.M. Pardalos et al. (eds.), Handbook of Combinatorial Optimization,DOI 10.1007/978-1-4419-7997-1 23, © Springer Science+Business Media New York 2013

1631

mailto:[email protected]



1632 M.T. Thai et al.

or performance guarantees to large-scale networks. To this end, this chapterintroduces two new optimization models to quantify the network vulnerability,which aim to discover the set of key node/edge disruptors, whose removal resultsin the maximum decline of the global pairwise connectivity. Results presentedin this chapter consist of the NP-completeness and inapproximability proofs ofthese problems along with pseudo-approximation algorithms.

1 Introduction

Complex network systems such as the Internet, WWW, MANETs, and the powergrids are often greatly affected by several uncertain factors, including externalnatural or man-made interferences (e.g., severe weather, enemy attacks, maliciousattacks). Additionally, they are extremely vulnerable to attacks, that is, the failuresof a few critical nodes (links) which play a vital role in maintaining the network’sfunctionality can break down its operation [1].

In order to assess the network vulnerability, we need to address several fun-damental questions such as the following: How do we quantitatively measure thevulnerability degree of a network? What are the key nodes (links) that play a vitalrole in maintaining the network’s functionality? What is the minimum number ofnodes (links) do we need to take down in order to break down an adversarialnetwork?

Despite numerous methods have been investigated on the network vulnerability,they neither reveal the global damage done on the network when multiple nodes(links) fail simultaneously nor correctly identify the key nodes who play a vitalrole in maintaining the network’s functionality [2–8]. (More details of related workare discussed later in Sect. 6.) In this chapter, two optimization models along withtheir four related problems to assess the network vulnerability are discussed asfollows:

Model 1: Critical Vertex (Edge) Disruptor [CVD (CED)] : Given a graphG D .V; E/ representing a network (where V is a set of nodes and E is a set oflinks in the network) and a positive integer k < jV j (k < jEj), determine a set ofvertices (edges) S � V (S � E) with jS j D k so as to minimize the total pairwiseconnectivity in the induced graph GŒV n S� (GŒE n S�), obtained by removing S

from G, where the total pairwise connectivity in G is defined as the total numberof connected pairs of vertices in G. A pair of nodes is connected if there is a pathbetween them in graph G.

Note that E is a set of links where links can be physical links or logical linksbetween two endpoints. For example, if there is a communication between nodesu and v in the network, there will be an edge .u; v/ in the representing graph G. IfG is used to represent the functional dependency between each node in a network,then there is an edge .u; v/ in G iff there is a functional relationship between u andv in the network. Therefore, the connectivity discussed in this chapter is not simplya physical connectivity in a given network.

Hardness and Approximation of Network Vulnerability 1633

Clearly, this model aims to discover vertices (edges) whose removal maximizesthe network fragmented and maximumly destroys the network. The ratio betweenthe size of S and the number of pairwise disconnectivity in GŒV nS� determines thedegree of vulnerability of G, that is, the smaller this ratio is, the more vulnerablethe network is. Thus, this model represents a new paradigm for quantitativelycharacterizing the vulnerability of an underlying network.

Model 2. ˇ�Vertex (Edge) Disruptor [ˇ�VD (ˇ�ED)] : Given a graph G D.V; E/ and 0 � ˇ < 1, find a subset of vertices (edges) S � V (S � E) with theminimum cardinality so that the total pairwise connectivity of GŒV n S� (GŒE n S�)is at most ˇ

�n

2

�.

This model also reveals the vulnerability degree of the networks. That is, themore key nodes (edges) there are (i.e., the more nodes (edges) whose deletion isrequired to meet the requirement of pairwise connectivity), the less vulnerabilitythe network is. Conversely, the fewer the key nodes (edges), the more vulnerabilitythe network will be to the attacks.

Model 1 can be used to maximally disrupt the network with a given cost, whereasModel 2 can be used in the study of breaking down an adversarial social network toa certain degree with the minimum cost. For example, this concept can be appliedto destroy, that is, arrest, a small number of key individuals in an adversarial socialnetwork (e.g., terrorist networks) in order to maximally disrupt the networks, abilityto deploy a coordinated attack. The set of nodes S obtained from the two modelspresent critical nodes in networks, such as a key person, a commander of militarynetworks, or a key computer components in computer networks.

Moreover, the two models present a deeper meaning and greater potentials onthe study of multiple disruption levels (different values of k and ˇ). Several recentstudies in the context of wireless networks have aimed to discover the nodes/edgeswhose removal disconnects the network, regardless of how disconnected it is [9–11].However, it is not reasonable to limit the possible disruption to only disconnectingthe graph, ignoring how fragmented it is since the giant connected component stillexists and the network may function well. For example, a scale-free network cantolerate high random failure rates [1], since the destructions to boundary verticesmay not significantly decline the network connectivity even though the whole graphbecomes disconnected. In addition, different disruption levels may require differentsets of disruptors on which these two models can differentiate, whereas existingmethods cannot. For example, the node centrality method always returns a set ofnodes with nonincreasing degrees regardless of the disruption level.

The study of these models also finds applications in several other fields, suchas network immunization, offering control of epidemic outbreaks, and containmentof viruses. For example, instead of an expensive mass vaccination, we can vac-cinate a small number of individuals (corresponding to key nodes of a network).The immunized nodes cannot propagate the virus, thus minimizing the overalltransmissibility of the virus.

This chapter presents the hardness complexity of the four problems, CED, CVD,ˇ�ED, and ˇ�VD, along with their pseudo-approximation solutions. The resultsare discussed in the following sections.


2 Hardness Results

2.1 NP-Completeness of Critical Edge Disruptor

In this section, the NP-completeness of CED in general graphs and in unit diskgraphs (UDGs) is shown. Furthermore, this section presents an NP-completeness ofCED in power-law graphs (PLGs), strengthening its intractability. UDGs are oftenused in the networking research field to model the homogeneous wireless networkswhere all nodes have the same transmission range where PLGs represent a classof complex networks whose nodes follow the power-law distribution such as theInternet and social and biological networks [12]. That is, the fraction of nodes inthe network having degree k is proportional to k�ˇ , where ˇ is a parameter whosevalue is typically in the range 2 < ˇ < 3.

The hardness proof of CED in general graphs uses a reduction from the p-multiway cut problem, which is defined as follows:

Definition 1 (p-Multiway Cut) Given an undirected graph G D .V; E/ and asubset of terminals T � V with jT j D p, find a minimum set of edges of G

whose removal disconnects all vertices in T . That is, each vertex in T belongs toeach connected component.

Note that p-multiway cut is NP-complete for any p � 3 according to E. Dahlhauset al. [13].

Fact 1 Given a graph G D .V; E/ with p-connected components, the total pairwiseconnectivity of G is lower bounded by p

�bn=pc2

�and upper bounded by

�n�pC1

2

�

where n D jV j.

Theorem 1 The critical edge disruptor (CED) problem is NP-complete.

Proof Consider the decision version of CED that asks whether G D .V; E/

contains a set of edges S � E of size k such that the total pairwise connectivityof GŒE n S� is at most c for a given positive integer c.

It is easy to see that CED 2 NP since a nondeterministic algorithm needs onlyguess a subset S of edges and checks in polynomial time whether S has theappropriate size k and the total pairwise connectivity of GŒE n S� is at most c

by using the depth-first search.The reduction from the 3-multiway cut (3-MC) problem to CED is used to prove

its NP-hardness. Let an undirected graph G D .V; E/ where jV j D n, a set T � V

of three terminals and a positive integer k � jEj be any instance of 3-MC. Thegraph G0 D .V 0; E 0/ is constructed as follows: For each terminal v 2 T , l D n2,more vertices are added onto v to construct a clique of size n2 C 1, as illustratedin Fig. 1. We now show that there is a 3-MC cut of size k in G iff G0 has a CED


V4

V3V2

Clique Kl

Edge Set El

a b

V6

...V5

V1

V4

V3V2 V6

V5

V1

Fig. 1 The reduction from 3-multiway cut instance to a CED instance. (a) Graph G with T Dfv2; v5; v6g. (b) Reduction from 3-multiway cut to CED

S of size k such that the pairwise connectivity of GŒE n S� is at most c where

2�

n2C12

�C �n2Cn�2

2

�< c <

�2n2C2

2

�.

First, suppose S � E is a 3-MC for G with jS j � k. By our construction andaccording to Fact1, the total pairwise connectivity of G0ŒE 0nS� is upper bounded by

2

n2 C 1

2

!

C

n2 C n � 2

2

!

(1)

which is less than c. Thus, S is also a CED of G0.Conversely, suppose that S � E 0 with jS j D k � jEj is a CED of G0, that is, the

total pairwise connectivity of G0ŒE 0nS� � c. We show that S indeed is also a 3-MCof G. Note that S \ .E 0 n E/ D ;, that is, for any e 2 S , e cannot be an edge inthe constructed cliques since removing k edges cannot disconnect the cliques; thus,

the total pairwise connectivity is�

nC3n2

2

�, which is larger than c. Therefore, S � E .

Now, we further state that set S also disconnects all the three terminals. Assumethat it is not, then the two cliques of these two terminals are connected; thus, the

total pairwise connectivity is at least�

2n2C22

�> c, contradicting to the fact the total

pairwise connectivity of G0ŒE 0 n S� � c. �

We now show that CED still remains NP-complete in UDGs by reducing fromthe planar independent set (PIS) with maximum degree 3. Given a planar graphG D .V; E/, the PIS problem asks us to find a subset S � V with the maximumsize such that no two vertices in S are adjacent.

Lemma 1 (L.G. Valiant [14]) A planar graph G with maximum degree 4 canbe embedded in the plane using O.jV j/ area in such a way that its vertices areat integer coordinates and its edges are drawn so that they are made up of linesegments of the form x D i or y D j , for integers i and j .


Theorem 2 The critical edge disruptor (CED) problem is NP-complete in UDGs.

Proof Consider the decision version of CED that asks whether a UDG G D .V; E/

contains a set of edges S � E of size k0 such that the pairwise connectivity in theinduced graph GŒE n S� is at most c for a given positive integer c. Clearly, CED 2NP in UDGs.

To prove that CED on UDGs is NP-hard, we reduce the planar independent set(PIS) with maximum degree 3 to it. Let a planar graph G D .V; E/ where jV j D n

with maximum degree 3 and a positive integer k � n be an arbitrary instance ofPIS. We must construct in polynomial time a UDG G0 D .V 0; E 0/ and positiveintegers k0 and c such that G has an independent set of size k iff G0 D .V 0; E 0/ hasa CED of size k0 where the pairwise connectivity of G0 after removing this CED isat most c.

The construction of G0 is as follows: Given a planar graph G, we first embed itinto the plane according to Lemma 1. For each node vi 2 V , convert it to a triangleTvi D fvi1; vi2; vi3g (dark gray triangles in Fig.2b). Each edge in Tvi is set to be a unit

length r Dp

33

. Since each node vi in G has at most three neighbors, say x, y, andz, we attach these neighbors to vi1, vi2, and vi3. That is, three edges .vi ; x/, .vi ; y/,and .vi ; z/ in G become .vi1; x/, .vi2; y/, and .vi3; z/. If the degree is less than 3,we only need to arbitrarily choose one or two from them to attach. Afterwards,for each embedded edge euv D .u; v/, we divide it into jeuvj “pieces” according toits length and replace each piece by a concatenation of two triangles as shown inFig. 2b. Thus, G0 is composed of 2

Pu;v2V jeuvj K3 cliques where jeuvj is the length

of edges between node u and v after embedded onto the plane. Note that G0 is a unit

disk graph with radius r Dp

33

. Finally, we set k0 D 2P

u;v2V jeuvj C 3.n � k/ andc D 3.

Pu;v2V jeuvj C k/.

First, suppose S � V is an independent set of G with jS j D k. For each piecein all edges of G, we remove two edges from two concatenation triangles such thateach piece has one triangle left. After doing this, all triangles Tvi correspondingto the node vi 2 S will be disconnected from other nodes in G0. For each nodevi 62 S , we remove all edges in the corresponding triangles Tvi in G0. Consider aset S 0 consisting of all the removed edges (2 edges per piece and 3 edges for eachTvi where vi 62 S ). Then clearly jS 0j D k0j and the pairwise connectivity in G0 afterremoving S 0 is c. Therefore, S 0 is a CED of G0.

Conversely, suppose that S 0 � V 0 with jS 0j D k0 is a CED of G0, that is, thepairwise connectivity of G0ŒE 0 n S 0� is at most c. First, it is easy to see that thepairwise connectivity will increase by

Pu;v2V jeuvj=2 if two triangles corresponding

to every other pieces are connected since there is one more connected node pair.Thus, either one of the two triangles for each piece in G0 will be disconnected. Itimplies that two edges will be removed from each piece. Apart from this, in orderto reduce the pairwise connectivity to c, we remove 3.n � k/ more edges to avoidthe connectivity between the nodes in different triangles. Note that for each triangleTvi , either all three edges are removed or none since the removal of edges in a


a

b

Fig. 2 The reduction from a planar independent set instance to a CED instance. (a) An instanceG. (b) A reduced graph G0

connected component does not help to reduce the pairwise connectivity if it cannotdisconnect the component. Some edges belonging to K2 are exchangeable with thenodes in other triangles within polynomial time. This implies the nodes vi (in V )corresponding to Tvi which has all their edges remained is an independent set of G.Since the number of Tvi is k, the IS also has size k. �

Now our attention is turned to the study of CED’s NP-completeness in power-law graphs. The P.˛; ˇ/ model mentioned in [15] is used to describe a power-lawgraph. In this model, the number of vertices of degree d is b e˛

dˇ c, where e˛ is thenormalization factor. To show CED’s hardness, we reduce from the clique separator(CS) problem, which is defined as follows:

Definition 2 (Clique Separator (CS)) Given a graph G D .V; E/, find a set ofedges S such that the induced graph GŒE n S� has the connected components to becliques and each clique has size at least 3. A subgraph of G is called a clique iffall its vertices are pairwise adjacent. The CS problem asks us to find such a cliqueseparator with the minimum size.

Before proceeding, let us first present the following fundamental lemma, whichstates that a PLG can be constructed from any given simple graph by choosing anappropriate value ˛:

Lemma 2 (Ferrante et al. [16]) Let G1 D .V1; E1/ be a simple graph with n nodesand ˇ � 1. For ˛ � maxf4ˇ; ˇ log n C log.n C 1/g, we can construct a power-law


graph G D G1 [ G2 with exponential factor ˇ and the number of nodes e˛�.ˇ/ byconstructing a bipartite G2 as a maximal component in G. Here �.�/ D P1

iD11i� is

the Riemann zeta function.

Since the hardness proof of CED is based on the reduction from CS, the hardnessof CS must be investigated first as follows:

Lemma 3 The CS problem is NP-hard.

Proof To prove that CS is NP-hard, we reduce the planar independent set (PIS)problem with maximum degree 3 to it. By using the same reduction in Theorem 2,it can be shown that G has an independent set of size k iff G0 has a CS of sizek0 D 2

Pu;v2V jeuvj C 3.n � k/.

First, suppose S � V is an independent set of G with jS j D k. To disconnectthe triangles from other nodes, two edges from two concatenation triangles for eachpiece are removed. For each Tvi where vi 2 S , since all three nodes will not overlapwith other cliques, the triangle Tvi will be selected as a connected component ofsize 3. For the other Tvi where vi 62 S , Tvi cannot be selected as a clique sinceat least one of its endpoints overlaps with other triangle. Thus, the removed edgeshave size k0. And after removing these edges, the obtained connected componentsare cliques of size 3. That said, the removed edges set is a CS of G0.

Conversely, suppose that S 0 � V 0 is a CS of G0 with S 0 D k0. For each piece, atleast two edges have to be removed to disconnect the triangle from other nodes. Foreach Tvi , its edges belongs to S 0 if at least one has been used by the other triangles.Since k0 D 2

Pu;v2V jeuvj C 3.n � k/, it is easily to modify S 0 to be an independent

set of G. �

Theorem 3 The critical edge disruptor (CED) problem is NP-complete in power-law graphs.

Proof Again, consider the decision version of CED that asks whether a power-lawgraph G D .V; E/ contains a set of links S � V of size k0 such that the pairwiseconnectivity in GŒE n S� is at most c for a given positive integer c.

To prove that CED in PLGs is NP-hard, we reduce the clique separator (CS) toit. Given G, a power-law graph G0 D G [ Gb where the bipartite graph Gb D.Ub; VbI Eb/ is a maximal component in G0 is constructed according to Lemma 2.We show that there is a CS of size k in G iff G0 has a CED S 0 of size k0 such that thepairwise connectivity of G0ŒE 0nS 0� is at most c, where k0 D kCjEb j�jMbj and c DjEj � k C jMbj. Here Mb denotes the set of edges in the maximum matching of Gb .

First, suppose S � V is a clique separator of G with jS j D k. The total pairwiseconnectivity of GŒE n S� is equal to jEj � k since all components in this graph arecliques. Since the maximum matching on Gb can be found in polynomial time usingHopcroft-Karp algorithm [17], the pairwise connectivity of G0 is c after removingadditional Eb n Mb edges. Therefore, the set S 0 D S [ .Eb n Mb/ with sizek0 D k C jEbj � jMbj is a CED of G0.


Conversely, suppose that S 0 � V 0 is a CED of G0 with size k0. Let S 0 D A [ Sb,where A and Sb are the removed edges (in S 0) in G and Gb , respectively. We showthat jSbj = jEbj � jMbj. If jSbj < jEbj � jMbj, the pairwise connectivity of Gb isincreased at least two when adding one more edge into the maximum matching. Onthe other hand, the removal of l edges on G can reduce the pairwise connectivityat most l after removing the CS k. Therefore, we have jSb � jEbj � jMbj in orderto maximally reduce the number of pairwise connectivity. If jSbj > jEbj � jMbj,the pairwise connectivity of Gb is reduced by one when removing one moreedge from the maximum matching. Meanwhile, an edge added into the residualgraph of G will increase the pairwise connectivity at least one if it connects totwo independent nodes and at least 3 if it has one endpoint belonging to somecomponent in the residual graph of G. Therefore, maximally reducing the numberof pairwise connectivity requires jSbj � jEbj � jMbj. Combining both conditions,we have Sb D jEbj � jMbj. Thus, jAj D k, which should be a CS of G in order forGŒE n A� having the pairwise connectivity of size jEj � k. �

2.2 NP-Completeness of Critical Vertex Disruptor

In this section, the NP-completeness of CVD on both UDGs and PLGs are shown.

Definition 3 (Minimum Vertex Cover) Given a graph G D .V; E/, find a subsetC � V with the minimum size such that for each edge in E at least one end vertexbelongs to C .

Lemma 4 (Garey et al. [18]) Minimum vertex cover remains NP-complete onplanar graphs with maximum degree 3.

Lemma 5 For any planar graph G D .V; E/ with maximum degree 3, it can beembedded into a UDG with the radius 1=2 such that there exists an independent-space �.v/ for each vertex v 2 V satisfying �.v/ ¤ ; and any new vertices in �.v/

will be incident to v and independent from all other vertices.

Proof After mapping G D .V; E/ into an integer coordinates based on Lemma 1,a Gu D .Vu; Eu/ UDG can be constructed as follows: Each edge e D .vi ; vj / 2 E

is replaced by a path .vi ; w1; w2; : : : ; w2pe ; vj / where pe is the length of e and theradius r is set to 1=2. Now, we prove that such independent-space �.v/ exists foreach v 2 V . Since G has the maximum degree 3, that is, for each vertex v 2 V , thereare at most 3 neighbors (in the form of wi ) in Vu. Thus, there exists a nonempty space�.v/ for each v such that the maximum distance between the vertices in this areato vertex v is at most 1/2 and the minimum distance between such vertices to allother vertices is larger than 1/2. As shown in Fig. 3, the shadow area correspondsto �.v1/. �


W2W''1 W2p12W1V1 V2

W'1

Fig. 3 An independent-space �.v1/

Theorem 4 The critical vertex disruptor (CVD) problem is NP-complete in UDGs.

Proof Consider the decision version of CVD that asks whether a UDG G D .V; E/

contains a set of vertices S � V of size k0 such that the total pairwise connectivityof GŒV n S� is at most c for a given positive integer c.

Again, it is easy to see that CVD 2 NP .Now, to prove that CVD is in NP-hard, we reduce the planar vertex cover (PVC)

with maximum degree 3 to it. Let a planar graph with maximum degree 3 G D.V; E/ where jV j D n and a positive integer k � n be an arbitrary instance of PVC.We must construct in polynomial time a UDG G0 D .V 0; E 0/ and positive integersk0 and c such that G has a vertex cover of size at most k iff G0 D .V 0; E 0/ has aCVD of size k0 and the pairwise connectivity of G0 after removing this CVD is atmost c.

Such a construction is described as follows: Given G, construct a Gu according toLemma 5. On Gu, for each independent-space �.vi /, a new gray vertex ui is addedsuch that there is an edge only between vi and ui to obtain G0 as shown in Fig. 4.Finally, set k0 D k CP

e2E.G/ pe and c D n � k.First, suppose S � V is a vertex cover of G with jS j � k. It is easy to verify that

G has a vertex cover S with size at most k iff Gu has a vertex cover Su with sizeat most k C P

e2E.G/ pe. That is, Su consists of S and a half of nodes wi on eachpath .vi ; w1; : : : ; w2p.vi ;vj /

; vj /. We show that Su is also a CVD of G0. RemovingSu from G0 returns all disjoint edges .vi ; ui / left where vi … S . Thus, the pairwiseconnectivity is n � k, which is equal to c.

Conversely, suppose that S 0 � V 0 with jS 0j D k0 is a CVD of G0, that is, the totalpairwise connectivity of G0ŒV 0 nS 0� is at most c. If ui 2 S 0, that is, S 0 contain a graynode, replacing ui with any vi or wi (black or white nodes) will further decrease thepairwise connectivity. Thus, it can be assumed that S 0 does not contain any graynode. Since jS 0j D k CP

e2E.G/ pe, it is easily to modify the set S 0 to be a vertexcover of Gu. In this case, the total pairwise connectivity is at most c D n � k. Thus,S 0 \ V is a VC of G. �


V1

V3V2

W112 W1

23

W223

W323W4

23

W213W1

13 W313 W4

13

W212

V1 V3

V2 W523W6

23

W112

W212

V1

U1

U2

U3

V2 W423W5

23W623

W213W1

13 W313 W4

13 V3

W123

W223

W323

a

c

b

Fig. 4 The reduction for a planar vertex cover instance to a CVD instance. (a) A planar graph G

(b) Embed G onto a UDG Gu. (c) Create G0 by adding gray modes at independent-spaces

Theorem 5 The critical vertex disruptor (CVD) problem is NP-complete in power-law graphs.

Proof Again, consider the decision version of CVD that asks whether a power-lawgraph G D .V; E/ contains a set of nodes S � V of size k0 such that the pairwiseconnectivity in GŒV n S� is at most c for a given positive integer c.

Now, to prove that CVD in power-law graphs is NP-hard, we reduce the vertexcover (VC) to it. Let a graph G D .V; E/ where jV j D n and a positive integerk � n be any instance of VC. A power-law graph G0 D .V 0; E 0/ is constructed


a

bFig. 5 The reduction from avertex cover instance to CVDinstance in power-law graphs.For simplicity, all edges in G

are omitted. (a) An instanceG. (b) Reduced graphG0 D G1 [ G2

as follows. First, for each node vi 2 V , one additional node ui is added onto it,denoted by G1 where V1 D V [ U and U D fuig. Then according to Lemma 2,a power-law graph G0 D .V 0; E 0/ can be constructed by embedding G1 and abipartite graph G2 D .V 1

2 ; V 22 I E2/ where V 1

2 and V 22 are two sets of disjoint nodes

in G2 and ˛ � maxf4ˇ; ˇ log.2n/ C log.2n C 1/g with some specific ˇ. Notethat V 1

2 and V 22 are marked gray and white separately as shown in Fig. 5. We show

that there is a VC of size k in G iff G0 has a CVD S 0 of size k0 such that thepairwise connectivity of G0ŒV 0 n S 0� is at most c, where k0 D k C minfjV 1

2 j; jV 22 jg

and c D n � k.First, suppose S � V is a vertex cover of G with jS j D k. Therefore, G has a

vertex cover S of size k iff G [G2 has a vertex cover S 0 of size k CminfjV 12 j; jV 2

2 jgsince VC is polynomially solvable in any bipartite graphs according to Konig’stheorem [19]. Then after removing S 0 from G0, only all disjoint links .vi ; ui / areleft where vi 62 S . Therefore, the pairwise connectivity in G0 is n � k, which isequal to c. That said, S 0 is a CVD of G0.

Conversely, suppose that S 0 � V 0 with jS 0j D k0 is a CVD of G0, that is, thetotal pairwise connectivity of G0ŒV 0 n S 0� is at most c. First, if ui 2 S 0, this ui canbe replaced with any vi without increasing the pairwise connectivity. Since jS 0j Dk0 D kCminfjV 1

2 j; jV 22 jg, it is easily to modify the S 0 to be a vertex cover of G[G2,

where the total pairwise connectivity on G0 is at most c D n � k. Thus, S 0 \ V is aVC of G. �

2.2.1 InapproximabilityAs the CVD is NP-complete, one will question how tightly one can approximatethe solution, leading to the theory of inapproximability. The inapproximabilityfactor gives us the lower bound of near-optimal solution with theoretical per-formance guarantee. That said, no one can design an approximation algorithm


with a better ratio than the inapproximability factor. In this section, the inap-proximability of CVD is further investigated in general graphs by showing thegap-preserving reduction from maximum clique problem, which is defined asfollows:

Definition 4 (Maximum Clique) Given a graph G D .V; E/, find a clique ofmaximum size.

Theorem 6 The CVD problem is NP-hard to be approximated within ‚�

kn�

�for

any � < 1 � logn 2 in general graphs unless P D NP .

Proof In the proof, a gap-preserving reduction from maximum clique problem toCVD is shown. According to Hastad [20], maximum clique is proven NP-hard tobe approximated within n1�� . Let a graph G D .V; E/ and a positive integer k �jV j D n be an instance of maximum clique problem (MC). Now we constructthe graph G0 D .V 0; E 0/ as follows: Firstly, a complement graph G D .V; E/

where E D f.vi ; vj /j.vi ; vj / 62 Eg is constructed. Next, for each vertex v in G,l � n1�� 1, more nodes are added to construct a clique of size l C 1 and obtainG0, as shown in Fig. 6. Let � be a feasible solution of MC on G and ' be a feasiblesolution of CVD on G0 where j'j D n � k. Define c D .n � k/

�l2

� C k�

lC12

�; the

completeness and soundness are shown respectively as follows:Completeness: In this step, we need to show that if OP T .�/ D k then

OP T .'/ D c

Let OP T .�/ be the maximum clique in graph G, then OP T .�/ is the max-imum independent set in G. Since the minimum vertex cover and the maximumindependent set are complementary on the same graph, removing the set of nodesV n OP T .�/ in G will result into a set of only isolated nodes. Thus, the pairwiseconnectivity in G0 after removing such n � k critical nodes is .n � k/

�l2

�C k�

lC12

�,

which is equal to c. It implies that OP T .'/ D c.Soundness: In this step, we need to show that if OP T .�/ < 1

n1�� k, then

OP T .'/ > ‚�

kn�

�c.

a b

Fig. 6 The gap-preservingreduction from a maximumclique instance to a CVDinstance. (a) A graph instance(b) Reduced graph G0

from MC


1-

1-

OneConnectedComponent

Fig. 7 The lower bound instance of CVD in G0 given that j'j D n� k and OP T .�/ < 1n1�� k

First of all, since OP T .�/ < 1n1�� k, we cannot remove as many as n � 1

n1�� k

critical nodes for any � < 1. Therefore, it is impossible to reduce the pairwiseconnectivity on G0 to

�n � 1

n1�� k� �

l2

� C 1n1�� k

�lC1

2

�by fragmenting G0 into n

components.For any � < 1 � logn 2, it is easy to verify that 1 � 1

n1�� > 1n1�� . In this case,

the lower bound of the pairwise connectivity in G0 after removing n� 1n1�� k critical

nodes is shown to be ‚�

kn�

�c. Note that the pairwise connectivity is minimized

when the number of connected components are maximized and each connectedcomponent has the same size as shown in Fig. 7. That is, since 1 � 1

n1�� > 1n1�� ,

the lower bound is achieved when the same number of cliques CG1 connects to oneCG2, where CG1 and CG2 denote the group of cliques of size l and cliques of sizel C 1, respectively, after removing n � 1

n1�� k nodes. Therefore, the following lowerbound is obtained:

0

@

�1� 1

n1��

1

n1��

C 1

�.l C 1/

2

1

A 1

n1��k

�

.n1�� 1/.l C 1/

2

!1

n1��k


D�

.n1��1/.lC1/2

�1

n1�� k

.n � k/�

l2

�C k�

lC12

�c

��

ln1��

2

�1

n1�� k

n�

l2

�C lkc

>

�ln1��

2

�1

n1�� k

n�

lC12

� c

D ‚

�k

n�

�c

Step 2 holds since bzc � z � 1 for any real number z; steps 4 and 5 hold sincel � n1�� 1 and k � n. �

2.3 NP-Completeness of ˇ-Edge Disruptor

In this section, the NP-completeness of ˇ-ED is shown by reducing from thebalanced cut problem. This section begins with several needed definitions asfollows:

Definition 5 A cut hS; V nSi corresponding to a subset S 2 V in G is the set ofedges with exactly one endpoint in S . Usually V nS is denoted by NS . The cost of acut is the sum of its edges’ costs (or simply its cardinality in the case all edges haveunit costs) and denoted by c.S; NS/.

Finding a min cut in the graph is polynomial solvable [21]. However, if one asksfor a somewhat “balanced” cut of minimum size, the problem becomes intractable.A balanced cut is defined as following:

Definition 6 (Balanced cut) Let f be a function from the positive integers to thepositive reals. An f -balanced cut of a graph G D .V; E/ asks us to find a cut

˝S; NS ˛

with the minimum size such that minfjS j; j NS jg � f .jV j/.

Abusing notations, for 0 < c � 12, c-balanced cut is used to find the cut

˝S; NS ˛

with the minimum size such that minfjS j; j NS jg � cjV j. The following results onbalanced cut shown in [22] will be used in the proofs:

Corollary 1 (Monotony) Let g be a function with

0 � g.n/ � g.n � 1/ � 1


Then f .n/ � g.n/ for all n, implies f -balanced cut is polynomially reducible tog-balanced cut.

Corollary 2 (Upper bound) ˛n�-balanced cut is NP-complete for ˛; � > 0.

It follows from Corollaries 1 and 2 that for every f D O.˛n�/; f -balanced cut isNP-complete. It is now ready to prove the NP-completeness of ˇ-edge disruptor:

Theorem 7 ˇ-edge disruptor is NP-complete.

Proof Theorem 7 is proven for a special case when ˇ D 12. For other values of ˇ,

the proof can go through with a slight modification of the reduction. In the proof, n

is considered to be a large enough number, say n > 103.Consider the decision version of the problem that asks whether an undirected

graph G D .V; E/ contains a 12-edge disruptor of a specified size:

1

2-ED D fhG; Ki j G has a

1

2-edge disruptor of size Kg

To show that 12-ED is in NP-complete, we must show that it is in NP and that all NP-

problems are polynomial time reducible to it. The first part is easy; given a candidatesubset of edges, it is easily check in polynomial time if it is a 1

2-edge disruptor of

size K . To prove the second part, we show that f -balanced cut is polynomial time

reducible to 12-ED where f D b n�

q2b n2

3 cCn

2c.

Let G D .V; E/ be a graph in which one seeks to find an f -balanced cut of sizek. Now construct the graph H.VH ; EH / as follows: VH D V 0 [ C1 [ C2 whereV 0 D f vi j i 2 V g and C1; C2 are two cliques of size b n2

3c. The total number of

nodes in H is, hence, N D 2b n2

3c C n. Beside edges inside two cliques, an edge

.vi ; vj / is added for each edge .i; j / 2 E . Next each vertex vi connects to b n2

4c C 1

vertices in C1 and b n2

4c C 1 vertices in C2 so that the degree difference of nodes in

the cliques are at most one. The construction of H.VH ; EH / is illustrated in Fig. 8.Now we need to show that there is a f -balanced cut of size k in G iff H has an12-edge disruptor of size K D n

�b n2

4c C 1

C k where 0 � k � b n2

4c. Note that the

cost of any cut˝S; NS ˛ in G is at most jS jj NS j � b .jS jCj NS j/2

4c D b n2

4c.

On the one hand, an f -balanced cut˝S; NS ˛ of size k in G induces a cut

˝C1 [ S; C2 [ NS ˛ with size exactly n

�b n2

4c C 1

C k. If the cut is selected as the

disruptor, the pairwise connectivity will be at most 12

�N2

�.

On the other hand, assume that H has a 12-edge disruptor of size K D

n�b n2

4c C 1

C k. Because separating n nodes in a clique from the other nodes


Fig. 8 Construction ofH D .VH ; EH / fromG D .V; E/

in that clique requires cutting at least n.b n2

3c � n/ > n

�b n2

4c C 1

C k edges, there

are at most n nodes separated from both the cliques. A direct consequence is thatat least .b n2

3c � n/ nodes remain connected in each clique after removing edges in

the disruptor. Denote the sets of those nodes by C 01 and C 02, respectively. C 01 and C 02cannot be connected otherwise the pairwise connectivity will exceed 1

2

�N2

�. Denote

by X1 and X2 the set of nodes in V 0 that are connected to C 01 and C 02 respectively.Since C 01 and C 02 are disconnected we must have X1 \ X2 D �.

The disruptor can be modified without increasing its size and the pairwiseconnectivity such that no nodes in the cliques are split. For each u 2 C1nC 01, weremove from the disruptor all edges connecting u to C 01 and add to the disruptorall edges connecting u to X2. This will move u back to the connected componentthat contains C 01 while reducing the size of the disruptor at least .b n2

3c � n/ � n.

At the same time, an arbitrary node v 2 X1 is selected and added to the disruptorall remaining v’s adjacent edges. This increases the size of the disruptor at most.b n2

4c C 1/ C n while making v isolated. By doing so, the size of the disruptor

decreases by .b n2

3c � n/ � n � ..b n2

4c C 1/ C n/ > 0. In addition, the pairwise

connectivity will not increase when connecting u to C 01 and disconnecting v fromC 01. If no nodes are left in X1, a node v 2 X2 can be selected (as in that casejC 02 [X2j > jC 01 [X1j) to make sure the pairwise connectivity will not be increased.The same process is repeated for every node in C2nC 02 and at the end of the process,C 01 D C1 and C 02 D C2.

We will prove that X1[X2 D V 0, that is, hX1; X2i induces a cut in V.G/. Assumenot, the cost to separate C1 [X1 from C2 [X2 will be at least (b n2

4cC1/.jV 0�X1jC

jV 0�X2j/ D .b n2

4cC1/.2n�jX1j�jX2j/ � .b n2

4cC1/.nC1/ > n

�b n2

4c C 1

Ck

that is a contradiction.Since X1 [ X2 D V 0, the disruptor induces a cut in G. To have the pairwise

connectivity at most 12

�N2

�, both .C1 [ X1/ and .C2 [ X2/ must have size


at least N�pN2

. If follows that X1 and X2 must have size at least f .n/ Db n�

q2b n2

3 cCn

2c. The cost of the cut induced by hX1; X2i in G will be n

�b n2

4c C 1

Ck � n.b n2

4c C 1/ D k. �

2.4 Hardness of ˇ-Vertex Disruptor

Theorem 8 ˇ-vertex disruptor is NP-complete.

Proof The details of the proof will be ignored. Instead, only the sketch is presentedby showing that the vertex cover is polynomial time reducible to ˇ-vertex disruptor.Let G D .V; E/ be a graph in which one seeks to find a vertex cover of size k. Notethat if we remove nodes in a vertex cover from the graph, the pairwise connectivityin the graph will be zero. Hence, by setting ˇ D 0, G has a vertex cover of size k

iff G has an ˇ-vertex disruptor of size k.One can also avoid using ˇ D 0 by replacing each vertex in G by a clique of

large enough sizes, say O.n/; that ensures no vertices in cliques will be selected inthe ˇ-vertex disruptor. �

Given the NP-hardness of the ˇ-vertex disruptor, the best possible result is apolynomial-time approximation scheme (PTAS) that given a parameter � > 0,produces a .1 C �/-approximation solution in polynomial time. Unfortunately, thisis impossible unless P = NP.

Theorem 9 Unless P = NP, ˇ-vertex disruptor has no polynomial-time approxima-tion scheme.

Proof In the case ˇ D 0, the problem is equivalent to finding the minimum vertexcover in the graph. In [23], Dinur and Safra showed that approximating vertex coverwithin constant factor less than 1.36 is NP-hard. Hence, a PTAS scheme for ˇ-vertexdisruptor does not exist unless P D NP. �

3 Approximation Algorithm for ˇ-Edge Disruptor

This section presents an O.log32 n/ pseudo-approximation algorithm for the ˇ-edge

disruptor problem in directed graphs. (Note that in directed graphs, a pair .u; v/ issaid to be connected if there exists a directed path from u to v and vice versa, from vto u.) Formally, the algorithm finds in a directed graph G a ˇ0-edge disruptor whosecost is at most O.log

32 n/OPTˇ�ED , where ˇ0

4< ˇ < ˇ0 and OPTˇ�ED is the cost

of an optimal ˇ-edge disruptor.As shown in Algorithm 1 , the approximation solution consists of two main

steps: (1) constructing a decomposition tree of G by recursively partitioning the


graph into two halves with a directed c-balanced cut and (2) solving the problemon the obtained tree using a dynamic programming algorithm and transform thissolution to the original graph. These two main steps are explained in the next twosubsections.

3.1 Balanced Tree Decomposition

Let us first introduce some definitions before describing the balanced tree decom-position algorithm.

Definition 7 Given a directed graph G D .V; E/ and a subset of vertices S � V .The set of edges outgoing from S and the set of edges incoming to S are denoted byıC.S/ and ı�.S/, respectively. A cut

˝S; NS ˛ in G is defined as ıC.S/. A c-balanced

cut is a cut˝S; NS ˛ s.t. minfjS j; j NS jg � cjV j. The directed c-balanced cut problem is

to find the minimum c-balanced cut. And finally, P.G/ denotes the total pairwiseconnectivity in G.

Note that a cut˝S; NS ˛ separates pairs .u; v/ 2 S � . NS/ so that there is no path u to

v, that is, there is no strongly connected component (SCC) containing vertices bothin S and NS .

The tree decomposition procedure is as follows: As shown in Algorithm 1 (lines3–13), the procedure starts with a tree T D .VT ; ET / containing only one rootnode t0. t0 is associated with the vertex set V of G, that is, V.t0/ D V.G/. Foreach node ti 2 VT whose V.ti / contains more than one vertex and V.ti / has notbeen partitioned, the subgraph GŒV.ti /� induced by V.ti / in G is partitioned using a

c-balanced cut algorithm [24] where constant c D 1 �q

ˇ

ˇ0. For each c-balanced

cut on GŒV.ti /�, two children nodes ti1 and ti2 of ti in VT are created. These twochildren nodes correspond to two sets of vertices returning by the cut. Finally, eachnode ti is assigned a cost cost.ti / which is equal to the cost of the cut performed onGŒV.ti /�. This procedure continues until V.ti / contains only a single vertex. That is,the leaves of T represent nodes in G. Therefore, T has n leaves and n � 1 internalnodes where each internal node in T represents a subset of nodes in V .

Note that the root node t0 is at level 1. If a node is at level l , all its children aredefined to be at level l C 1. Note that all collections of vertices corresponding tonodes in a same level form a partition in V .

Lemma 6 The height of T obtained in the above balanced tree decompositionprocedure is at most O.� log1�c0 n/

Proof Note that the directed c-balanced cut algorithm [24] finds in polynomialtime a c0-balanced cut within a factor of O.

plog n/ from the optimal c-balanced

cut for c0 D ˛c and fixed constant ˛. Thus, when separating GŒV.ti /� using c-balanced cut, the size of the larger part is at most .1 � c0/jV.ti /j. By induction


method, it can be shown that if a node ti is on level l , the size of the correspondingcollection V.ti / is at most jV j�.1�c0/l�1. It follows that the tree’s height is at mostO.� log1�c0 n/. �

3.2 Pseudo-approximation Algorithm and Analysis

This subsection presents the second main step which uses the dynamic programmingto search for the right set of nodes in T such that the cuts to partition thosecorresponding sets of vertices in G have the minimum cost and the obtainedpairwise connectivity is at most ˇ0

�n2

�where ˇ < ˇ0 < 1. The details of this step

are shown in Algorithm 1 (lines 14–23).Denote a set F D ft1; t2; : : : ; tkg � VT such that V.t1/; V .t2/; : : : ; V .tk/ is a

partition of V.G/ , that is, V.G/ Dk]

iD1

V .ti / and for any pair ti and tj 2 F ,

V.ti / \ V.tj / D ;. Such a subset F is called G-partition. Denote by A.ti / the set

Algorithm 1 ˇ-edge disruptor1: Input: A directed graph G D .V; E/ and 0 � ˇ < ˇ0 < 1

2: Output: A ˇ0-edge disruptor of G

fConstruct the decomposition treeg3: c 1�

qˇ

ˇ0

4: T .VT ; ET / .ft0g; �/, V .t0/ V .G/; l.t0/ D 1

5: while 9 unvisited ti with jV .ti /j � 2 do6: Mark ti visited, create new child nodes ti1; ti2 of ti7: l.ti1/; l.ti2/ l.ti /C 1

8: VT VT [ fti1; ti2g9: ET ET [ f.ti ; ti1/; .ti ; ti2/g

10: Separate GŒV .ti /� into two using directed c-balanced cut11: Assign two obtained partitions to V .ti1/ and V .ti2/

12: cost.ti / The cost of the balanced cut13: end whilefFind the minimum cost G-partitiong

14: for ti 2 T in reversed BFS order from root node t0 do15: for p 0 to ˇ0

�n2

�do

16: if P.GŒV .ti /�/ � p then17: cost.ti ; p/ 0

18: else19: cost.ti ; p/ minfcost.ti1; �/C cost.ti2; p � �/C cost.ti / j � � pg20: end if21: end for22: end for23: Find F with P.F / D minfcost.t0; p/ j p � ˇ0

�n2

�g24: Return union of cuts used at A.F / during tree construction


Fig. 9 A part of a decomposition tree. F D ft2; t3; t5; t6g is a G-partition. The correspondingpartition fV .t2/; V .t3/; V .t5/; V .t6/g in G can be obtained by using cuts at ancestors of nodes inF , that is, t0; t1; t4

of nodes corresponding to ancestor of ti in T and A.F / D[

ti2F

A.ti /. It is clear that

an F is G-partition iff F satisfies:1. 8ti ; tj 2 F W ti … A.tj / and tj … A.ti /

2. 8ti 2 VT ; ti is a leaf: A.ti / \ F ¤ �

In case F D ft1; t2; : : : ; tkg is G-partition, V.t1/, V.t2/, : : :, V.tk/ can beseparated by performing the cuts corresponding to ancestors of nodes in F duringthe tree construction. For example, a decomposition tree with a G-partition setft2; t3; t5; t6g in Fig. 9. The corresponding partition fV.t2/; V .t3/; V .t5/; and V.t6/gin G can be obtained by cutting GŒV.t0/�, GŒV.t1/�, and GŒV.t4/� successivelyusing c-balanced cuts in the tree construction. The cut cost, hence, will becost.t0/ C cost.t1/ C cost.t4/. In general, the total cost of all the cuts to separateV.t1/; V .t2/; : : : ; V .tk/ is equal to

cost.F / DX

ti2A.F /

cost.ti /


And the pairwise connectivity in G is equal to

P.F / DX

ti2F

P.GŒV .ti /�/

Our goal now is to find a G-partition F 2 VT so that P.F / � ˇ0�

n2

�with a

minimum cost.F / since the cut associated to F on T is the ˇ0-edge disruptor of G.Clearly, finding such a set F has an optimal substructure; thus, it can be found inO.n3/ using dynamic programming as described next.

Let cost.ti ; p/ denote the minimum cut cost to make the pairwise connectivityin GŒV.ti /� be less than or equal to p using only c-balanced cuts corresponding tonodes in the subtree rooted at ti . The minimum cost for a G-partition subset F thatinduces a ˇ0-edge disruptor of G is then minfcost.t0; p/ j p � ˇ0

�n2

�g where t0 isthe root node in T .

The value of cost.ti ; p/ can be calculated using following recursive formula:

cost.ti ; p/ D8<

:

0 if P.GŒV .ti /�/ � p

min��p

cost.ti1; �/ C cost.ti2; p � �/ C cost.ti / otherwise

where ti1; ti2 are children of ti .In the first case, whenP.GŒV .ti /�/ � p, no cut is required; thus, cost.ti ; p/ D 0.

Otherwise, all possible combinations of pairwise connectivity � in V.ti1/ and p ��

in V.ti2/ for all � � p can be investigated. The combination with the smallest cutcost is then selected.

Based on the above recursive formula, a dynamic programming (shown in lines14–23 of Algorithm 1 ) can find a minimum cost G-partition set F that induces aˇ0-edge disruptor in G.

The only part left is to prove that such a set F exists in T and the cost of theˇ0-edge disruptor induced from F found in the dynamic programming algorithm isno more than O.log

32 n/Optˇ-ED. The proof is shown as follows:

Lemma 7 There exists a G-partition subset F in T that induces a ˇ0-edge

disruptor whose cost is no more than O�

log32 n

Optˇ-ED.

Proof We first show that such a set F exists. That is, we are able to find a setG-partition set F such that P.F / � ˇ0

�n2

�. Denote Dˇ an optimal ˇ-edge disruptor

in G. Removing Dˇ from G, we obtain a set of strongly connected components(SCCs), denote as Cˇ D fC1; C2; : : : ; Ckg.

We construct a G-partition subset F based on Cˇ as shown in the Algorithm 2 .Nodes in T are visited in a top-down manner, that is, every parent must be visitedbefore its children. This can be done by visiting nodes in breadth first search (BFS)order from the root node t0. For each node ti , if there exists some component Cj 2Cˇ such that V.ti / contains more than .1 � c/jV.ti /j nodes in Cj (all leaves in T


satisfies this condition) and no ancestors of ti has ever been selected into F , thenwe select ti as a member of F . It is clear to see that F is a G-partition as it satisfiestwo conditions mentioned earlier.

The total pairwise connectivity of G induced by F is bounded as

P.F / �X

ti2F

jV.ti /j

2

!

D 1

2

X

Cj2Cˇ

X

jV.ti /\Cj j�.1�c/jV.ti /jjV.ti /j2 � n

2

� 1

2

X

Cj2C

0

BBB@

X

jV.ti /\Cj j�r

ˇ

ˇ0jV.ti /j

jV.ti /j

1

CCCA

2

� n

2

� 1

2

X

Cj2C

sˇ0ˇ

jCj j!2

� n

2

<ˇ0

2ˇ

0

@X

Cj2CjCj j2 � n

1

A � ˇ0

n

2

!

Thus, such a set F exists.Next, we show that cost.F / � O.log

32 n/Optˇ-ED. Let h.T / and Lu

T denote theheight of T and the set of nodes at level u of T , respectively. We have

cost.F / Dh.T /X

uD1

X

ti2.LuT\A.F //

cost.ti / (2)

If ti 2 A.F /, then ti is not selected to F . Hence, there exists Cj 2 Cˇ so thatjV.ti / \ Cj j < .1 � c/jV.ti /j (otherwise ti was selected into F as it satisfies theconditions in the line 3, Algorithm2 ). To guarantee c < 1�c we constrain c < 1=2,that is, ˇ >

ˇ0

4.

The edges in the optimal ˇ-edge disruptor Dˇ separate Cj from the otherSCCs. Hence, Dˇ also separates Cj \ V.ti / from the V.ti /nCj in GŒV.ti /�.Denote sep.ti ; Dˇ/ the set of edges in Dˇ separating Cj \ V.ti / from the rest inGŒV.ti /�. Clearly, sep.ti ; Dˇ/ is a directed c-balanced cut of GŒV.ti /�. Since, thecut algorithm we used in the tree construction has a pseudo-approximation ratio ofonly O.

plog n/, we have cost.tu/ � O.

plog n/jsep.ti ; Dˇ/j.


Algorithm 2 Find a good G-partition set F of T that induces a ˇ0-edge disruptorin G

F �

for ti 2 T in BFS order from t0 doif .9Cj 2 Cˇ W jV .ti /\ Cj j � .1� c/jV .ti /j/ and (A.ti /\ F D ;) then

F F [ ftigend if

end for

Recall that if two nodes ti and tj are on the same level, then V.ti / and V.tj / aretwo disjoint subsets. It follows that sep.ti ; Dˇ/ and sep.tj ; Dˇ/ are also disjointsets. Therefore, we have

X

ti 2.LuT \A.F //

cost.ti /

� O.p

log n/X

ti 2.LuT \A.F //

jsep.ti ; Dˇ/j

� O.p

log n/j [

ti 2.LuT \A.F //

sep.ti ; Dˇ/j

D O.p

log n/Optˇ-ED

Since h.T / is at most O.log n/ (Lemma 6), it follows from Eq. 2 that cost.F / �O.log

32 n/Optˇ-ED. It completes the proof. �

Since there exists a G-partition subset of T that induces a ˇ0-edge disruptor

whose cost is no more than O�

log32 n

Optˇ-ED as shown in Lemma 7 and the

dynamic programming is always able to find such a set F , the following theoremfollows immediately:

Theorem 10 Algorithm 1 achieves a pseudo-approximation ratio of O.log32 n/ for

the ˇ-edges disruptor problem.

4 Approximation Algorithm for ˇ-Vertex Disruptor

This section presents a polynomial-time algorithm (shown in Algorithm3 ) that findsin a directed graph G D .V; E/ a ˇ0-vertex disruptor whose the size is at mostO.log n log log n/ times the optimal ˇ-vertex disruptor where 0 < ˇ < ˇ02.

At the high level, Algorithm3 consists of two main phases: (1) vertex conversionand (2) size constraint cut. In the first phase, a given graph G is converted into G0in a way that removing v 2 G has the same effects as removing edge in G0. Inthe second phase, G0 is cut into SCCs capping the sizes of the largest component


while minimizing the number of removed edges. The constraint on the size ofeach component is kept relaxing until the set of cut edges induces a ˇ0-vertexdisruptor of G.

4.1 Algorithm Description

Phase 1: Vertex Conversion. Given a directed graph G D .V; E/ for which wewant to find a small ˇ0-vertex disruptor, a new directed graph G0 D .V 0; E 0/ isconstructed as follows:1. V 0 construction: For each vertex v 2 V , create two vertices v� and vC in V 0.

Thus, V 0 D f v�; vC j v 2 V g.2. E 0 construction: For each vertex v 2 V , add a directed edge .v� ! vC/ to E 0.

And for each directed edge .u ! v/ 2 E , add a directed edge .uC ! v�/ 2 E 0.That is, E 0 D f.v� ! vC/ j v 2 V g [ f.uC ! v�/ j .u ! v/ 2 Eg.

3. Edge cost assignment: Assign 1 for all edges .v� ! vC/ and C1 to other edgesin E 0.An example of this construction is shown in Fig. 10.Based on this conversion, a careful edge cut on G0 will return a vertex disruptor

on G. Indeed, let E 0V D f.v� ! vC/ j v 2 V g and consider a cut set D0e � E 0that contains only edge in E 0V , we have a one-to-one correspondence betweenD0e and Dv D fv j .v� ! vC/ 2 D0eg which is a vertex disruptor set of G.However, G and G0 have different maximum pairwise connectivity, .n�1/n

2for G and

.2n�1/2n

2for G0, the fractions of pairwise connectivity remaining in G and G0 after

removing Dv and D0e are, therefore, not simply related to each other. Thus, we nextdescribe what a “careful edge cut” is and how good a vertex disruptor of G can beobtained.

Phase 2: Size Constraint Cut. Observe that if edges are removed in orderto separate a graph into SCCs, then there is a relation between the pairwiseconnectivity in the remaining graph and the maximum size of SCC. The smallerthe maximum size of SCC, the smaller pairwise connectivity in the graph. However,the smaller the maximum size of each SCC, the more edges are needed to be cut.Therefore, a good range of the maximum size of SCC in G0 needs to be foundin order to return a good vertex disruptor of G with the minimum cut. Along thisdirection, a binary search can be performed to find a right upper bound ˇjV 0j andlower bound ˇjV 0j of the size of each SCC in G0. As shown in Algorithm 3 , at eachstep, we find in G0 D .V 0; E 0/ a minimum cost edge set whose removal partitionsthe graph into strongly connected components, each has size at most QjV 0j, whereQ D b ˇ C ˇ

2�c � �. The value of Q is rounded to the nearest multiple of � so that the

number of steps for the binary search is bounded by log 1�. The problem of finding

a minimum cost edge set to decompose a graph of size n into strongly connectedcomponents of size at most n is known as -separator problem. At this step, the


baFig. 10 Phase 1: Vertexconversion which converts agraph G into G0 such thatremoving a vertex v 2 G hasthe same effects as removingan edge in G0

Algorithm 3 ˇ0-vertex disruptorInput: Directed graph G D .V; E/ and fixed 0 < ˇ0 < 1

Output: A ˇ0-vertex disruptor of G

G0.V 0; E 0/ .�; �/

8v 2 V W V 0 V 0 [ fvC; v�g8v 2 V W E 0 E 0 [ f.v� ! vC/g; c.v�; vC/ 1

8.u! v/ 2 E W E 0 E 0 [ fuC ! v�g; c.uC; v�/ 1ˇ 0; ˇ 1

DV V .G/

while .ˇ � ˇ > �/ do

Q b ˇC ˇ

2�c � �

Find De � E 0 to separate G0 into strongly connected components of sizes at most QjV 0jusing algorithm in [25]Dv fv 2 V .G/ j .vC ! v�/ 2 Degif P.GŒV nDv�/ � ˇ

�n2

�then

ˇ D QRemove nodes from Dv as long as P.GŒV nDv�/ � ˇ

�n2

�

if jDV j > jDvj thenDV D Dv

end ifelse

ˇ D Qend if

end whileReturn DV

algorithm presented in [25] which finds a -separator in a directed graph G witha pseudo-approximation ratio of O

�1�2 : log n log log n

�for a fixed � > 0 is used.

In our context, D QjV 0j. By the cost assignment step in the vertex constructionphase, the edge cut obtained in this step must be a subset of E 0V as other edges e0not in E 0V have a cost of C1; hence, e0 never got selected. Finally, the cut edges inG0 is converted to vertices in G to obtain the ˇ0-vertex disruptor. Simply, for eachedge .v� ! vC/ in a cut set, we have a corresponding vertex v 2 V .


4.2 Theoretical Analysis

Lemma 8 Algorithm 3 always terminates with a ˇ0-vertex disruptor.

Proof We show that whenever Q � ˇ0 then the corresponding Dv found inAlgorithm 3 is a ˇ0-vertex disruptor of G. Consider the edge separator D0e of G0induced by Dv. We first show the mapping between SCCs in GŒV nDv� and SCCs inG0ŒE 0nD0e�, the graph obtained by removing D0e from G0. Partition the vertex set V

of G into (1) Dv: the set of removed nodes; (2) Vsingleton: the set of SCCs whose sizesare one, that is, each SCC has only one node; and (3) Vconnected: the set of remainingSCCs whose sizes are at least two, denote Vconnected D Ul

iD1 Ci ; jCi j � 2. Verticesin Vconnected belong to at least one cycle in G, whereas vertices in Vsingle are allsingleton.

We have the following corresponding SCCs in G0ŒE 0nD0e�:1. v 2 Dv $ SCCs fvCg and fv�g in G0ŒE 0nD0e�. Because after removing .v� !

vC/, vC does not have incoming edges and v� does not have outgoing edges.2. v 2 Vsingleton $ SCCs fvCg and fv�g. Assume vC belong to some SCC of size

at least 2 , that is, vC lies on some cycle in G0. Because the only incoming edgeto vC is from v�, it follows that v� is preceding vC on that cycle. Let u� anduC be the successive vertices of vC on that cycle. Then u and v should belong tothe same SCC in G which yields a contradiction as v 2 Vsingleton. Similarly, v�cannot lie on any cycle in G0.

3. SCC Ci � Vconnected $ SCC C 0i D fv�; vC j v 2 Ci g. This can be shown using asimilar argument as above.Since D0e is a Q-separator, the sizes of SCCs in G0ŒE 0nD0e� are at most Q 2n. It

follows that the sizes of SCCs in GŒV nDv� are bounded by Qn.Denote the set of SCCs in GŒV nDv� by C with the convention that vertices in Dv

become singleton SCC in GŒV nDv�. Therefore, we have

P.GŒV nDv�/ DX

Ci2C

jCi j2

!

D 1

2

0

@X

Ci2CjCi j2 � jV j

1

A

� 1

2

0

@X

Ci2CQjV j/jCi j � jV j

1

A

D 1

2

� QjV j2 � jV j

� Q

jV j2

!

< ˇ0

jV j2

!

This guarantees that the binary search always finds a ˇ0-vertex disruptor andcompletes the proof. �

Theorem 11 Algorithm 3 achieves a pseudo-approximation ratio ofO.log n log log n/ for the ˇ-vertex disruptor problem.


Proof We will show that Algorithm3 always finds a ˇ0-vertex disruptor whose sizeis at most O.log n log log n/ times the optimal ˇ-vertex disruptor for ˇ02 > ˇ > 0.First, it follows from the Lemma 8 that Algorithm 3 returns a ˇ0-vertex disruptorDv. At some step, the size of Dv equals to the cost of Q-separator D0e in G0 where Qis at least ˇ0 � � according to Lemma 8 and the binary search scheme. The cost ofthe separator is at most O .log n log log n/ times the Opt

. Q��/-separator using thealgorithm in [25].

Consider an optimal .ˇ02 � 9�/-vertex disruptor D0v of G and its correspondingedge disruptor D0e in G0. Denote the cost of that optimal vertex disruptor byOpt.ˇ02�9�/-VD. If there exists in GŒV nDv� a SCC Ci so that jCi j > .ˇ0 � 2�/n,then

P.GŒV nDv�/ >1

2..ˇ0 � 2�/n � 2/..ˇ0 � 2�/n � 1/ > .ˇ02 � 9�/

n

2

!

when n >20.ˇ0C1/

�. Hence, every SCC in G0ŒV nD0v� has size at most .ˇ0 � 2�/.2n/,

that is, D0e is an .ˇ0 � 2�/-separator in G0. It follows that Opt.ˇ02�9�/-VD �Opt.ˇ0�2�/-separator in G0.

Because Q � � � ˇ0 � 2�, we have

Opt. Q��/-separator � Opt.ˇ0�2�/-separator � Opt.ˇ02�9�/-VD

The size of the vertex disruptor jDvj D jD0ej is at most O .log n log log n/

times Opt. Q��/-separator. Thus, the size of found ˇ0-vertex disruptor Dv is at most

O.log n log log n/ times the optimal .ˇ02 � 9�/-vertex disruptor. As an arbitrarysmall � can be chosen, setting ˇ D ˇ02 � 9� completes the proof. �

5 A Second Look

Another look at the solutions to ˇ-ED and ˇ-VD in order to obtain a better algorithmis discussed in this section. As can be seen earlier, a c-balance cut for a treedecomposition is used to solve the ˇ-ED problem. This results into a

plog n ratio

for a c-balance cut and another log n for the height of the decomposition tree,leading to a log1:5 n ratio. If a better cut without a need of decomposing G intoa tree is devised, a log n factor can be saved, thus obtaining a

plog n ratio. In order

to do so, let us first describe the following definitions which are the key concepts toa better algorithm by approximating the best “local” cut at each step:

Cut Ratio. Given a graph G D .V; E/, for any cut˝S; NS ˛ where S � V and

NS D V n S , the cut ratio of hS; NSi, denoted by ˛.S/, is defined as


˛.S/ D c.S; NS/

jS jj NS j ; (3)

where c.S; NS/ is the total cost of edges in the cut.The cut ratio metric intuitively allows to find “natural” partitions: The numerator

captures the minimum cost criterion, while the denominator favors a balancedpartition. Assume that G is connected, the cut disrupts at least jS jjV n S j pairs,those with exact one endpoint in S ; hence, ˛.S/ can also be seen as the averagecost to disrupt pairs.Sparsest Cut. The sparsest cut is a cut with the smallest cut ratio. Let ˛.G/ denotethe minS�V ˛.S/, the minimum cut ratio of all cuts in G. Thus, the sparsest cutobtains a cut with the cut ratio ˛.G/. As expected, finding the sparsest cut is also anNP-hard problem [26].

5.1 O.p

log n/ Pseudo-approximation for ˇ-Edge Disruptor

As the cut ratio can be seen as the average cost to disrupt the pairwise connectivity,iteratively finding and applying the sparsest cut until obtaining a ˇ-edge separatormay result in a small cost ˇ-edge disruptor. We describe a new algorithm using thesparsest cut in Algorithm 4 and show this algorithm is indeed a

plog n pseudo-

approximation algorithm for the ˇ-ED problem.The recursive bisection algorithm (RBA) iteratively selects a connected com-

ponent in the graph and calls a subroutine SPARSEST CUT to cut the componentinto two smaller ones, until the pairwise connectivity in the graph is no morethan ˇ

�n2

�. The subroutine SPARSEST CUT function works as follows: Assume

that it takes a connected graph G0 D .V 0; E 0/ as the input it will find asmall ratio cut S 0 in G0, and returns the pair of

�hS 0; NS 0i; ˛.S 0/�, that is, the

edges in the cut and the cut ratio. At each iteration, a component C � in G

that has the minimum cut ratio is selected, and the edges in the sparsest cutof C � are removed from G. After that, C � is no longer a component of G,and new components are introduced into G. The SPARSEST CUT is used again

Algorithm 4 Recursive bisection algorithm (RBA)Eˇ ;for each connected component C in G

.CE; C˛/ SPARSEST CUT.C / [27]while P.G/ > ˇ

�n2

�

Find a component C � of G with minimum cut ratio C �

˛

Eˇ Eˇ [ C �

E

Remove edges in C �

E from G

for each new component C 0 in G�C 0

E; C 0

˛

� SPARSE CUT.C 0/

return Eˇ


to find cuts and the corresponding cut ratios for the newly created components.This procedure is repeated until the pairwise connectivity in G is no morethan ˇ

�n2

�.

Clearly, the performance of the SPARSEST CUT subroutine directly impacts onthe performance of RBA. Here the SPARSEST CUT is selected as the algorithm thatguarantees the best theoretical result in literature for the min ratio cut problem [27].The result in [27] gives us a polynomial-time algorithm to find a cut hS; NSi given agraph G D .V; E/ with the cut ratio at most O.

plog n/˛.G/. This selection yields

an O.p

log n/ pseudo-approximation algorithm for the ˇ-edge disruptor problemas analyzed below.

Lemma 9 Given a graph G D .V; E/ and a subset of edges M! � E , if ! DP.G/ � P.GŒE n M!�/ > 0, then c.M!/

!� ˛min.G/, where

˛min.G/ D minf˛.C / j C is a connected component of Gg:

Proof We first prove for the case G is connected, that is, ˛min.G/ D ˛.G/ andP.G/ D �

n2

�. Let C1; C2; : : : ; Ck be connected components in GŒE n M!� and let

Ci .V / denote the set of vertices in component Ci . By definition of ˛.G/, we have˛.G/ � ˛.Ci .V //. Thus,

c�Ci.V /; Ci .V /

� ˛.G/jCi .V /jjV n Ci .V /j

Take the sum over all components Ci , we have

X

i

c�Ci .V /; Ci .V /

� ˛.G/

X

i

jCi.V /j.n � jCi.V /j/

Simplify both sides and note thatX

i

jC.Vi/j D n, we have

2c.M!/ � ˛.G/

�nX

i

jC.Vi /j �X

i

jC.Vi/j2�

� 2˛.G/

��n2

� �X

i

P.Ci/

�� 2˛.G/ !

Hence, the lemma holds when G is connected.Now, if G is not connected, let T1; T2; : : : ; Tl be connected components of G,

M.j /! be the intersection of M! and the edges in Tj , and T 0j be the subgraphs

obtained from Tj after removing M.j /! . Applying the above results to the case that

the graph is connected on each connected component, we have


c.M!/ DX

j

c.M .j /! / �

X

j

˛.Tj /�P.Tj / � P.T 0j /

� ˛min.G/X

j

�P.Tj / � P.T 0j /

D ˛min.G/ .P.G/ � P.GŒE n M!�//

Thus, the lemma holds for every graph G. �

Theorem 12 If SPARSEST CUT is an approximation algorithm with a factor f .n/

for the min cut ratio problem, then RBA returns a ˇ-edge disruptor of cost at mostf .n/ OPTe

ˇ0 , for any 0 < ˇ0 < ˇ, where OPTeˇ0 denotes the cost of a minimum

ˇ0-edge disruptor.

Proof By Lemma 9, if removing a set of edges M! � E from G disrupts ! pairsof vertices, then c.M!/

!� ˛min.G/. Thus, at any round in the while loop of RBA,

since a set of edges E �0 in a minimum ˇ0-edge disruptor, for some 0 < ˇ0 < ˇ, can

disrupt at least .ˇ � ˇ0/�

n2

�more pairs in G, we have

OPTeˇ0

.ˇ � ˇ0/�

n2

� � ˛min.G/: (4)

From the hypothesis that SPARSE CUT is an f .n/ factor approximation algorithmfor the min cut ratio problem, the average cost to disrupt a pair by removing C �Eis upper bounded by f .n/˛min.G/. By (4), the average cost to disrupt pairs in thegraph at any step is at most

f .n/OPTe

ˇ0

.ˇ � ˇ0/�

n2

�

Therefore, even when Eˇ disrupt all�

n2

�pairs in G, the total cost is no more than

f .n/OPTe

ˇ0

.ˇ � ˇ0/�

n2

�

n

2

!

� f .n/

ˇ � ˇ0OPTe

ˇ0 :

Hence, the theorem follows. �

Remarks If the underlying graph is directed, the directed sparsest cut algorithm in[24] can be used in the place of the sparsest cut algorithm in [27] to obtain a pseudo-approximation algorithm with the same ratio. Note that for directed graphs, a pairof vertices .u; v/ is said to be connected if there exists a path from u to v and from vto u (i.e., connected in both directions).


5.2 O.p

log n/ Pseudo-approximation for ˇ-Vertex Disruptor

For the ˇ-vertex disruptor problem, the first phase – vertex conversion – as discussedin Sect. 4 still can be used. However, in order to obtain an O.

plog n/ pseudo-

approximation, a relationship between ˇ-VD and ˇ-ED needs to be defined in orderto take advantage of Algorithm 4 instead of using the phase two – size constraintcut. Recall that given a directed graph G D .V; E/ for which we want to find a smallˇ-VD, after the vertex conversion, we should obtain a directed graph G0 D .V 0; E 0/which will have twice the number of vertices in G, that is, jV 0j D 2jV j D 2n.

Consider a directed edge disruptor set D0e � E 0 that contains only edge in E 0V .There is a one-to-one correspondence between D0e to a set Dv D fv j .v� ! vC/ 2D0eg in G.V; E/ which is a vertex disruptor set in G. Since G and G0 have differentmaximum pairwise connectivity, .n�1/n

2for G and .2n�1/2n

2for G0, the fractions

of pairwise connectivity remaining in G and G0 after removing Dv and D0e are,however, not exactly equal to each other.

Lemma 10 A ˇ-edge disruptor set in the directed graph G0 induces the same costˇ-vertex disruptor set in G.

Proof Let us denote Dv and D0e for vertex disruptor in G and edge disruptor in G0.Given P.G0ŒE 0 n D0e�/ � ˇ

�2n

2

�we need to prove that: P.GŒV nDv �/ � ˇ

�n

2

�

where n D jV j.Assume GŒV nDv� has l SCCs of size at least 2, say Ci ; i D 1 : : : l . The

corresponding SCCs in G0ŒE 0 n D0e� will be C 0i ; i D 1 : : : l where jC 0i j D 2jCi j.Since .2k

2 /.2n

2 /� .k

2/.n

2/D k.n�k/

.n�1/n.2n�1/� 0; for all 0 � k � n. We have

P.GŒV nDv�/�n2

� DlX

iD1

�jCi j2

�

�n2

� �lX

iD1

�jC 0

i j2

�

�2n2

� � ˇ �

Lemma 11 A ˇ-vertex disruptor set in G induces the same cost .ˇ C �/-edgedisruptor set in G0 for any � > 0.

Proof The same notations in the proof of Lemma 10 are used here. GivenP.GŒV nDv�/ � ˇ

�n2

�we need to prove that P.G0ŒE 0 n D0e�/ � .ˇ C �/

�2n2

�. We have

P.G0ŒE 0nD0e�/�2n2

�

DlX

iD1

jCi j.n � jCi j/.n � 1/n.2n � 1/

C P.GŒV nDv�/�n2

�


D P.GŒV nDv�/�n2

��

1 � 1

2n � 1

�CPl

iD1 jCi jn.2n � 1/

< ˇ C 1

2n � 1< ˇ C � (5)

when n � b 1C�2�

cC1. �

We now prove the main theorem for conversion from directed edge disruptor tovertex disruptor.

Theorem 13 Given a factor f .n/ polynomial-time bicriteria approximation al-gorithm for ˇ-edge disruptor in directed graphs, there exists a factor f .n/

polynomial-time bicriteria approximation algorithm that finds a .ˇ C �/-vertexdisruptor whose cost at most f .n/ time the cost of the optimal ˇ-vertex disruptorwhere � > 0 is an arbitrary small constant.

Proof Let G be a directed graph with uniform vertex costs in which we wish tofind a ˇ-vertex disruptor. Construct G0 as described at the beginning of this section.

For ˇ0 < ˇ, apply the given approximation algorithm to find in G0 a ˇ-edge disruptor, denoted by D0e , with the cost at most f .n/ � Optˇ0�ED.G0/, whereOptˇ0�ED.G0/ is the cost of a minimum ˇ-edge disruptor in G0. From Lemma 10,D0e induces in G a ˇ-vertex disruptor Dv of the same cost. We shall prove that

Optˇ0�ED.G0/ � Optˇ0�VD.G/ C �0;

where Optˇ0�VD.G/ is the cost of a minimum ˇ0-vertex disruptor in G and �0 issome positive constant. It follows that the cost of Dv will be at most

f .n/ � .Optˇ0�VD.G/ C 0/ � .1 C �/f .n/Optˇ0�VD.G/

Here, assume that Optˇ0�VD.G/ >�0

�; otherwise, Optˇ0�VD.G/ can be found in time

O.n�0� C2/ (polynomial time).

From an optimal ˇ-vertex disruptor of G, construct its corresponding edgedisruptor D�e in G0. If P.G0ŒE n D�e � � ˇ

�2n2

�, then Optˇ�ED.G0/ � Optˇ�VD.G/,

and we yield the proof. Thus, only the case P.G0ŒE n D�e � > ˇ�

2n2

�is considered.

Among SCCs of G0ŒE n D�e �, there must be a SCC of size at least ˇ2n or else

G0ŒE n D�e � � ˇ�1�

ˇ2n2

� � ˇ�

2n2

�(contradiction). Remove �0 D

l1ˇ

mvertices

from that SCC. The pairwise connectivity in G0ŒE n D�e � will decrease at least�ˇ2n � 1

ˇ

1ˇ

D 2n � 1ˇ2 � n for sufficient large n. From Lemma 11, the pairwise

connectivity after removing vertices will be less than


�ˇ C 1

2n � 1

� 2n

2

!

� n � ˇ

2n

2

!

Therefore, after removing at most �0 vertices from D�e , a ˇ-edge disruptor isobtained. Hence,

Optˇ�ED.G0/ � Optˇ�VD.G/ C �0:

From the above theorem and the O.p

log n/ pseudo-approximation algorithm fordirected ˇ-edge disruptor (shown in Algorithm 4 ), we have the following result: �

Theorem 14 For any 0 � ˇ0 < ˇ, there is an algorithm that finds a ˇ-vertexdisruptor of cost at most O.

plog n/�.OPTv

ˇ0/, where OPTvˇ0 is the optimal ˇ-vertex

disruptor.

6 Literature

Several existing works on network vulnerability and survivability have been inves-tigated, roughly categorized into two periods. Researches during the early periodmainly focused on investigating the node’s centrality and prestige, using functionsof node degree [3, 5, 28, 29] as the only measure. The main measures of centralitycan be classified as degree centrality, betweenness, closeness, and eigenvectorcentrality. Readers are referred to [30] to find more details about this centralitymeasurement.

As wireless ad hoc networks came forth, several recent studies have aimedto discover the critical nodes whose removal causes the network disconnected,regardless of how fragmented it is! Several centralized, distributed, and localizedheuristics have been proposed. Some representatives are centralized DFS-basedsearch [7–9, 31], distributed disjoint path [10, 32], and localized k-critical [11].Most of these algorithms suffer from high communication overhead to discover thenetwork partition and could not efficiently locate the critical nodes. In addition, noneof these methods have been proven theoretically.

Unfortunately, none of the above work reveals the global damage done on thenetwork in the case of multiple nodes/links failing simultaneously, thus inaccuratelyassessing the network vulnerability. None of these methods prove the performancebound of their solutions either.

This chapter presents a novel paradigm via two new optimization modelsfor quantitatively characterizing the vulnerability of networks. The proposedapproaches aim to identify the key nodes who play a vital role in thenetwork overall performance, such as whose removal maximizes the networkfragmented. This study defines the objective function differently in termsof the connected components rather than a traditional direct measure ofalgebraic connectivity, thereby providing an excellent way to assess the networkvulnerability.


7 Conclusion

This chapter presents two models along with their complexity hardness andsolutions to the network vulnerability. There are a couple of notes that we wouldlike to discuss further as follows:

It is easy to see that when k is large enough, the CVD problem cannot beapproximated within a factor of ˛.n/ for any polynomial-time computable function˛.n/ unless P D NP by reducing from the vertex cover problem. To avoid the caseof the optimal total pairwise connectivity after deleting S equal to 0, k can be set ask < jEj=� where � is the maximum degree of a given graph G. Approximationsolutions to this problem are still open.

The two optimization models can be modified to further assess the networkvulnerability under other objective functions. For example, in the context ofnetworks, QoS (quality of services) is an important concept. Therefore, we canconsider maximizing the QoS instead of total pairwise connectivity. Other objectivefunctions include the following: minimize the number of pairwise nodes such astheir shortest paths in GŒV n S� � d for some distance d minimize the connectedvertex disruptor. All these problems can be studied on a probabilistic graph G D.V; E/ where each edge e 2 E has a failure probability pe 2 Œ0; 1�, which models adynamic network.

As many complex networks such as the Internet, WWW, biological networks,and social networks can specifically be modeled as power-law graphs, where thenumber of nodes of degree i is be˛=iˇc for some constants ˛ and ˇ, it would beinteresting to investigate the proposed problems (ˇ-ED, ˇ-VD, CVD, and CED)in power-law networks. As shown early, these problems still remain NP-completeon power-law graphs. Even though, does the power-law distribution play any keyfactor on the inapproximability? Are we able to obtain a smaller such as a constantapproximation algorithm for these problems on power-law graphs?

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HardnessandApproximationofNetwork Vulnerabilitymythai/files/vulchap.pdf · or performance guarantees to large-scale networks. To this end, this chapter introduces two new optimization