Source Location Problems Considering Vertex-Connectivity and Edge-Connectivity Simultaneously

Hiro ItoDepartment of Communications and Computer Engineering, School of Informatics, Kyoto University,Kyoto 606-8501, Japan

Motoyasu ItoSemiconductor & Integrated Circuits Group, Hitachi, Ltd., Tokyo, Japan

Yuichiro ItatsuSystem Development Division, Fujitsu Kansai-Chubu Net-Tech Ltd., Osaka, Japan

Kazuhiro Nakai, Hideyuki Uehara, Mitsuo YokoyamaDepartment of Information and Computer Sciences, Toyohashi University of Technology, Aichi, Japan

Let G � (V, E) be an undirected multigraph, where V andE are a set of vertices and a set of edges, respectively.Let k and l be fixed nonnegative integers. This paperconsiders location problems of finding a minimum-sizevertex-subset S �� V such that for each vertex x �� V thevertex-connectivity between S and x is greater than orequal to k and the edge-connectivity between S and x isgreater than or equal to l. For the problem with edge-connectivity requirements, that is, k � 0, an O(L(�V�, �E�, l))time algorithm is already known, where L(�V�, �E�, l) is thetime to find all h-edge-connected components for h � 1,2, . . . , l and O(L(�V�, �E�, l)) � O(�E� � �V�2 � �V�min{�E�,l�V�}min{l, �V�}) is known. In this paper, we show that theproblem with k ≥≥ 3 is NP-hard even for l � 0. We thenpresent an O(L(�V�, �E�, l)) time algorithm for 0 ≤≤ k ≤≤ 2 andl ≥≥ 0. Moreover, we prove that the problem parameter-ized by the size of S is fixed-parameter tractable (FPT)for k � 3 and l ≥≥ 0. © 2002 Wiley Periodicals, Inc.

Keywords: graph; location problem; connectivity; source;fixed-parameter tractable

1. INTRODUCTION

Problems of selecting the best location for facilities in agiven network that satisfy a certain property are calledlocation problems [10]. Such problems have been studiedextensively. However, in most of these problems, the ob-

jective is to minimize the sum or the maximum value ofdistances between a facility and a vertex. Recently, locationproblems with objective functions involving connectivity,or flow-amount, have been treated and several polynomialtime algorithms have been developed [2, 7–9, 11, 13, 14].

Connectivity and/or flow-amount are very important fac-tors in applications to control and design multimedia net-works. In a multimedia network, some vertices of the net-work may have functions of offering several types ofservice for users. Let us call a vertex that can offer a certainservice i a source, and let S be a set of sources, where wecan locate more than one source in a network. A user at avertex x can use service i by communicating with source sthrough a path between s and x (or a set of paths between Sand x). The flow-amount (which is the capacity of the pathsbetween S and x) affects the maximum amount of data that canbe transmitted from S to a user at vertex x. Also, the edge-connectivity or the vertex-connectivity between S and x mea-sures the robustness of the service against network failures.

Source location problems, which are described as fol-lows, were introduced to find the best location of sourcesunder connectivity and/or flow-amount requirements [11]:Given a graph G � (V, E) with a set V of vertices and a setE of edges having nonnegative real capacities, a cost func-tion w: V 3 R� (where R� denotes the set of nonnegativereals), and a demand function d: V 3 R�, we want to

Minimize �x�S

w�x�

subject to S � V

��S, x� � d� x� for all x � V � S,

Received April 2001; accepted April 2002Correspondence to: H. Ito; e-mail: itohiro@kyoto-u.ac.jpPreliminary results of this paper appeared in the 11th Annual InternationalSymposium on Algorithms and Computation (ISAAC2000) [6]Published online 00 Month 2002 in Wiley InterScience(www.interscience.wiley.com). DOI 10.1002/net.10034© 2002 Wiley Periodicals, Inc.

NETWORKS, Vol. 40(2), 63–70 2002

where �(S, x) is a measurement based on the edge-connec-tivity, the vertex-connectivity, or the flow-amount betweenS and a vertex x in the graph G. For such measurements�(S, x), one may consider the minimum size �(S, x) of anedge-cut C � E that separates x from S or the minimumsize �(S, x) of a vertex-cut C � V � S � x that separatesx from S.

Source location problems with �(S, x) � �(S, x) inundirected graphs were treated by Tamura et al. [13, 14], Itoet al. [8, 9], and Arata et al. [2]. They gave polynomial timealgorithms for uniform costs w( x) � 1, x � V, while theproblem with general costs w( x), x � V was shown to beweakly NP-hard [2]. Recently, Ito et al. [7] considered asource location problem with �(S, x) � �(S, x) for adigraph and showed that the problem with uniform costsw( x) � 1 and uniform demands d( x) � k can be solved inpolynomial time if k is fixed.

In this paper, we study source location problems subjectto both vertex-connectivity and edge-connectivity measure-ments, that is, for a given multigraph G (i.e., a graph whichmay have parallel edges) and fixed integers k, l � 0, theobjective is to find the minimum-size vertex-subset S � Vsuch that for every x � V the vertex-connectivity betweenS and x is greater than or equal to k and the edge-connec-tivity between S and x is greater than or equal to l. We callthis problem the (k, l )-connectivity source location prob-lem or the (k, l )-CONLOC for short.

We establish a minimax theorem for the (k, l )-CONLOC.Based on the theorem, we present an O(L(n, m, l )) timealgorithm for solving the (k, l )-CONLOC with k � 2 andan arbitrary l, where L(n, m, l ) is the time to compute allh-edge-connected components for h � 1, 2, . . . , l. Thealgorithm can find all optimal solutions in the same timecomplexity. We show that the (k, l )-CONLOC is NP-hardif k � 3. Furthermore, we prove that the (k, l )-CONLOCwith k � 3 is fixed-parameter tractable (FPT), by presentingan O(3�S�n � L(n, m, l )) time algorithm.

The contents of the paper are as follows: Section 2introduces some definitions. Section 3 defines a dual solu-tion (k, l )-deficient-family and a lower bound on the opti-mal value and derives necessary and sufficient conditionsfor a source set S to be optimal in the (2, 0)- and (3,0)-CONLOC. Section 4 presents an O(L(n, m, l )) timealgorithm for the (k, l )-CONLOC with k � 2 and anarbitrary l. Sections 5 and 6 show results on NP-hardnessand FPT of the (k, l )-CONLOC, respectively. Section 7describes the conclusions and some remaining problems.

2. DEFINITIONS

Let G � (V, E) be an undirected multigraph, where Vand E denote the vertex set and the edge set, respectively. Itmay have multiple edges but no self-loops. Denote n � �V�and m � �E�. Unless confusion arises, ( x, y) denotes anedge with the end points x, y � V. For x � V, N( x)denotes the set of vertices adjacent to x, that is, N( x) � { y� V�( x, y) � E}.

For a pair of vertex-subsets X, Y � V, E(X, Y) denotesthe edge set {( x, y) � E�x � X, y � Y}. An edge-cut isa set of edges of the form E(X, V � X), which may bewritten simply as E(X). Vertices x and y are said to beconnected if there is a path which contains x and y. We saythat two vertex-subsets X and Y are l-edge-connected ifthere is no edge-cut E(W) with �E(W)� � l such that X �W and Y � V � W. The edge-connectivity between X andY, denoted by �(X, Y), is defined to be the maximumnumber l such that X and Y are l-edge-connected. Define�(X, Y) � � if X � Y � A. A graph is called l-edge-connected if �( x, y) � l for all x, y � V.

A vertex-subset W is called a vertex-cut if three vertex-subsets W, X, Y � V satisfy X � A, Y � A, W � X � W� Y � X � Y � A, W � X � Y � V, and E(X, Y) � A.Further, for X � X and Y � Y, we say that W separatesX and Y. If a singleton set {w} is a vertex-cut, w is calleda cut-vertex. Two vertex-subsets X and Y are k-vertex-connected if there exists no vertex-cut W such that �W� � kand W separates X and Y. The vertex-connectivity betweenX and Y, denoted by �(X, Y), is the minimum number ksuch that X and Y are k-vertex-connected. If X � Y � A orE(X, Y) � A, then we define �(X, Y) � �. A singleton set{ x} may be written as x for notational simplicity. A graphis called k-vertex-connected if �( x, y) � k for all x, y � V.

In this paper, we consider the following problem:

(k, l )-connectivity source location problem((k, l )-CONLOC)

Input: G � (V, E),Output: S � V,Objective function: minimize �S�,Constraints: �(S, x) � k and �(S, x) � l for all x � V.

Let be a set of symbols and let * be the set of strings ofsymbols in . A problem instance of a parameterized prob-lem is expressed as ( x, p) � * � *, where p is calleda parameter. A parameterized problem is called FPT if it hasan algorithm whose running time is at most f( p)�x�c, wheref is an arbitrary function and c is a constant (independent ofp). By using the idea of parameterized problems, Downeyand Fellows [3] defined detailed classes over the class ofNP-complete problems such that FPT � W[1] � W[2]� . . . � W[SAT] � W[P]. Moreover, they conjecturedthat inclusions between two classes are all strict. Followingtheir classification, we can say that FPT is the most tractableclass among NP-complete problems. We consider the fol-lowing parameterized version of the (k, l )-CONLOC:

(k, l )-CONLOC (parameterized version)Input: G � (V, E),Output: S � V,Parameter: an integer p,Constraints: “�S� � p,” and “�(S, x) � k and �(S, x)� l for all x � V.”

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To design our algorithm for solving the (k, l )-CONLOC,we use the k-vertex- or l-edge-connected components in agiven graph. A 2-vertex-connected component in a graph isa maximal induced subgraph that has no longer a cut-vertex[1].

The definition of 3-vertex-connected components isslightly lengthy. We here give an intuitive definition (see [5]for the exact definition). Any 3-vertex-connected graph G isa 3-vertex-connected component itself. We apply the fol-lowing argument to each of the 2-vertex-connected compo-nents in a given graph G. Thus, assume that G is 2-vertex-connected but not 3-vertex-connected. A vertex-cut of ordertwo is called a cut-pair. Then, G has at least one cut-pair.Let { x, y} be a cut-pair of G. The edges of G can bedivided into equivalent classes E1, E2, . . . , Eh such thattwo edges which lie on a common path not containing anyvertex of { x, y} except as an end-point are in the sameclass. The classes Ei are called the separation classes of Gin respect to { x, y}. Let E � �i�1

h Ei and E� � �i�h�1h

Ei be such that �E� � 2, �E�� � 2. Let V (respectively,V�) be the set of end-vertices of edges in E (respectively,E�). Let G � (V, E � {( x, y)}) and G� � (V�, E� �{( x, y)}). The graphs G and G� are called split graphs,and the edges ( x, y) in G and G� are called virtual edges.Replacing a graph G by two split graphs is called splittingG. A split graph is also split when it has a cut-pair. The3-vertex-connected components of a graph G are obtainedby applying such splitting operations to G iteratively alongall cut-pairs. A 3-vertex-connected component is typically3-vertex-connected. However, there are two special cases:The first is a bond of the form ({ x, y}, {( x, y), ( x, y), ( x,y), . . .}) (there are at least three edges), and the second isa cycle. The latter is not 3-vertex-connected. A bond (re-spectively, a cycle) of size p is called a p-bond (respec-tively, a p-cycle). Figure 1 shows an example of 3-vertex-connected components in a graph. The 3-vertex-connectedcomponents of the graph G in (a) are D1, D2, . . . , D7. In(b), broken lines denote virtual edges and two virtual edgescreated by the same splitting operation are labeled with thesame number. D2 and D6 are bonds, D3, D4, and D7 arecycles, and the others are 3-vertex-connected graphs. Notethat a bond or a cycle can be split into smaller components,that is, a p-cycle (respectively, a p-bond) can be split into aq-cycle (respectively, a q-bond) and a ( p � q � 2)-cycle[respectively, a ( p � q � 2)-bond] for 3 � q � p � 1.

However, such splittings are prohibited in the definition of3-vertex-connected components (the reason is that the re-sultant set of 3-vertex-connected components is not fixeduniquely if such splittings are allowed). For example, al-though D4 can be split into two 3-cycles, it will never bedone.

An l-edge-connected component is a subgraph inducedby a maximal vertex-subset W � V such that �( x, y) � lholds for all x, y � W in G [12]. For h � 1, define

��h� :� C�C is an h-edge-connected component of G�.

It is known that �(h) for all h � 1, 2, . . . , l can beconstructed in O(m � n2 � n min{m, ln}min{l, n}) time[12]. All 2-vertex-connected components and all 3-vertex-connected components can be obtained in linear time [1, 5].In this paper, the vertex set of a k-vertex-connected com-ponent (respectively, an l-edge-connected component) maybe called a k-vertex-connected component (respectively, anl-edge-connected component) unless confusion arises.

3. A LOWER BOUND ON SOME CONDITIONS

First, we introduce a (k, l )-deficient-family, which is adual solution to the (k, l )-CONLOC. Based on this, a lowerbound on the optimal value to the problem will be derived.

For a vertex-subset X � V of a graph G � (V, E), theboundary of X is defined as

B�X� :� x � X�E�x, V � X� � A�.

A vertex-subset X � V is called a (k, l )-deficient-set if “X� B(X) � A and �B(X)� � k” or “�E(X)� � l.” A family� � {X1, X2, . . . , Xp} of (k, l )-deficient-sets is called a(k, l )-deficient-family if Xi � Xj � A for i � j.

Theorem 1. Let S be a feasible solution to the (k, l)-CONLOC and � � {X1, X2, . . . , Xp} be a (k, l)-deficientfamily. Then,

�S� � ���.

Proof. Assume that Xi � S � A for some Xi � �. IfXi � B(Xi) � A and �B(Xi)� � k, then �(S, x) � k for x

FIG. 1. An example of 3-vertex-connected components.

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� Xi � B(Xi). On the other hand, if �E(Xi)� � l, then �(S,x) � l for x � Xi. In any case, Xi � S � A holds for allXi � �. By noting that X1, X2, . . . , Xp are mutuallydisjoint, we obtain �S� � i�1

p �S � Xi� � ���. �

The inequality of Theorem 1 presents a lower bound on�S�. By the theorem, a (k, l )-deficient-family can be viewedas a dual solution to the (k, l )-CONLOC. The inequalitywill be used for proving the optimality of our algorithm for(2, l )-CONLOC presented in Section 4, that is, the algo-rithm finds a primal-dual solution.

There are exponentially many (k, l )-deficient-sets. Tocompute a (k, l )-deficient-family with the maximum size,we characterize some necessary and sufficient conditionsfor a set S to be optimal to (0, l )-CONLOC, (2, 0)-CONLOC, and (3, 0)-CONLOC, respectively. Before this,we introduce some more notation:

Let l � 1 and k � {1, 2, 3}. An h-vertex-connectedcomponent D (h � k) is called a (k, 0)-leaf-component ifit is a (k, 0)-deficient-set [i.e., �B(D)� � k and D � B(D)� A]. For example, D1, D3, D5, and D7 of Figure 1 are (3,0)-leaf-components. Note that D2 and D6 are not (3, 0)-leaf-components because D2 � B(D2) � D6 � B(D6)� A. A vertex-subset C � �h�1

l �(h) is called a (0,l )-leaf-component if it is a (0, l )-deficient-set [i.e., �E(C)�� l]. Let �k denote the set of (k, 0)-leaf-components and�l denote the set of (0, l )-leaf-components.

The next theorem was proved by Ito and Yokoyama [9]:

Theorem 2 [9]. For a graph G � (V, E) and a vertex-subset S � V, �(S, x) � l for all x � V if and only if C �S � A for all (0, l)-leaf-components C. �

In this paper, we show the following new property:

Theorem 3. Let G � (V, E) be a graph and let S be asubset of V.

(1) �(S, x) � 2 for all x � V if and only if D � S � A

for all (2, 0)-leaf-components D.(2) �(S, x) � 3 for all x � V if and only if the

following conditions (i) and (ii) hold:(i) D � S � A for all (3, 0)-leaf-components D.

(ii) If there is a vertex x such that �N(x)� � 2,then ({x} � N(x)) � S � A. �

We need condition (ii) of (2) due to the existence of3-vertex-connected components of cycles, which are not3-vertex-connected. For example, a p-cycle Cp � ({ x0,x1, . . . , xp�1}, {( x0, x1), . . . , ( xp�2, xp�1), ( xp�1,x0)}) ( p � 4) is a unique 3-connected-component itself.Hence, S � { x0} satisfies condition (i). However, �(S, x)� 2 for every x � { x2, . . . , xp�2}. In this case, condition(ii) is not satisfied, that is, �N( x)� � 2 and ({ x} � N( x))� S � A for x � { x2, . . . , xp�2}. This shows thenecessity of condition (ii).

For proving Theorem 3, the following lemma is used:

Lemma 4 [6]. For a graph G � (V, E), a vertex-subset X� V, and an integer k � {1, 2, 3}, if X � B(X) � A and�B(X)� � k, then (1) or (2) holds:

(1) There is a (k, 0)-leaf-component D � X.(2) There is a vertex x � X such that �N( x)� � k. �

The proof of Lemma 4 is slightly lengthy and is omittedhere; see [6] for the proof.

Proof of Theorem 3. The necessity is clear. The suf-ficiency follows from Lemma 4. �

4. ALGORITHM FOR (k, l)-CONLOC WITHk ≤ 2

In this section, we present an algorithm for solving the(2, l )-CONLOC. By Theorem 1, we can find an optimal setS by computing a maximum size (2, l )-deficient-family.For this purpose, we first find minimal elements of �l � �2.

Let �(2, l ) � �l denote the set of vertex-subsets C suchthat no Y � �l � �2 is a proper subset of C. Let �(2, l )� �2 denote the set of vertex-subsets D such that no Y � �l

� �2 is a proper subset of D. Two sets �(2, l ) and �(2, l )may have common elements and, hence, we define �(2, l ):� �(2, l ) � �(2, l ).

Any two elements in �(2, l ) are disjoint. However, �(2,l ) may contain D, D � �(2, l ) such that B(D) � B(D)(�{ z}). Let Z � V denote the set of vertices z such that{ z} � B(D) � B(D) for some two D, D � �(2, l ).Define �(2, l ) :� {D � �(2, l )�z � Z(B(D) � { z})}.Let

��2, l � C1, . . . , Cc�,

��2, l � D1, . . . , Dd�,

Z z1, . . . , zh�. (1)

Now, we construct a (2, l )-deficient-family � � {X1,X2, . . . , Xp} ( p � c � d � h) by defining Xi as follows:

Xi Ci, for i 1, 2, . . . , c, (2)

Xi Di�c, for i c 1, c 2, . . . , c d, (3)

Xi � D � ��2, l ��B�D� zi�c�d��,

for i c d 1, . . . , c d h. (4)

Lemma 5. Family � � {X1, X2, . . . , Xp} of components in(2)–(4) is a (2, l)-deficient-family. �

To prove the lemma, we use the next result due to Ito andYokoyama [9].

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Lemma 6 [9]. For a graph G � (V, E) and a vertex-subsetW � V, if �E(W)� � l, then there is a (0, l)-leaf-componentC contained in W. �

Proof of Lemma 5. From the definition, it is clear that“Xi � B(Xi) � A and �B(Xi)� � 2” or “�E(Xi)� � l” foreach 1 � i � c � d � h. Then, we prove that Xi and Xj

are disjoint for 1 � i � j � c � d � h. If 1 � i � j � cor c � 1 � i � j � c � d � h, then Xi and Xj are clearlydisjoint. Assume that there are Xi and Xj such that 1 � i� c � j � c � d � h and Xi � Xj � A. From thedefinitions of �(2, l ) and �(2, l ), none of Xi and Xj is asubset of the other. Hence, Xi � Xj � A and Xj � Xi � A.Let B(Xj) � { z}.

● By Lemma 6, z � Xi � Xj implies that �E(Xi � Xj)� ��E(Xi)� � k, and, hence, there is a (2, l )-leaf-componentC � Xi � Xj � Xi, that is, Xi � �(2, l ), a contradiction.

● By Lemma 6, z � Xi � Xj implies that �E(Xj � Xi)� ��E(Xi)� � k, and, hence, there is a (2, l )-leaf-componentC � Xj � Xi � Xj, that is, Xj � �(2, l ), a contradiction. �

From Theorem 1 and Lemma 5, �S� � c � d � h holdsif S is a feasible solution to the (2, l )-CONLOC. Moreover,we show that a feasible solution S that achieves the lowerbound can be constructed from a (2, l )-deficient-family �� {X1, . . . , Xp} of (2)–(4) by using the following algo-rithm:

Procedure COVER(2, l )begin

S :� A.For each Xi (i � 1, 2, . . . , c � d), choose an arbitraryx � Xi and S :� S � { x}.For each Xi (i � c � d � 1, c � d � 2, . . . , c � d� h), S :� S � { zi�c�d}.

end.

Lemma 7. A vertex-subset S constructed by COVER(2, l)is a feasible solution to the (2, l)-CONLOC.

Proof. Assume that S is not feasible. From Theorem 2and (1) of Theorem 3, there is a C � �l such that S � C� A or a D � �k such that S � D � A. Thus, there mustbe a C � �(2, l ) such that S � C � A or a D � �(2,l ) such that S � D � A.

(1) Assume that �(2, l) contains a C such that S � C� A. Then, C is equal to some Xi � � (i � {1,2, . . . , c}), and, hence, a vertex from C has beenchosen to be added to S.

(2) Assume that �(2, l) has a D such that S � D � A. IfD � �(2, l), then D is equal to some Xi � � (i � {c� 1, c � 2, . . . , c � d}), and, hence, a vertex fromD has been chosen to be added to S. If D � �(2, l),then z � B(D) is in Z, implying that D � S � A. �

Lemmas 5 and 7 mean that the maximum size p � ��� ofthe dual solution � is equal to the minimum size �S� of theprimal solution S. Therefore, we establish the next result:

Theorem 8. The (k, l)-CONLOC with k � 2 can be solvedin O(L(n, m, l)) time, where L(n, m, l) is the time requiredfor finding all h-edge-connected components for h � 1,2, . . . , l.

Proof. If k � 1, the problem is equivalent to the (0,l )-CONLOC and can be solved in O(L(n, m, l )) time [9].For k � 2, an S constructed by procedure COVER(2, l ) isan optimal solution from Theorem 1 and Lemmas 5 and 7.It takes O(L(n, m, l )) time to find all h-edge-connectedcomponents for h � 1, 2, . . . , l. All 2-vertex-connectedcomponents can be found in linear time [1]. Other parts ofthe algorithm can be executed in O(n � m) time. Therefore,the entire running time is O(L(n, m, l )). �

It follows from Theorem 8 that COVER(2, l ) actuallyprovides a minimax theorem for feasible sets in (2, l )-CONLOC and (2, l )-deficient-families. Since L(n, m, l )� O(m � n2 � n min{m, kn}min{k, n}) is known [12],we have the following corollary:

Corollary 9. The (k, l)-CONLOC with k � 2 can besolved in O(m � n2 � n min{m, ln}min{l, n}) time. More-over, it can be solved in linear time if l � 3. �

From the discussions so far, optimal solutions S can becharacterized as follows:

Corollary 10. A set S � {x1, x2, . . . , xp} of vertices is anoptimal solution to the (k, l)-CONLOC with k � 2 if andonly if xi � Ci (�Xi) for 1 � i � c, xi � Di�c (�Xi) for c� 1 � i � c � d, and xi � zi�c�d for c � d � 1 � i � c� d � h, where {C1, . . . , Cc}, {D1, . . . , Dd}, {z1, . . . , zh},and � � {X1, . . . , Xc�d�h} are defined by (1)–(4).

Proof. If S is feasible, then S contains a vertex fromeach Xi. Hence, if zi � S for some i � {1, . . . , h}, thenthere would be at least one D � �(2, l ) such that D � S� A, a contradiction. �

By Corollary 10, we can represent all optimal solutionsin the form (�(2, l ), �(2, l ), Z). This property is veryuseful for designing a real network, which needs to take intoaccount several other restrictions in practice such as flow-amount, distance, and cost. One may determine the bestlocation from the many candidates of solutions S to the (k,l )-CONLOC.

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5. NP-HARDNESS OF (k, 0)-CONLOC WITHk ≥ 3

Theorem 11. The (k, l)-CONLOC is NP-hard for each k� 3 and l � 0.

Proof. We prove the NP-hardness of the (3, 0)-CONLOC. For other k and l, the proof is easily extended.We reduce to the (3, 0)-CONLOC the vertex cover problem(VC) [4], which is a well-known NP-hard problem.

Vertex cover problem (VC)Input: G � (V, E);Output: U � V;Objective function: minimize�U�.Constraints: { x, y} � U � A for all ( x, y) � E.

Let G0 � (V0, E0) be an instance of the VC. Weconstruct an instance G � (V, E) of the (3, 0)-CONLOC asfollows (Fig. 2 illustrates the reduction from the VC to the(3, 0)-CONLOC):

V V0 � V � V�,

V u1, u2, u3�,

V� vij1, vij

2, vij3�� xi, xj� � E0�,

E E � E� � E�,

E � xi, vij1�, � xi, vij

2�, � xi, vij3�, � xj, vij

1�, � xj, vij2�,

� xj, vij3�, �vij

1, vij2�, �vij

2, vij3�, �vij

3, vij1��� xi, xj� � E0�,

E� � xi, u1�, � xi, u2�, � xi, u3��xi � V0�,

E� �u1, u2�, �u2, u3�, �u3, u1��.

Observe that G is 2-vertex-connected. There are �E0� � 13-vertex-connected components in G, and each 3-vertex-connected component is denoted by Dij or D0, where Dij

consists of { xi, xj, vij1 , vij

2 , vij3 } and D0 consists of V0 �

{u1, u2, u3}.The equivalence of the two instances G0 and G is clear.

(Note that it is sufficient to consider a vertex-set S such thatS � V0 for solving G.) Thus, the (3, 0)-CONLOC isNP-hard. �

6. FPT ALGORITHM FOR (3, l)-CONLOC

In this section, we prove that the parameterized (3,l )-CONLOC is FPT.

Theorem 12. The parameterized (3, l)-CONLOC can besolved in O(3pn � L(n, m, l)) time. �

We present an O(3pn � L(n, m, l )) time algorithm forsolving the parameterized (3, l )-CONLOC as follows: De-fine �3

� :� �3 � {{ z} � N( z)�z � V, �N( z)� � 2}. Let�(3, l ) � �l denote the set of vertex-subsets C such that noY � �l � �3

� is a proper subset of C. Let �(3, l ) � �3�

denote the set of vertex-subsets C such that no Y � �l ��3

� is a proper subset of C. There may be common elementsof �(3, l ) and �(3, l ), and, hence, we define �(3, l ) :��(3, l ) � �(3, l ). Note that an element in �(3, l ) is eithera 3-vertex-connected component or a set { z} � N( z) with�N( z)� � 2. We call the former type 1 and the latter type 2.We have the following lemmas:

Lemma 13. For C � �(3, l) and D � �(3, l), if C � D� A, then C � B(D) � A.

Proof. Assume that C � D � A and C � B(D) � A.Then, E(C � D) � E(C) and, hence, �E(C � D)� � l,contradicting the minimality of C and D. �

Lemma 14. Let S be a feasible solution and D � �(3, l)be of type 1. If (D � B(D)) � S � A, then for {x, y} � B(D)and any z � (D � B(D)) � S, one of S � S � {x} � {z}or S� � S � {y} � {z} is feasible.

Proof. Assume that S is not feasible. Thus, there is aC � �(3, l ) � �(3, l ) such that C � S � A. From the

FIG. 2. Illustration of reduction from the VC to the (3, 0)-CONLOC.

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feasibility of S, we have z � C. From the minimality of D,C � �(3, l ), and, hence, C � �(3, l ). From Lemma 13and x � C, y � C. Therefore, S� is feasible. �

Proof of Theorem 12. From Lemma 14, a feasiblesolution can be obtained by choosing a vertex from B(D)for each D � �(3, l ) of type 1 and a vertex from D � { z}� N( z) � �(3, l ) of type 2. Note that �B(D)� � 2 and�{ z} � N( z)� � 3. We can check whether or not there isa feasible solution by using a search tree of height at mostp, which is defined as follows: Each vertex of the tree islabeled with (S, L), where S is a vertex-subset and L is a listof elements of D � �(3, l ) with D � S � A. The root ofthe tree is labeled by (A, �(3, l )).

Each vertex labeled (S, L) of the tree has at most threechildren defined as follows: A vertex-subset S covers avertex-subset D if �(S, x) � 3 and �(S, x) � l for all x� D.

(1) If L contains at least one type 1 element, then choose anarbitrary type 1 element D � L. Let B(D) � {x, y}.By Lemma 14, S � {x} or S � {y} is sufficient tocover D. So, we let the current vertex have two chil-dren: The first child is labeled with (S � {x}, L(x)),where L(x) � L includes D such that x � D (L(x)can be computed by removing D � L such that x� D from L), and the second is labeled with (S �{y}, L(y)).

(2) If L contains no type 1 element (hence, it contains atleast one type 2 element). Choose an arbitrary {z} �N(z) � L. Let N(z) � {x, y}. Clearly, S � {x}, S �{y}, or S � {z} is sufficient to cover D. So, we let thecurrent vertex have three children: the first, the second,and the third child are labeled with (S � {x}, L(x)), (S� {y}, L(y)), and (S � {z}, L(z)), respectively.

For a given p, we search vertices at height at most p in thetree. Assume that a vertex labeled with (S, A) is generatedin the tree of height at most p (i.e., �S� � p). Every D� �(3, l ) is covered by S. However, some C � �(3, l )may not be covered yet. Let q be the number of suchcomponents [q is easily calculated from S and �(3, l )]. If�S� � q � p, then a feasible solution is easily constructedfrom S by adding a vertex arbitrarily selected from eachuncovered C � �(3, l ). If �S� � q � p, then we cannotconstruct a feasible solution including S. If no vertex islabeled by (S, A) with �S� � q � p in the tree of at heightat most p, we can conclude that the given instance isinfeasible.

The number of vertices in the tree is at most 3p�1. Then,the computation time is O(3pn � L(n, m, l )). Therefore,Theorem 12 is proved. �

7. CONCLUDING REMARKS

In this paper, we studied the (k, l )-CONLOC, the loca-tion problem of finding a minimum-size vertex-subset Ssuch that �(S, x) � k and �(S, x) � l for all x � V. By

using the primal-dual property, we obtained an O(L(n, m,l )) time algorithm for finding all optimal solutions to the (k,l )-CONLOC with 0 � k � 2 and l � 0. Moreover, weproved that the (k, l )-CONLOC is NP-hard for each k � 3and l � 0, and the parameterized problem is FPT if k � 3and l � 0 by showing an O(3�S�n � L(n, m, l )) timealgorithm for a parameter �S�.

There are several remaining problems. It has been settledwhether the (k, l )-CONLOC is NP-hard or P for all k andl. On the other hand, it is remains open whether or not it isFPT for k � 4.

In this paper, only undirected graphs are treated. How-ever, the (k, l )-CONLOC in digraphs is also an importantproblem class. Ito et al. [7] studied the (k, l )-CONLOC indigraphs and gave an polynomial time algorithm for k � 1and a fixed l. If k � 3, the problem is obviously NP-hardfrom the results of this paper. It is left open to determine thecomplexity status of the problem with k � 2 or arbitrary l.

Acknowledgments

We would like to express our appreciation to ProfessorNagamochi Hiroshi of Toyohashi University of Technologyfor his valuable comments.

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