Locating Sources to Meet Flow Demands in Undirected Networks 1

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1 Journal of Algorithms 42, (2002) doi: /jagm ,available online at on Locating Sources to Meet Flow Demands in Undirected Networks 1 Kouji Arata Matsushita Communication Industrial Co., Ltd., Yokohama , Japan Satoru Iwata Department ofmathematical Informatics, Graduate School ofinformation Science and Technology, University oftokyo, Tokyo , Japan iwata@sr3.t.u-tokyo.ac.jp Kazuhisa Makino Division ofsystems Science, Graduate School ofengineering Science, Osaka University, Toyonaka, Osaka , Japan makino@sys.es.osaka-u.ac.jp and Satoru Fujishige Division ofsystems Science, Graduate School ofengineering Science, Osaka University, Toyonaka, Osaka , Japan fujishig@sys.es.osaka-u.ac.jp Received March 8,2000 This paper deals with the problem of finding a minimum-cost vertex subset S in an undirected network such that for each vertex v we can send d v units of flow from S to v. Although this problem is NP-hard in general,h. Tamura,H. Sugawara, M. Sengoku,and S. Shinoda (IEICE Trans. Fund. J81-A (1998), ) have 1 A preliminary version of this paper appears in Proceedings of 7th Scandinavian Workshop on Algorithm Theory (SWAT2000) [2]. The authors gratefully acknowledge the partial support of a Scientific Grant in Aid from the Ministry of Education,Science,Sports and Culture of Japan. This work was done while K. Arata and S. Iwata were with the Division of Systems Science,Graduate School of Engineering Science,Osaka University /02 $ Elsevier Science All rights reserved. 54

2 source location in networks 55 presented a greedy algorithm for solving the special case with uniform costs on the vertices. We give a simpler proof on the validity of the greedy algorithm using linear programming duality and improve the running time bound from O n 2 M n m to O nm n m,where n is the number of vertices in the network and M n m denotes the time for max-flow computation in a network with n vertices and m edges. We also present an O n m + n log n time algorithm for the special case with uniform demands and arbitrary costs Elsevier Science Key Words: location problem; edge-connectivity; max-flow; greedy algorithm; maximum adjacency ordering. 1. INTRODUCTION Let = G u d c be an undirected network on the underlying graph G = V E with a vertex set V of cardinality n and an edge set E of cardinality m. It has a capacity function u E R +,a demand function d V R +,and a cost function c V R +,where R + denotes the set of nonnegative reals. This paper addresses the problem of finding a minimumcost vertex subset S V such that the maximum flow value between S and v is at least d v for each v V. For a pair of disjoint subsets X Y V,we denote by λ X Y the maximum flow value between X and Y in. For any v V we write the singleton set v as v. For convenience,we assign λ X Y =+ if X Y. Note that λ is independent of the demand function d. Now,the problem is formulated as follows. Minimize c v v S subject to S V (1) λ S v d v v V We call this problem the Source Location problem. We say that a vertex set S V covers a vertex v if λ S v d v. Namely, Source Location asks for a minimum-cost subset S V that covers all the vertices in V. A motivation of this problem is given as follows. Suppose we are asked to locate multiple servers S which can provide a certain service in the multimedia network. A user at a vertex v can receive the service by connecting to a server in S through a path in. To ensure the quality of the service to v even if d v links become out of order,we should select S so that the edge-connectivity between S and v is at least d v. Thus the problem of finding a minimum-cost fault-tolerant setting of servers is an instance of Source Location with unit edge capacities. A special case of this problem with a uniform cost function was introduced by Tamura et al. [12,13]. They called it the plural cover problem. They

3 56 arata et al. first considered the case in which both d and c are uniform and described an algorithm that runs in O n 2 M n m time [12],where M n m denotes the time complexity of computing an s-t maximum flow in an undirected network with n vertices and m edges (see Goldberg and Rao [5,6] for recent development after the book of Ahuja et al. [1] was published). Later Tamura et al. [13] showed that a simple greedy algorithm solves problem Source Location in O n 2 M n m time even if the demand function d is arbitrary while the cost function c is still uniform. Ito et al. [7] described another algorithm to improve the time complexity to O npm n m,where p is the number of distinct values of d v v V ; i.e., p = d v v V. In this paper,we analyze the greedy algorithm of Tamura et al. [13] to give a simpler proof based on the linear programming duality. We then improve the greedy algorithm to run in O nm n m time. In the case of uniform demand and arbitrary cost,we give an O n m + n log n time algorithm. The algorithm makes use of maximum adjacency (MA) ordering,which was used by Nagamochi and Ibaraki for the minimum cut problem [8] and the edge-connectivity augmentation problem [9] for undirected networks. Finally,we show that Source Location is in general NP-hard,by reducing the knapsack problem to Source Location. Thus,it remains open to prove the NP-hardness in the strong sense or to devise a pseudo-polynomial time algorithm. We summarize the time complexity of Source Location in Table I, where bold letters indicate the complexity results obtained in this paper. The rest of the paper is organized as follows. Section 2 formulates Source Location as an integer programming problem,section 3 considers Source Location when c is uniform but d is arbitrary,and Section 4 discusses the case in which d is uniform but c is arbitrary. In Section 5,we show that Source Location is in general NP-hard. TABLE I Summary of the Results on Source Location c: uniform c: arbitrary d: uniform O n 2 M n m * Tamura et al. [12] O nm n m Ito et al. [7] O n m + n log n O n m + n log n d: arbitrary O n 2 M n m Tamura et al. [13] O npm n m ** Ito et al. [7] NP-hard O nm n m *M n m : the time complexity for computing a maximum s-t flow. **p : the number of distinct values of d v v V.

4 source location in networks INTEGER PROGRAMMING FORMULATION In this section,we formulate Source Location as an integer programming problem with an exponential number of constraints. A cut is a proper nonempty subset of V.ForacutX,let X denote the set of edges that cross X; i.e., X = e e = v w E v X w V X,and let κ X denote its capacity; i.e., κ X = u e e X If X = v,we write κ v instead of κ v. For a pair of disjoint vertex subsets X Y V,we denote κ X Y = e X Y u e. Forv V,we simply write κ X v instead of κ X v. We also denote by d W the maximum demand in W ; i.e., d W =max d v v W We say that a vertex v W attains the maximum demand in W if d v =d W. A cut W is called deficient if κ W <d W. IfacutW is deficient but any other nonempty subset X W is not, W is called a minimal deficient set. It is easy to see that the max-flow min-cut theorem implies the following lemma. Lemma 2.1 [13]. Let = G = V E u d c be an undirected network. Then S V covers all vertices in V if and only if S W holds for every minimal deficient set W. Let = W 1 W 2 W l be the family of all minimal deficient sets and let V = v 1 v 2 v n. Define an l n matrix A = A ij by A ij = 1if v j W i and A ij = 0 otherwise. From Lemma 2.1, Source Location can be written as the 0 1 integer programming problem, Minimize subject to n c j x j j=1 n A ij x j 1 i = 1 2 l (2) j=1 x j 0 1 j = 1 2 n where c j = c v j j = 1 2 n,and the 0 1 vector x = x 1 x 2 x n is the characteristic vector of a subset of V.

5 58 arata et al. Remark 2.1. Note that l = might be exponential in n and m. For example,let us consider a network = G = V E u d c,where V = v 1 v 2 v n, E = v 1 v i i = 2 3 n, u e =1 e E, { n if i = 1 d v i = 2 0 otherwise, and c is an arbitrary cost function. Then we can see that { } n = W W = + 1 W v 2 1 and hence = ( ) n 1 n 2 3. THE UNIFORM COST CASE 3.1. A Greedy Algorithm In this section,we consider Source Location with a uniform cost function. Specifically we assume that c v =1 v V. Tamura et al. [13] proposed the following greedy algorithm to solve Source Location. Algorithm Greedy Step 0. Order the vertices v 1 v 2 v n in V such that d v 1 d v 2 d v n. Step 1. Initialize j = 1 and S = V. Step 2. If S v j covers all vertices in V,then S = S v j. Step 3. If j = n then output S and halt. Otherwise, j = j + 1 and go to Step 2. Example 3.1. Let us apply Greedy to the network = G = V E u d c given in Fig. 1,where the values of u e (e E) and d v (v V ) are,respectively,attached to edges and vertices in Fig. 1 and c v =1 for all v V. The results are illustrated in Fig. 2. We initially include all vertices in S and check whether the set S v 1 covers all vertices or not. Since it covers v 1 (and hence all vertices in V ),we update S = S v 1 (see (i) in Fig. 2). We next check whether S v 2 covers all vertices or not. Since it covers both v 1 and v 2,we update S = S v 2 (see (ii)). By repeating this argument to the current S = v 3 v 4 v 7 (see (iii) (vii)),we finally obtain S = v 4 v 5 v 7 shown in (viii).

6 source location in networks 59 FIG. 1. A network = G = V E u d c,where c v =1 v V. FIG. 2. Algorithm Greedy applied to the network in Fig. 1.

7 60 arata et al. In order to show the correctness of algorithm Greedy,we consider the linear programming relaxation of (2), n Minimize c j x j and its dual, subject to Maximize subject to j=1 n A ij x j 1 i = 1 2 l (3) j=1 x j 0 l y i i=1 j = 1 2 n l A ij y i c j j = 1 2 n (4) i=1 y i 0 i = 1 2 l Recall that c j = 1 j = 1 2 n is assumed in this section. We also replace Steps 1 and 2 in algorithm Greedy as follows. Step 1. Initialize j = 1, S = V,and y i = 0 for i = 1 2 l. Step 2. (2-1) If S v j covers all vertices in V,then S = S v j. (2-2) Otherwise,choose a W i with W i S = v j,and set y i = 1. Obviously,the revised algorithm always keeps a feasible solution S (i.e., S covers all vertices in V ). Assume that Step 2 finds a vertex w that is not covered with S v j in some execution of Step 2 and Step 3. Then by the max-flow min-cut theorem,there exists a W i such that W i contains w and W i S v j =. Since S covers w,we have W i S = v j. Therefore,Step 2 always finds a W i with W i S = v j if S v j does not cover a vertex in V. Moreover,since S contains v l, l = j j + 1 n, j is the maximum number k with v k W i ; i.e., d v j =d W i. Let us also note that we need not actually perform Step 1 (initialization of y) in the revised version. It suffices to keep only non-zero y i s in Step 2. Let x and y be the primal and dual variables obtained at the end of the revised greedy algorithm. Note that x is the characteristic vector of the output S of the algorithm. The algorithm does not delete v j from S if and only if it updates y i as y i = 1 for some i with W i S = v j. Hence,at the termination,we have n l x j = (5) j=1 yi i=1

8 source location in networks 61 Therefore, x is a 0 1 solution satisfying (3) and (5). By the weak duality of linear programming problems (3) and (4),we only need to prove the feasibility of y in (4) to show the correctness of the algorithm Greedy. The feasibility of y will be proved by Lemmas 3.1 and 3.3 given below. Recall that the cut capacity function κ satisfies κ X +κ Y κ X Y +κ Y X X Y V (6) We call a set function satisfying (6) posi-modular [10]. The following lemma was originally proved by Tamura et al. [13]. Here we give a simple proof of it. Lemma 3.1 [13]. Let = G = V E u d c be an undirected flow network. Let W 1 and W 2 be minimal deficient sets in, and for each i = 1 2, let v i W i be a vertex that attains the maximum demand in W i.ifw 1 W 2, then we have v 1 W 1 W 2 or v 2 W 1 W 2. Proof. Suppose,to the contrary,that both v 1 W 1 W 2 and v 2 W 2 W 1 hold. Since W 1 and W 2 are deficient sets, d v 1 >κ W 1 and d v 2 > κ W 2 hold. It follows from (6) that d v 1 +d v 2 >κ W 1 +κ W 2 κ W 1 W 2 +κ W 2 W 1 This means that d v 1 >κ W 1 W 2 or d v 2 >κ W 2 W 1 holds. Since we have v 1 W 1 W 2 and v 2 W 2 W 1 by the assumption,it follows that W 1 W 2 or W 2 W 1 is deficient,which contradicts the minimality of W 1 or W 2. Arrange the columns of A in such a way that d v 1 d v 2 d v n. For each index i with 1 i l,let k i denote the maximum number k with v k W i. Then Lemma 3.1 implies the following. Lemma 3.2. as a submatrix. The matrix A does not contain ( j k i 1 k i 2 ) i i (7) Proof. If A contains such a submatrix,we have W i1 W i2, k i 1 W i1 W i2,and k i 2 W i2 W i1,which contradicts Lemma 3.1. Lemma 3.3. The dual variables y obtained by the revised greedy algorithm give a feasible solution to (4).

9 62 arata et al. Proof. Let S be the solution obtained by the revised algorithm. Suppose that for a pair of distinct rows i 1 i 2 we have yi 1 = yi 2 = 1. It follows from the discussion after Steps 1 and 2 that W il S = k i l l = 1 2 (8) Note that k i 1 k i 2 holds,since otherwise yi 1 and yi 2 must be updated in the same iteration in Step 2,a contradiction. Hence,from (8) and Lemma 3.2,we have W i1 W i2 =. We have thus shown the following. Theorem 3.1. If the cost function c is uniform, then algorithm Greedy produces an optimal solution of Source Location An Efficient Implementation We now analyze the time complexity of algorithm Greedy. Steps 0,1, and 3 are clearly executed in O n log n, O 1,and O n time,respectively. As for Step 2,Tamura et al. [13] checked if S v j covers all vertices in V by computing λ S v j v i (i.e.,a max flow from S v j to v i ) for all v i. Clearly,this requires O nm n m time. Since Step 2 is iterated n times,the required time is O n 2 M n m in total [13]. However,the following lemma implies that Step 2 can be replaced by Step 2. If S v j covers v j,then S = S v j. Lemma 3.4. If S v j covers v j in Step 2 of algorithm Greedy, then S v j covers v i for all i j. Proof. Assume that some v i with i<jis not covered with S v j. Then there exists a cut X with v i X, V X S v j,and κ X < d v i. Then, S X v j clearly holds. Moreover,we have S X = v j since otherwise S does not cover v i,which contradicts the property that Greedy always keeps a feasible set S. Hence, X separates v j and S v j. Since κ X <d v i d v j,it follows that S v j does not cover v j,a contradiction. Thus we have improved the time complexity by use of Step 2 instead of Step 2. Theorem 3.2. If the cost function c is uniform, then problem Source Location can be solved in O nm n m time.

10 source location in networks THE UNIFORM DEMAND CASE In this section,we consider Source Location with a uniform demand function d. We assume that d v =g (a fixed positive real) holds for all v V. We show that,in this setting,source Location can be solved in O n m + n log n time. Our algorithm uses the maximum adjacency ordering and does not use any maximum flow computations. An ordering v 1 v 2 v n of all vertices in V is called a maximum adjacency (MA) ordering if it satisfies κ v 1 v 2 v i v i+1 κ v 1 v 2 v i v j for 1 i<j n The MA ordering plays a crucial role in this section through the following lemma. Lemma 4.1 [8, 11]. Let G = V E be an undirected graph with a nonnegative capacity function u. Then, the following statements hold. (i) An MA ordering v 1 v 2 v n can be computed in O m + n log n time. (ii) The last two vertices v n 1 and v n for every MA ordering in G satisfy λ v n 1 v n =κ v n (9) We can choose the first vertex v 1 arbitrarily. If the demand function d is uniform,minimal deficient sets are pairwise disjoint by the posi-modularity 6 of κ; i.e.,for every pair of W 1 and W 2 in, W 1 W 2 = holds. Therefore,in order to solve Source Location,we try to find all minimal deficient sets W and construct a minimum-cost source set S V by choosing from each W a vertex v W with the minimum cost c v among W. Since any source set S must contain all v V such that κ v <g,we initialize S as S = v V κ v <g. If S n 1,then we output S and halt. Otherwise,if S,we construct a network from by merging all vertices in S into a single vertex s,and if S =,we construct a network from by adding a new isolated vertex s. Then we compute an MA ordering v 1 = s v 2 v h 1 v h starting from s in. Denote by λ and κ,respectively,the maximum flow value and the cut capacity in. Since all vertices v except for s in satisfy κ v g,lemma 4.1 implies λ v h 1 v h =κ v h g

11 64 arata et al. Namely,every cut X in that separates v h 1 and v h satisfies κ X g. We claim that no minimal deficient set W in separates v h 1 and v h. Assume to the contrary that there exists such a minimal deficient set W. Then W contains a vertex w S,since otherwise κ W =κ W g a contradiction. Then it follows from w and the minimality of W that W is given as W = w. However,this does not separate v h 1 and v h, which contradicts the assumption. From our claim,we can update by merging the vertices v h 1 and v h into a single vertex ˆv. We then check if ˆv satisfies κ ˆv g. Since κ ˆv <gimplies that W = v h 1 v h is a minimal deficient set in the original network,if κ ˆv <g,we again update the network by merging s and ˆv into s and update S by adding v h 1 if c v h 1 <c v h ; otherwise,we add v h to S. Now we have κ ˆv g for all vertices except for s in. By repeating the above argument for (i.e.,we compute an MA ordering v 1 = s v 2 v h 1 v h in,merge the last two vertices v h 1 and v h,and so on),we can compute a minimum-cost source set S. Formally it can be written as follows. Algorithm Contract Input. A network = G = V E u d c,where d v = g for all v. Output. A minimum-cost vertex set S V which covers all vertices in V. Step 0. Initialize S = v V κ v <g and α v =v for all v V. Step 1. If S V 1,then output S and halt. Otherwise,construct a network from by merging all vertices in S into a single vertex s. Step 2. (2-I) Compute an MA ordering v 1 = s v 2 v h 1 v h starting from s in. (2-II) Merge the last two vertices v h 1 and v h into a single vertex ˆv. Denote the resulting network by again. (2-III) If c α v h 1 <c α v h,then α ˆv =α v h 1 ; otherwise, α ˆv =α v h. (2-IV) If κ ˆv <g,then update by merging s and ˆv into s, and S = S α ˆv. Step 3. If V 2,where V denotes the vertex set of,then output S and halt. Otherwise go to Step2.

12 source location in networks 65 FIG. 3. A network = G = V E u d c,where d v =8 v V. Note that the algorithm uses α for computing,from each W,a vertex v W with the minimum cost c v among W. Formally, α v with v s stores the vertex v in the original network having the minimum cost c v among P v,where P v is the set of all vertices v in V which are merged to v. Example 4.1. Let us apply algorithm Contract to the network = G = V E u d c given in Fig. 3,where u e (e E) and c v (v V ) are,respectively,attached to edges and vertices in Fig. 3 and d v =8 for all v. The results are illustrated in Fig. 4. In Step 0,the algorithm initializes S = v b and α (see (i) in Fig. 4),and since S < V 1,Step 1 updates the network (i) to (ii). Step 2 then computes an MA ordering v 1 = s v 2 = v e v 3 = v d v 4 = v a v 5 = v c (see (iii-1)) and merges v 4 = v a and v 5 = v c into v ac (see (iii-2)). Since c α v a <c α v c, Step 2 puts α v ac =α v a. Moreover,by κ v ac < 8,Step 2 updates by merging s and v ac into s,and S = S α v ac = v a (see (iii-3)). Since V =3,the algorithm returns to Step 2. Step 2 again computes an MA ordering v 1 = s v 2 = v d v 3 = v e (see (iv-1)) and merges v 2 = v d and v 3 = v e into v de (see (iv-2)). Since c α v d <c α v e,step 2 puts α v de =α v d. Moreover,by κ v de < 8,Step 2 updates by merging s and v de into s,and S = S α v de = v d (see (iv-3)). Now,since V =1,Step 3 outputs S = v a v b v d whose cost is 6. Theorem 4.1. Problem Source Location can be solved in O n m + n log n time if the demand function d is uniform. Proof. Since the above discussion shows the correctness of algorithm Contract,we only consider its time complexity. Clearly Steps 0,1,and 3 take O n + m time. Step 2 can be executed in O n m + n log n time since it has n 1 iterations and each iteration takes O m + n log n time by Lemma 4.1. Therefore,in total,it requires O n m + n log n time. In the uniform demand case,minimal deficient sets are so-called extreme sets explored by Watanabe and Nakamura [14]. An extreme set is a vertex subset X such that κ Y >κ X for any proper nonempty subset Y X. Any efficient algorithm for finding all the extreme sets can be used to solve our problem. In particular,a recent randomized algorithm of Benczúr and

13 66 arata et al. FIG. 4. Algorithm Contract applied to the network in Fig. 3. Karger [3] leads to an Õ n2 algorithm that solves our problem with high probability. 5. NP-HARDNESS OF GENERAL CASE In this section,we show the NP-hardness of Source Location with nonuniform cost and demand functions. Theorem 5.1. Problem Source Location is NP-hard, even if the undirected graph G = V E is a star; i.e., E = v w w V \ v for some v V. Proof. We transform the NP-hard problem Knapsack to this problem [4]. Problem. Knapsack.

14 source location in networks 67 Input. A finite set Z = z 1 z 2 z n associated with a size function σ Z Z + and a value function τ Z Z + and a positive integer β z i Z σ z i,where Z + denotes the set of all nonnegative integers. Output. A subset X Z that is an optimal solution of Maximize τ z subject to z X σ z β (10) z X X Z It is easy to see that Knapsack is polynomially equivalent to the problem of computing a subset Y Z that solves Minimize τ z z Y subject to σ z σ z β z Y z Z (11) Y Z by identifying Y with Z X. Therefore,in the following we consider (11) instead of (10). For this problem instance,we consider an undirected network = G = V E u d c with V = Z z 0, E = z 0 z i z i Z, u z 0 z i = σ z i for i = 1 2 n,and σ z i β if i = 0 d z i = z i Z 0 otherwise, τ z i +1 if i = 0 c z i = z i Z τ z i otherwise. Note that d z i =0 for all z i Z. Therefore S V covers all vertices in V if and only if it covers z 0 ; i.e., λ S z 0 d z 0 = σ z i β (12) Moreover,since z 0 and Z covers z 0 and since c z 0 > z i Z c z i,an optimal S is contained in Z. This implies that λ S z 0 = u z 0 z i = σ z i (13) z i S z i S and hence (12) is equivalent to the constraint in (11). Since c z i =τ z i for all z i Z, S Z is an optimal solution for the instance of problem (11) if and only if it is optimal for the corresponding instance for Source Location. z i Z

15 68 arata et al. 6. CONCLUSION In this paper,we have analyzed the greedy algorithm of Tamura et al. [13] for Source Location with a uniform cost function and given a simpler proof based on the linear programming duality. We have also improved the Greedy algorithm to run in O nm n m time. Moreover,we have given an O n m + n log n time algorithm for Source Location with a uniform demand function. Finally,we have shown that Source Location is in general NP-hard by reducing Knapsack to Source Location. Thus, it remains open to prove the NP-hardness in the strong sense or to devise a pseudo-polynomial time algorithm. ACKNOWLEDGMENTS The authors thank the anonymous referees for their helpful and constructive comments, which improved the presentation of this paper. In particular,one of the reviewers suggested us a simpler version of algorithm Contract. REFERENCES 1. R. K. Ahuja,T. L. Magnanti,and J. B. Orlin, Network Flows, Prentice Hall,Englewood Cliffs,NJ., K. Arata,S. Iwata,K. Makino,and S. Fujishige,Locating sources to meet flow demands in undirected networks, in SWAT2000 (M. M. Halldorsson,Ed.),Springer Lecture Notes in Computer Science,Vol. 1851,pp ,Springer-Verlag,Berlin/New York, A. A. Benczúr and D. R. Karger,Augmenting undirected edge connectivity in Õ n2 time, J. Algorithms, 37 (2000), M. R. Garey and D. S. Johnson, Computers and Intractability, Freeman,New York, A. V. Goldberg and S. Rao,Beyond the flow decomposition barrier,j. ACM 45 (1998), A. V. Goldberg and S. Rao,Flows in undirected unit capacity networks,siam J. Discrete Math. 12 (1999), H. Ito,H. Uehara,and M. Yokoyama,A faster and flexible algorithm for a location problem on undirected flow networks, IEICE Trans. Fund. E83-A (2000), H. Nagamochi and T. Ibaraki,Computing edge-connectivity of multigraphs and capacitated graphs, SIAM J. Discrete Math. 5 (1992), H. Nagamochi and T. Ibaraki,Deterministic Õ nm time edge-splitting in undirected graphs, J. Combin. Optim. 1 (1997), H. Nagamochi and T. Ibaraki,A note on minimizing submodular functions,inform. Process. Lett. 67 (1998), M. Stoer and F. Wagner,A simple min cut algorithm,j. ACM 44 (1997) H. Tamura,M. Sengoku,S. Shinoda,and T. Abe,Some covering problems in location theory on flow networks, IEICE Trans. Fund. E75-A (1992), H. Tamura,H. Sugawara,M. Sengoku,and S. Shinoda,Plural cover problem on undirected flow networks (in Japanese), IEICE Trans. Fund. J81-A (1998), T. Watanabe and A. Nakamura,Edge connectivity augmentation problems,j. Comput. System Sci. 53 (1987),