Search for all d-mincuts of a limited-flow network

Computers&Operations Research29(2002)1843–1858/locate/dsw Search for all d-Mincuts of a limited- ow networkWei-Chang Yeh∗Department of Industrial Engineering,Feng Chia University,P.O.Box67-100,Taichung407,TaiwanReceived1May2000;received in revised form1February2001AbstractIn this presentation,we developed an intuitive algorithm based on some simple concepts that were found in this study.It is more e cient than the best-known existing algorithm.The computational complexity of the proposed algorithm is analyzed and compared with the existing methods.One example is illustrated to show how all d-MCs are generated by our proposed algorithm.As evidence of the utility of the proposed approach,extensive computational results on random test problems are presented.Our results compare favorably with previously developed algorithms in the literature.Scope and purposeMany real-world systems are multistate systems composed of multistate components in which the reliability can be computed in terms of the lower bound points of level d,called d-Mincuts(d-MCs). Such systems(electric power,transportation,etc.)may be regarded as ow networks whose arcs have independent,discrete,limited and multivalued random capacities.In this study,all MCs are assumed to be known in advance and we focused on how toÿnd the entire d-MCs before calculating the reliability value of a network.Analysis of our algorithm and comparison to existing algorithms shows that our proposed method has the following advantages:(1)the number of d-MC candidates is less than those in the existing methods,(2)the d-MC candidates are simpler toÿnd and verify,which makes the method more e ective than the existing methods,and(3)our method is easier to understand and implement.? 2002Elsevier Science Ltd.All rights reserved.Keywords:Reliability;Limited- ow network;d-MC;Max- ow1.IntroductionMultistate systems are more practical and reasonable in many real-life situations.Analysis of multistate systems extends to binary-state systems and has become a new subject in system ∗Tel.:+886-4-2321-6168;fax:886-4-2208-4168.E-mail address:wcyeh@.tw(W.-C.Yeh).0305-0548/02/$-see front matter?2002Elsevier Science Ltd.All rights reserved.PII:S0305-0548(01)00062-41844W.-C.Yeh/Computers&Operations Research29(2002)1843–1858reliability.A limited- ow network is a multistate system composed of multistate components. For example,an electric power transmission and distribution system may supply power at dif-ferent levels and only certain levels will satisfy the load(or demand,stress)level.The reliability evaluation of a multistate system is NP-hard[11–13]but practical.Several authors have presented some approaches to evaluate the probability that the system capacity level is d[1–10,14–21].Most of these methods are formulated in terms of either d-MPs or d-MCs.However,it seems that only Xue[5,6],Jane et al.[1,17],and Yeh[7,8]have presented algorithms to search for all d-MPs and=or d-MCs for a general multistate case.Many of the steps in their method seem to be super uous when applied to limited- ow networks[1,5,6,8]. Even for a smaller sized problem,it is still impossible to evaluate the reliability of a multistate system in a reasonable time using their algorithm.The need for an e cient and intuitive method to search for all d-MCs thus arises.The purpose of this paper is to develop an intuitive algorithm based on some simple concepts that were found in this study to search for all d-MCs.It is more e cient than the best-known existing algorithm.The computational complexity of the proposed algorithm is analyzed and compared with the existing methods.As evidence of the utility of the proposed approach,we present extensive computational results on random test problems.The results compare favorably with previously developed algorithms in the literature.The paper is organized as follows.Section2describes the notation,nomenclature and as-sumptions required.Section3contains the formulation for d-MCs in a limited- ow network, and some lemmas and theorems are discussed.In Section4,the best-known algorithm[8] (Algorithm0)is discussed.Section5describes our main algorithm(Algorithm1)which is an improvement upon the best-known algorithm(Algorithm0)using some simple concepts that are discussed in Section3.Section6illustrates how to generate all d-MCs in a network through a numerical example using putational experiments are provided in Section7. Concluding remarks are given in Section8.2.Notation,nomenclature and assumptions2.1.NotationG(V;E;W)a ow network with the set of nodes V={v i|16i6n},the set of arcs E={e i|16i6m},and W denotes the max-capacity of arcsn;m the number of nodes in V and arcs in E,respectivelys;t s;t∈V are the speciÿed source node and sink node,respectivelye i the i th arc in E(v i;v j)the arc(in E)from node v i to node v j,where v i and v j∈VO(e i)a system vector in which the capacity level is1for e i and0for other arcsW(e);w(e)the max-capacity and capacity level of arc e∈E in G(V;E;W),respec-tivelyC i the i th MC in G(V;E;W)W.-C.Yeh /Computers &Operations Research 29(2002)1843–18581845C ij ˙d (C i )C ij =(x 1;x 2;:::;x m )is the j th d -MC candidate (system-state)vector gen-erated from C i in G (V;E;W ),where x k =w (e k ), x k =d for e k ∈C i and x l =W (e l )for e l ∈C i C ij ˙d (C i )and C ij is (not)a d -MC G (V;E;C ij )the network corresponds to G (V;E;W )except C ij (e k )=x k 6W (e k )for e k ∈C i and C ij (e k )=W (e k )=x k for e k ∈C i ,if C ij =(x 1;x 2;:::;x m )G d;k (V;E;C ij )the remaining ow network after sending d capacity through node s to node t in G (V;E;C ij ),and then increase one capacity level of the k th arc in C i C i (e k )the max-capacity level of e k ∈E in G (V;E;C ij ),where ∀C ij ˙d (C i )M (X )the system max- ow from node s to node t under the system-state X X 6(=;¡)YX =(x 1;x 2;:::;x m )6(=;¡)Y =(y 1;y 2;:::;y m )i.f.f.x i 6(=;¡)y i for all i =1;2;:::;m MC(MP)mincut(minpath)2.2.Nomenclatured -MC for 06d ¡the value of max- ow in G (V;E;W ),a system-state vector,whose M (X )=d ,is said to have the capacity level d .A system-state vector X is a d -MC if and only if (1)its capacity level is d and (2)M (X +o (e i ))¿d for each e i ∈E and w (e i )¡W (e i )Demand (or load)level dthe demand level d is a non-negative integer-valued load or stress requirement for a given ually it is a random variable whose distribution { d }can be determined through con-tinuous observation and forecasting System reliability if the demand level d is a deterministic constant,then R d =Pr {C :M (C )¿d }is the system reliability.If the de-mand d is random,then the system reliability R d is deÿned as Pr {X :M (X )¿d } d .These two are consistent with the general reliability deÿnition viz.R d =Pr {strength ¿stress }Assumptions.The limited- ow network satisÿes the following assumptions [1,7,8,22–24]:Each node is perfectly reliable.The capacity of each arc is a non-negative integer-valued random variable according to a given distribution.The capacities of di erent arcs are statistically inde-pendent.All ows in the network obey the conservation law.3.Model formulation and preliminariesIf the ow network is series–parallel,then each d -MC candidate is a real d -MC [1].There-fore,we only focus on a non-series–parallel ow network in the remainder of this study.The relationship between the max- ow and the min s –t cut in Theorem 1is a very fundamental concept.Its proof can be found in any textbook about network ow theory [22–24].1846W.-C.Yeh /Computers &Operations Research 29(2002)1843–1858Theorem 1.The value of max- ow from source node s to sink node t equals the value of the min s–t cut in G (V;E;W ).The following theorem is originated and proved from Jane et al.[1].It is described that an MC satisfying three conditions simultaneously in Theorem 2is a d -MC candidate.It is used in the Implicit Algorithm of our paper to generate d -MC candidates.Theorem 2.Let C be a d-MC ;then there exists at least one MC C i ;such that the following three conditions hold : C i (e )=d for all e ∈C i ,C i (e )6W (e )for all e ∈C i ;C i (e )=W (e )for all e ∈C i .At least one arc exists in common between any d -MC candidate,say C ij ,and any min s –t cuts in G (V;E;C ij ).Lemma 1.Let C ij ˙d (C i ).Then C i ∩C ∗=∅;if C ∗is any one of min s–t cuts in G (V;E;C ij ).Proof.If there exists a min s –t cut C ∗in G (V;E;C ij )such that C i ∩C ∗=∅,then C ∗is also a min s –t cut in G (V;E;W ),i.e.this is contradictory to the case in which d is less than the value of max- ow in G (V;E;W ).Therefore,the proof follows.The following discusses some special properties about the d -MC candidates.Lemma 2.Let C ij ˙d (C i );S ij be a set of min s–t cuts in G (V;E;C ij );and C ∗∈S ij −{C i }.If C ik (e )¡ C ij (e );where ∀e ∈C i ∩C ∗and ∀C ik ˙d (C i );then C ik #d (C i ).Proof.Since C ij (e ∗)=C ik (e ∗)=W (e ∗)for all e ∗∈C ∗−C i ,if C ik (e )¡ C ij (e )for all e ∈C i ∩C ∗,then C ik (C ∗)6 C ik (e )+ C ik (e ∗)¡ C ij (e )+ W (e ∗)6d .Therefore,we have M (C ik )¡d ,i.e.C ik #d (C i ).From Lemma 2,we have the following result.Corollary1.Let C ij ˙d (C i );S ij be a set of min s–t cuts in G (V;E;C ij );and C ∗∈S ij −{C i }.If C ik ∼=d (C i );then C ik (e )¿ C ij (e );where ∀e ∈C i ∩C ∗.Lemma 3.Let C ij ˙d (C i );S ij be a set of min s–t cuts in G (V;E;C ij );and C ∗∈S ij −{C i }.If C ik ∼=d (C i )and W (e ∗)¿C ij (e ∗);then C ik (e )¿ C ij (e );where ∀e ∈C i ∩C ∗and ∃e ∗∈C i −C ∗.Proof.From Corollary 1,we already have C ik (e )¿ C ij (e ),where ∀e ∈C i ∩C ∗and C ik ∼=d (C i ).IfC ik (e )= C ij (e )and W (e ∗)¿C ij (e ∗),where ∀e ∈C i ∩C ∗and ∃e ∗∈C i −C ∗,thenW.-C.Yeh /Computers &Operations Research 29(2002)1843–18581847M (C ik +o (e ∗))=C ik (C ∗)=d .This is in contrast to the case in which C ik ∼=d (C i ).Thus,the proof follows.Corollary2.Let C ij ˙d (C i )with M (C ij )=d .If there is only an arc ;e ∈C i ;such that C ij (e )¡W (e )and e is also in a min s–t cut in G (V;E;C ij );then C ij is a d-MC .The following theorem is directly obtained from Theorem 2and Lemma 3.It is one of the most important results in this study.It will be employed in our algorithm to reduce the number of d -MC candidates.It also reduces the total time to verify which d -MC candidate is indeed a d -MC.Theorem 3.If S i is a set of min s–t cuts in G (V;E;C ij )for all C ij ˙d (C i );then ;C i (e )=d f or all e ∈C i ;C i (e )6W (e )f or all e ∈C i ;C i (e )=W (e )f or all e ∈C i ; C i (e )¿ C ij (e );where ∀e ∈C i ∩C ∗;∀C ∗∈S i −C i =∅and ∃e ∗∈C i −C ∗such that W (e ∗)¿C ij (e ∗):From the deÿnition of d -MC,a d -MC candidate is not a d -MC,if its capacity level is less than d .The following two theorems will be put in our algorithm to reduce the time of verifying which d -MC candidate with capacity level d is indeed a d -MC.Theorem 4.If C ij ˙d (C i );M (C ij )=d;and there is no other min s–t cut in G (V;E;C ij );then C ij ∼=d (C i ).Proof.C ij is the only one min s –t cut in G (V;E;C ij ).Thus,M (C ij +o (e ))¿d for all e ∈C i and C ij (e )¡W (e ).Therefore,C ij ∼=d (C i ).Theorem 5.Let C ij ˙d (C i );M (C ij )=d;S ij be a set of min s–t cuts in G (V;E;C ij );and E c ={e |e ∈C i −C ∗;∀C ∗∈S ij −{C i }}.If E c =∅or C ij (e )=W (e )for all e ∈E c ;then C ij ∼=d (C i ).Proof.Since C ij ˙d (C i )and M (C ij )=d ,if M (C ij +o (e ))¿d for all e ∈C i and C ij (e )¡W (e ),then C ij ∼=d (C i )from the deÿnition of d -MC.Any arc in C i −E c belongs to at least one min s –t cut in S ij −{C i },i.e.M (C ij +o (e ))¿d for all e ∈C i −E c while C ij (e )¡W (e ).If E c =∅,then C i −E c =C i .If C ij (e )=W (e )for all e ∈E c ,then M (C ij +o (e ∗))¿d for all e ∗∈C i −E c and Cij (e ∗)¡W (e ∗).The theorem now follows.Two consecutive d -MC candidates are generated from the same MC with nearly the same system vector.From this special structure,the max- ow of one of the two consecutive d -MC candidates may be easy to ÿnd without using any existing max- ow algorithm [25–27],if the1848W.-C.Yeh/Computers&Operations Research29(2002)1843–1858max- ow of the other candidate is known.The following theorem is another important result in our study.It originates from the above simple concept.It is also more e cient than any other max- ow algorithm[25–27]and will be implemented in our algorithm to reduce the run time forÿnding the max- ow.Theorem6.Let C ij˙d(C i);C ik˙d(C i);and C ik=C ij−o(e x)+o(e y).1unit capacity can be sent back from node t to node s through e x but not e y in G(V;E;C ij).After that;let C ij(e y)=C ij(e y)+1;if only1more unit capacity can be sent from node s to node t through e y but not e x;then M(C ij)=M(C ik);otherwise M(C ij)=M(C ik).Proof.Trivial.4.The best-known algorithmTo compare the e ciency and the time complexity between the proposed algorithm and best-known algorithm[8],the best-known algorithm(Algorithm0)is discussed in this sec-tion.Assume that all MCs also have been known in advance,the family of all d-MCs of an MC C i can then be found and checked using the best-known algorithm in the following steps:Algorithm0:[8]0.Let j←1and S i←∅.e the Implicit Algorithm toÿnd a d-MC candidate,say C ij(∈S i),of the followingmathematical model:C i(e)=d for all e∈C i;C i(e)6W(e)for all e∈C i;C i(e)=W(e)for all e∈C i:If no such d-MC candidate exists,then halt.2.If M(C ij)=d then k←1.Otherwise,j←j+1and go to Step1.3.If the capacity level of e k∈E in G(V;E;C ij)is equal to W(e k),then go to Step5.4.If M(C ij+o(e k))=d then j←j+1and go to Step1.5.If k¡m,then k←k+1and go to Step3.Otherwise C ij is d-MC,S i←S i∪{C ij};j←j+1and go to Step1.Algorithm0is based only on the deÿnition of d-MC with the time complexity O(m 0) [8],where 0is the total number of d-MC candidates and O( )is the time complexity to ÿnd the max- ow.However,there is still ample room to improve Algorithm0.For example, the number of d-MC candidates,which is found using the time-consuming and very burden-some Implicit Algorithm,can be reduced if the proposed properties discussed in Theorem3 are implemented.Next,Theorems4and5also can be implemented in Algorithm0to checkW.-C.Yeh /Computers &Operations Research 29(2002)1843–18581849if a d -MC candidate is a real d -MC while its capacity level is d .The max-capacity of the arcs between two consecutive d -MC candidates found in the Implicit Algorithm is nearly the same,except if there are only two arcs:one is 1unit capacity larger and the other is 1unit capacity smaller than that in the other d -MC candidate.Therefore,Theorem 6may be a better way to ÿnd the max- ow of each d -MC candidate than the most e cient max- ow algorithm.According to the above discussions,we propose a more e cient algorithm in the next section.5.The proposed algorithmA simpler but more e cient method (Algorithm 1)than Algorithm 0to ÿnd all d -MCs,where d is less than the max- ow in G (V;E;W ),generated from an MC is presented next.Suppose that all MCs have been precomputed,the family of all d -MCs of each MC can then be derived using the following steps:Algorithm 1:1.0.Let j ←0;C ij ←∅;S i ←∅,and V c ←{a |a ∈V ,and (v;a )or (a;v )in C ij ,where v ∈V }.1.1.(Implement Theorem 3)Let j ←j +1and use the Implicit Algorithm to ÿnd a d -MC candidate C ij of the following mathematical model: C i (e )=d for all e ∈C i ;C i (e )6W (e )for all e ∈C i ;C i (e )=W (e )for all e ∈C i ; C i (e )¿ C ij (e );where e ∈C i ∩C ∗k ;for all C ∗k ∈S i =∅:If no such d -MC candidate exists,then halt.1.2.If C ij =C i;j −1+o (e x )−o (e y ),then calculate M (C ij )by using Algorithm 2,otherwise use the max- ow e the max- ow algorithm to ÿnd M (C ij ).If M (C ij )¡d ,then C ij #d (C i )and go to Step 1.1.1.4.Check C ij and ÿnd a min s –t cut set S i by using Algorithm 3.Go to Step 1.1.Algorithm 2:(Implement Theorem 6)2.0.Send 1unit capacity from node t to node s through e x but not e y .2.1.Send 1unit capacity from node s to node t through e y but not e x .If this is impossible,then C ij #d (C i )and return to Step 1.1,otherwise M (C ij )=d and return to Step 1.4.Algorithm 3:(Implement Theorems 4and 5under the assumption of M (C ij )=d )3.0.Let V s ←V c ;V r ←V;E c ←∅, ag ←0,and E r ←E −(∪{e }),where ∀e ∈E and w (e )=C ij (e ).1850W.-C.Yeh /Computers &Operations Research 29(2002)1843–18583.1.If ∃v s ∈V s ,then V s ←V s −{v s };V r ←V r −{v s };V ∗←{v s },and E s ←∅,otherwise go to Step 3.6.3.2.Let v ∈V ∗and V ∗←V ∗−{v }.If A ={a |a ∈V r with (v;a )or (a;v )in E r }=∅,then V ∗←V ∗∪A;V r ←V r −A;V s ←V s −A ,and repeat Step 3.2.3.3.Let E s ←E s ∪{e |e =(v;a )or (a;v )in E −E r and a ∈V r }.If V ∗=∅,then go to Step 3.2.3.4.If E s =C i ,then C ∗ij ←(C i −E s )∪(E s −C i )and ag ←1,otherwise go to Step 3.1.3.5.If C ij (e )=d for ∀e ∈C ∗ij ,then S i ←S i ∪{C ∗ij },E c ←E c ∪(C i −E s ),and go to Step 3.1.3.6.(Implement Theorem 4)If ag =0,then C ij ∼=d (C i )and return to Step 1.1.3.7.(Implement Theorem 5)If E c =∅or C ij (e )=W (e )for all e ∈E c ,then C ij ∼=d (C i )and return to Step 1.1,otherwise,let E ∗c ←E c −{e |e ∈E c with C ij (e )=W (e )}.3.8.If ∃e ∈E ∗c ,then E ∗c ←E ∗c −{e }and C ij (e )←C ij (e )+1,else C ij ∼=d (C i )and return to Step 1.1.3.9.If 1unit capacity can be sent send from node s to node t through e ,then C ij #d (C i )and return to Step 1.1,otherwise,C ij (e )←C ij (e )−1and go to Step 3.9.From Algorithm 2,it takes O (2m )to send 1unit capacity from node s back to node t through two di erent directed paths.We have thus proved the following lemma.Lemma 4.Algorithm 2determines the capacity level of a d-MC candidate in O (m )time .It is not necessary for S i to be the set of all min s –t cuts in Algorithm 3.Otherwise,its number of elements will grow exponentially with the number of nodes.If C ∗∈S i =∅in Algorithm 3,then C ∗is a min s –t cut in G (V;E;C ij ),where ∃C ij ˙d (C i ).Moreover,C ∗∩C ij =∅from Lemma 1.Lemma 5.Algorithm 3runs in O (n )time to ÿnd the subset of S i (from Steps 3:0to 3:5)in G (V;E;C ij ).Proof.The construction of Algorithm 3investigates each node at most once only.Thus this lemma is implied.Lemma 6.Algorithm 3runs in O (m ·|C i |)time from Steps 3:8to 3:9.Proof.In the worst case,M (C ij +o (e ))must be larger than d for all e ∈C i from the deÿnition of d -MCs.It also takes O (m )time in Step 3.8to make sure that 1unit capacity can be sent from node s to node t through e .This result establishes the lemma.Followings directly from Lemmas 4–6,the following theorem summarizes the implication of the related result.Theorem 7.Algorithm 1ÿnds and checks the family of d -MC candidates of an MC in O ((mn +m ·|C i |) 1)=O (m 2 1)time ;where 1is the total number of d -MC candidates generated inW.-C.Yeh/Computers&Operations Research29(2002)1843–18581851 Step1:1of Algorithm1.Since ¿m; 0¿ 1,and from Theorem7,Algorithm1is more e cient than Algorithm0.6.An exampleThe next example illustrates our algorithm for the d-MCs problem.For the sake of conve-nience,we use lower bar(“”)in the number to denote the max-capacity of a relative arc in a d-MC candidate.Example1.There are4MCs in the following network(see Fig.1):C1={e1;e4};C2={e2;e5}, C3={e1;e3;e5}and C4={e2;e3;e4}.Use Algorithm1toÿnd and verify all2-MCs(d=2)in Fig.1,where nodes1and2are the source and sink nodes,respectively.The arc e is marked by W(e).Solve:(theÿrst d-MC candidate C11that generates from C1)1.0.Let j←0;C1j←∅;S1←∅; ag←0,and V c←{1;2;4}.1.1.By using the Implicit Algorithm(see Fig.2a),one of the feasible solu-tions of C1={e1;e4}that satisÿes C1(e1)+C1(e4)=2;(C1(e1);C1(e4))6(3;1),and (C1(e2);C1(e3);C1(e5))=(2;1;2)is C11=(2;2;1;0;2).1.2.According to the max- ow algorithm,the M(C11)=2in G(V;E;C11).1.3.From Algorithm3,there is no other min s–t cut in G(V;E;C11),i.e. ag=0,so C11isa2-MC.::(Generate from C3)1.0.Let j←0;C3j←∅;S3←∅,and V c←{1;2;3;4}.1.1.By using the Implicit Algorithm(see Fig.2b),one of the feasible solutions ofC3={e1;e3;e5}that satisÿes C3(e1)+C3(e3)+C3(e5)=2;(C3(e1);C3(e3);C3(e5)) 6(3;1;2)and(C3(e2);C3(e4))=(2;1)is C31=(2;2;0;1;0).1.2.According to the max- ow algorithm,the max- ow is2in G(V;E;C31).Fig.1.The network in Example1.1852W.-C.Yeh/Computers&Operations Research29(2002)1843–1858Fig.2.The2-MCs generated from MC by using the Implicit Algorithm.1.3.From Algorithm3,because S3←S3∪{C∗31};E c←C3−C∗31={e1;e3},and it isimpossible to send1more unit capacity from node s to node t,where C∗31={e2;e5}is another min s–t cut in G(V;E;C31),we have that C31is not a2-MC.1.1.By using the Implicit Algorithm(see Fig.2b)again,C32=(1;2;0;1;1)isanother of the feasible solutions of C3={e1;e3;e5}that C3(1)+ C3(e3)+C3(e5)=2;(C3(e1);C3(e3);C3(e5))6(3;1;2);(C3(e2);C3(e4))=(2;1)and C3(e5)¿C31(e5)=0(⇒C3(e1)=1).1.2.M(C32)=1from Algorithm2.Therefore,C32is not a2-MC.1.3.From Algorithm3,S3←S3∪{C∗32},where C∗32={e1;e4}is a min s–t cut in G(V;E;C32).1.1.By using the Implicit Algorithm again,there is an other2-MCcandidate of C3={e1;e3;e5}that satisÿed C3(e1)+C3(e3)+ C3(e5)=2;(C3(e1);C3(e3);C3(e5))6(3;1;2);(C3(e2);C3(e4))=(2;1);C3(e1)=1,and C3(e1)¿C32(e1)=1(⇒C3(e1)=2).::The result forÿnding and verifying all d-MCs by using Algorithm1is listed in Table1.putational resultTo compare the e ciency(running time)of the entire algorithm(Algorithm1)proposed in this paper against the best-known existing method(Algorithm0),these two algorithms were implemented in C++and carried out on a personal computer with Pentium III microproces-sor of1G MHz.The unit of the running time was(CPU)second.The capacity on each arc was randomly generated in discrete uniform distribution U[0;9].All of the non-integer values obtained from this experiment were rounded down to two digits,e.g.9:354and9:355were rounded down to9:35and9:36,respectively.The notations involved in the computation results from this experiment are listed below.•n;m;N MC;N Can and N d-MC:the average numbers of nodes,arcs,MCs,d-MC candidates and real d-MCs,respectively.Table1Final results of Example1i C i j C ij=(e1;e2;e3;e4;e5)A2-MC?E c Other min s–t cut(s) 1{e1;e4}1(2;2;1;0;2)Yes a∅no2(1;2;1;1;2)Yes a∅no2{e2;e5}1(3;2;1;1;0)Yes a∅no2(3;1;1;1;1)Yes a∅no3(3;0;1;1;2)Yes b{e5}{e2;e3;e4}3{e1;e3;e5}1(2;2;0;1;0)No c{e1;e3}{e2;e5}2(1;2;0;1;1)No d{e3;e5}{e1;e4}4{e2;e3;e4}1(3;2;0;0;2)Yes a∅no2(3;1;1;0;2)Yes a∅no3(3;1;0;1;2)Yes a∅no4(3;0;1;1;2)Yes b{e5}{e2;e5}a Since ag=0,i.e.no other min s–t cut in G(V;E;C ij)in Step3.6.b Since C ij(e)=W(e)in Step3.7for all e∈E c.c Since1unit capacity cannot be sent from node s to node t through e1after C31(e1)←C31(e1)+1.C3(e5)¿0⇒C3(e5)=1or2.d Since M(C32)=1¡2,we have C3(e1)¿1⇒C3(e1)=2.Table2Comparison of the results from n(n=5;6;:::;10)for Algorithms0and1n m N MC N Can N d-MC T1T0T0=T1N Can=N MC N d-MC=N MC N d-MC=N Can 59.177.40719.53203.540.000.0197.2327.510.28 613.6715.008419.732767.340.050.112.43561.32184.490.33 718.8030.1087,789.0035,754.600.92 2.933.182916.581187.860.41 825.7362.80860,672.04365,181.2317.6559.543.3713,704.975814.990.42 932.33125.2912,180,625.997,403,043.92554.432258.184.0797,222.7859,089.290.61 1040.95252.8727,319,651.5019,026,105.282943.5022,000.237.47108,037.9175,240.370.70•T0and T1:the average running time of Algorithms0and1,respectively.•T0=T1;N Can=N MC;N d-MC=N MC and N d-MC=N Can:the ratios of the values of T0and T1;N Can and N MC;N d-MC and N MC,and N d-MC and N Can,respectively.Due to the characteristics of NP-hard problem and the limitations of personal computers, the proposed algorithm solves only medium sized problems.Therefore,in the experiment,only six levels of n were employed;namely,n=5;6;:::;10.For each level of n;30problems were generated.Each set of problems was run for various values of d=2;3;:::;9.This totals up to 1440problems.parison of the e ciency of both algorithms for n=5;:::;10The results obtained by both algorithms for n=5;6;:::;10are presented in Table2.Table2 provides the average CPU times for the two algorithms for each n,the average numbers of arcs,MCs,d-MC candidates and real d-MCs.Three ratios are listed in the last three columnsof Table2to show the relationships among the number of MCs,d-MC candidates and real d-MCs.The ratio T0=T1is also shown in Table2to demonstrate the e ciency of the proposed algorithm.If there is a non-zero capacity arc between each pair of nodes,then there are n(n−1)=2 arcs in this network[23].The capacity on each arc was randomly generated in discrete uniform distribution U[0;9]in this experiment,i.e.the capacity of1=10arcs is expected to be zero. Therefore,m is close to9n(n−1)=20in Table2.The numbers of MCs,d-MC candidates and real d-MCs increase exponentially with n [1,7,8,13].The average running time also increases while the number of d-MC candidates and real d-MCs are increased.This can be seen from Table2where all of the values of T0;T1;N MC;N Can,and N d-MC tend to increase as n increases.However,an interesting observation is that the increasing speeds of the above values tend to decrease as n increases. For example,before rounding into two digits,the values of T1are0:0019494;0:0461359,and 0:9193073for n=5;6,and7,respectively,but the ratios of the two consecutive values of T1:0:0461359=0:0019494=23:67¿0:9193073=0:0461359=19:93decrease.The val-ues of N Can=N MC;N d-MC=N MC,and N d-MC=N Can also tend to increase as n increases.This results from(the increasing speed of the number of real d-MCs)¿(the increasing speed of the number of d-MC candidates)¿(the increasing speed of the number of MCs)while n is increasing. The results presented in Table2indicate that T0and T1are all less than one CPU second for n=5and6,while T0quickly climbs to almost1min for n=8.As n increases,the running times for both algorithms become prohibitively long.The maximum average running time T0, which occurs when n=10;is more than6h,but still less than50min for T1.From the same table,as n increases,T0=T1increases.Therefore,the larger the value for n,the more e cient Algorithm1is over Algorithm0.parison of the e ciency of both algorithms for d=2;:::;9Next,the in uence of d on the problem di culty is analyzed.From the same experiment, there are a total of30×6=180di erent data sets for each d.The corresponding results for di erent values of d from2,3to9,are collected and compared in Table3.The values of T0;T1;N Can and N d-MC all tend to increase as d increases in Table3.An inter-esting observation,similar to Section7.1,is that the increasing speeds of the above valuesTable3Comparison of the results from d(d=2;3;:::;9)for Algorithms0and1d N Can N d-MC N d-MC=N Can T1T0T0=T1 212,475.3311,974.730.96 2.34 4.17 1.78 378,765.6374,212.730.9413.8343.04 3.11 4381,541.34346,795.980.9163.75157.82 2.48 51,576,096.331,405,574.530.89252.89693.46 2.74 65,948,653.865,328,844.910.90583.812368.62 4.06 716,831,240.872,324,673.610.141481.236206.24 4.19 833,182,680.614,103,301.390.123019.3312,449.99 4.12 945,536,136.394,735,861.490.102696.5710,197.90 3.78tend to decrease as d increases.For example,the values of T1are2:34;13:83,and63:75 for d=2;3,and4,respectively,but the ratios of the two consecutive values of T1:13:83=2:34¿63:75=13:83decrease.The value of N d-MC=N Can decreases as d increases.The computational results presented in Table3indicate an increase in the problem di culty with an increase in d except at d=9.This may be due to the fact that the maximal capacity of each arc is only9.From T0=T1in Table3,Algorithm1is more e cient than the best-known algorithm, implemented in Algorithm0.parison of the e ciency of both algorithms for both d and nFinally,how the factors of both d and n simultaneously a ect the number of d-MC candidates, real d-MCs and average running time is discussed as follows.From the same experiment, the corresponding result of T0;T1,the number of d-MC candidates and real d-MCs for each (n;d)pair(d=2;3;:::;9and n=5;6;:::;10)are summarized in Table4.There are in total,30 di erent problems for each(n;d)pair.It is very obvious that T1is less than T0in all theTable4Comparison of the results from n=5;6;:::;10and d=2;3;:::;9for Algorithms0and1n\d2*******T150.000.000.000.000.000.000.000.0060.010.010.020.030.050.070.080.0970.020.060.170.400.79 1.46 2.19 2.2580.150.66 2.15 5.7515.0832.2350.8534.329 1.06 5.4422.3980.11258.80696.951377.711993.021012.8176.83357.751431.073228.148156.6516685.1114149.72T050.000.000.000.010.000.010.010.0060.010.020.050.090.150.200.190.1970.050.210.63 1.41 2.85 5.237.61 5.4180.40 2.107.5321.4660.24130.55193.9360.119 2.7416.8278.45308.601070.452997.645834.937755.851021.82239.10860.263829.1813078.0034103.8168663.2553365.86N Can5102.90216.80371.77571.43802.501048.771263.771378.30 6416.331120.632473.974899.108593.0313178.5016851.3719824.93 71526.905366.1314918.1733337.5366378.37125501.60198667.47256615.80 85416.9725070.0083772.37222739.13574213.471268862.732243057.502462244.13 917203.0093843.97397820.071465486.004837389.7713472080.4329162976.3047998208.37 1050185.90346976.271789891.707729544.8030204546.0086106773.20167473267.29222478546.80N d-MC587.80162.33237.27293.23308.00292.83200.4746.37 6351.43843.201631.202902.904255.275042.333768.333344.03 71416.804615.5711688.0022907.4740769.2068652.1790190.8745796.73 85079.7022654.4369537.40171457.83441638.40860704.131164663.10185714.83 916546.8086927.90354779.571277369.934097960.8710688676.5719257684.2023444405.50 1048365.87330072.971642902.436958515.8327388137.7068615613.8097438486.9335791005.80。