7 Max-Flow Problems. Business Computing and Operations Research 608

Transcription

1 7 Max-Flow Problems Business Computing and Operations Researh 68

2 7. Max-Flow Problems In what follows, we onsider a somewhat modified problem onstellation Instead of osts of transmission, vetor now indiates a maximum apaity that has to be obeyed Again, we onsider a networ with two speifially assigned verties s and t The objetive is to find a maximum flow from soure s to sin t E.g., this flow may be a transport of materials from an origin to a destination of onsumption Business Computing and Operations Researh 69

3 Flow Inflow and outflow 7.. Definition N = ( V E ) f E [ ] ( s,t) Assuming a networ,, is given as above. A mapping :, is denoted as an flow if and only if the following attributes apply: ( ) ( ) ( ) f ( i j). f e e, e E 2., ( s, i) ( ) (( )) ( ) (( )) = f j, i, i V : i s i t j V : i, j E j V : j, i E Outflow from node i Inflow of node i f = f s, i is denoted as the amount of flow. f is denoted E as the maximum flow if and only if f is maximally hosen. Business Computing and Operations Researh 6

4 Observation We an transform the equalities (2), whih are itemized above, as follows ( i, j ) f (( i, j) ) j V : E Outflow from node i ( j, i ) f (( j, i) ) j V : E Inflow of node i = f i = s f i = t otherwise ~ Let E = E matrix of { e },e = ( t,s) ~ and ( ) m V, E. Then, A f = f = f. A the vertex - ar adjaeny Business Computing and Operations Researh 6

5 Conlusions For what follows, we renumber the ars, beginning with, i.e., we obtain n ars with the numbering,2,3,,n Note that this inludes the artifiial ar (now ), onneting terminal t with soure s We now that,..., A =,..., A f = =,..., A f = ( ) ( ) ( ) ( ) A f,..., A f,..., A f ( ) ( ) ( ) ( ),..., A f = f = A f = A f = A f A f A f = Business Computing and Operations Researh 62

6 Max-Flow Problem Maximize f, s.t., A f f f. m A I.e., E n f, min (, j), ( i, n ) n j V: j> i V: i< n E n Maximum outflow from s= Maximum inflow to n= t Business Computing and Operations Researh 63

7 The dual of Max-Flow Now, we onsider π ɶ = π,γ,δ, with ( ) ( ) ( ) and δ ( δ,...,δ ) π = π,...,π,γ = γ,...,γ, = m n n n Minimize T s.t., ( ) l γ l, A π + γ δ = e π,γ,δ l= Business Computing and Operations Researh 64

8 Interpreting the dual This time, the dual is given in standard form, i.e., the Simplex Algorithm an be diretly applied to it Thus, we want to analyze it beforehand Let us onsider the equalities that have to be fulfilled Then, we an transform as follows n Minimize γ, s.t., l= l l ( ) ( ) ( ) if e = t,s E πi π j + γ δ = if e = i, j E e t,s E ( ) ( ) ( ) π = π,...,π,γ = γ,...,γ, and δ = δ,...,δ m n n Business Computing and Operations Researh 65

9 Business Computing and Operations Researh 66 The dual tableau Obviously, by onduting the alulation of the Primal Simplex, we obtain a tableau as follows n B n B T B B T T T T T T n B n B T B B n B T B B T B T T B T B B T B n n T T E A E A A A e A f f A f e f E A E A A A e A E A A A A e A E E A e +

10 Applying the simplex The top row of the dual tableau provides omprehensive information about the urrent state of the alulation Speifially, it allows a diret lin to the orresponding primal problem whih has to be solved originally More preisely, we have the following data in the row with: T T T T T T T B B B n B n ( ) T f e f : Objetive funtion value of P T T f A : Flow balane in the verties, i.e., is = for feasible f T T f : f e f A f f = A e A A A E A E Remaining apaity of the ars ( ) T f : Current orresponding solution to P Business Computing and Operations Researh 67

11 Business Computing and Operations Researh 68 A simple example = = A

12 Business Computing and Operations Researh 69 Applying the Simplex Step

13 Business Computing and Operations Researh 62 Applying the Simplex Step

14 Business Computing and Operations Researh 62 Applying the Simplex Step 2. [ ] ( )

15 Business Computing and Operations Researh 622 Applying the Simplex Step

16 Business Computing and Operations Researh 623 Applying the Simplex Step 3. [ ] ( )

17 Business Computing and Operations Researh 624 Applying the Simplex Step

18 Business Computing and Operations Researh 625 Applying the Simplex Step 4. [ ] ( )

19 Applying the Simplex Step Business Computing and Operations Researh 626

20 Applying the Simplex Step 5. [ ] ( ) Business Computing and Operations Researh 627

21 Applying the Simplex Step ( ) ( ) ( ) ( ) γ ( ) δ = ( ) f =,,,,, πɶ = π,γ,δ = π, i.e., = = Business Computing and Operations Researh 628

22 7.2. Definition: 7.2 Min-Cut Problems ( V E s t) Assuming N =,,,, is a networ with two labeled nodes s and t. A partition V = W W is denoted as an s - t ut if and only if s W and t W. ( i, j) is denoted as the apaity of the ut. ( ) i, j E with i W j W ( W,W ) ( i, j) A ut is denoted as a minimum ut if is minimal. ( ) i, j E with i W j W Business Computing and Operations Researh 629

23 Illustration i j s W i 2 j 2 W t j 3 i 3 j 4 i 4 j 5 Business Computing and Operations Researh 63

24 ( ) Problem definition T if i W We introdue π = ( π,...,πm ), with πi = if i W and T γ = γ,...,γn, with γ = ( ) if e = i, j i W j W otherwise Sine i W j W π = π = π π = and i W j = ( ) ( ) ( ) i j i j W π = π = π π =, we obtain the following problem: Minimize n γ i j i j, s.t., e = i, j t, s E : π π + γ π π + γ i j t s n T Minimize l γl, s.t., A π + γ δ = e ( π,γ,δ) l= Business Computing and Operations Researh 63

25 Observation The Min-Cut Problem orresponds to the dual of the Max-Flow Problem Thus, there is a diret onnetion between Min- Cut and Max-Flow Clearly, sine it is required that s and t belong to different parts of the ut, the Max-Flow is idential to the Min-Cut This beomes diretly oneivable by the fat that the Min-Cut is somehow the bottlene for the Max-Flow that may run through the entire networ Business Computing and Operations Researh 632

26 Consequene Lemma: ( W,W ) To every s - t ut, there exists a feasible solution to the dual of the Max-Flow Problem with the objetive funtion value ( ) W,W Business Computing and Operations Researh 633

27 Proof of Lemma Consider the following solution to the dual problem that has been generated aording to a given s-t ut π i if = if i γ = i W W if e = i, j i W j W if δ = ( ) otherwise ( ) ( ) e = i, j t,s i W j W otherwise Business Computing and Operations Researh 634

28 π i if i W = if i W Proof of Lemma ( ) if e = i, j i W j W γ = otherwise ( ) ( ) if e = i, j t,s i W j W δ = otherwise Let us onsider the possible ars of the networ. Speifially, we have to distinguish ( ) ( ) ( ) ( ) ( ) ( ). e = t,s π π + γ δ = + = t s 2. e = i, j t,s, with i W j W π π + γ δ = + = i j 3. e = i, j, with i W j W π π + γ i j δ = + = 4. e = i, j, with i W j W π π + γ δ = + = i j 5. e = i, j, with i W j W π π + γ δ = + = i j Business Computing and Operations Researh 635

29 The objetive funtion value We alulate the total weight of ars rossing the ut from W to W Thus, we may onlude ( ) = ( e ) W,W e = ( i,j),i W j W = e = ( i,j), γ ( e ) = = e E γ ( e ) Business Computing and Operations Researh 636

30 Diret onsequenes In what follows, our primal problem is n Minimize T s.t., ( ) l γ l, A π + γ δ = e π,γ,δ l= and the orresponding dual Maximize f, s.t., A f f f Business Computing and Operations Researh 637

31 Max-Flow-Min-Cut Theorem Theorem: ( ) ( ) ( ). For eah feasible s-t-flow f and eah feasible s-t ut W,W it holds: f W,W 2. A feasible s-t-flow f is maximal and the s-t ut W,W that is onstruted as defined in the Proof of Lemma is minimal if it holds: ( ) ( ) ( ) if e = i, j i W j W = if e = i, j i W j W 3. To a feasible Max-Flow f, there exists a Min-Cut with f f = W,W ( W,W ) Business Computing and Operations Researh 638

32 Proof of Theorem Part Sine the objetive funtion value of eah dual solution (Max-Flow) is a lower bound to eah feasible solution to the primal problem (Min-Cut), the proposition follows immediately. Business Computing and Operations Researh 639

33 Proof of Theorem Part 2 In order to prove the proposition 2, we mae use of the Theorem of the omplementary slaness, i.e., Theorem 5.. Speifially, we have to analyze the rows where the dual program leaves no sla at all. For this purpose, let us onsider the following alulations Sine f is assumed to be feasible, we now by the results obtained in Setion 7. that A f =. Consequently, the orresponding primal variables, i.e., π, may be defined arbitrarily. Business Computing and Operations Researh 64

34 Proof of Theorem Part 2 Let us now onsider ( ) ( ) ( ) if e = i, j i W j W En f f, e E f = if e = i, j i W j W Corresponding variables are γ. These variables are defined aordingly, i.e., if e = i, j i W j W γ = otherwise Thus, whenever there is no gap in the dual (this is the ase if one-value of the primal does not disturb. Other way round, if there is a gap in the dual (this is the ase if f ), the = ), the primal fixes it by zero-values. f = Business Computing and Operations Researh 64

35 Proof of Theorem Part 2 Finally, we onsider ( ) ( ) ( ) if e = i, j i W j W En f f, e E f = if e = i, j i W j W Corresponding variables are δ. These variables are defined just reversely, i.e., δ if e = i, j i W j W = otherwise Thus, whenever there is no gap in the dual (this is now the ase = (!)), the one-value of the primal does not disturb. f Other way round, if there is a gap in the dual (this is now the ase the primal fixes it by zero-values. f = (!)), Business Computing and Operations Researh 642

36 Proof of Theorem Part 3 This proof is temporarily postponed until we have introdued the algorithm of Ford and Fulerson that generates a Min-Cut aording to a given Max-Flow This is provided in Setion 7.4 Business Computing and Operations Researh 643

37 7.3 A Primal-Dual Algorithm We ommene with the dual problem A Maximize f, s.t., A f f f, i.e., E f -E Obviously, an initial feasible solution is f= By using a feasible dual solution, we get the set J that omprises three groups of indies. Speifially, we have: J = J J J, Sine π γ δ { ( ) }, { }, { } i A f = for all feasible f, we obtain J = {,2,3,..., m} J = i A f = J = f = J = f = π γ δ π Business Computing and Operations Researh 644

38 n ( ) Minimize α, s.t., The redued primal (RP) α π ( ( ) ( ) ) T Jγ J δ α,π,γ,δ E,A,E, E e. ( Jγ ) ( Jγ ) γ = ( Jγ ) δ ( Jδ ) Note that ( Jγ ) E ( J ) is generated out of matrix E by erasing all olumns that do not belong to set E δ J T is generated out of belong to set γ J δ n matrix by erasing all olumns that do not E n Business Computing and Operations Researh 645

39 The dual of the redued primal (DRP) E n A m Maximize ( ( ) ) T g, s.t., J γ E g, J γ ( ( ) ) T J Jδ δ E i.e., g A g g, i J g, i J i γ i δ Business Computing and Operations Researh 646

40 Updating f As provided by the design of primal-dual algorithm, an optimal solution of DRP may either indiate that f is already optimal or allow an improvement of f Thus, we have to find an appropriate λ whih ensures an improved but still feasible dual solution Speifially, ( DRP)... assuming gɶ as the optimal solution of, we update f by f : = f + λ gɶ new old Business Computing and Operations Researh 647

41 ( old λ ɶ ) ( ) Ensuring feasibility I In order to ensure feasibility, we have to guarantee the following:. A f + g. We already now A f + λ gɶ = A f + A λ gɶ = + A λ gɶ old = λ A gɶ, for all λ 2. old Sine gɶ is feasible, A gɶ ( f + λ gɶ ) ( f + λ gɶ ), old f f λ, gɶ > λ, gɶ < gɶ gɶ f λ, gɶ > gɶ Sine λ and f feasible, this is always fulfilled Business Computing and Operations Researh 648

42 Business Computing and Operations Researh 649 Ensuring feasibility II ( ) ( ) ( ) ~, ~ ~, ~ ~, ~, ~ ~ 3. we have to guarantee the following : And finally, this is always fulfilled feasible, and Sine < < > + f old g g f g g f g g f g f g f λ λ λ λ λ λ

43 Interpreting DRP Obviously, DRP an be interpreted as a speifially defined aessibility problem, i.e., a path is searhed in a redued graph This redued graph restrits the searhing proess as follows Ars that are already used up to apaity may only be used in baward diretion, i.e., the flow is redued Ars that are unused, i.e., f =, may only be used in forward diretion All other ars an be used in any diretion All indued flows are restrited by, i.e., a flow of maximum apaity is sought Business Computing and Operations Researh 65

44 Augmenting the flow Obviously, by solving DRP, we are aspiring an augmenting path Hene, it is not feasible to augment an already saturated flow or to derease a zero flow along some edge Consequently, if there is an augmentation possible, we are able to generate a flow f that indues only, -, or values at the respetive edges This onsiderably simplifies the updating of the dual solution in the Primal-Dual Algorithm Business Computing and Operations Researh 65

45 Ensuring feasibility with g=,,- In order to ensure feasibility, we have to guarantee the following: ( λ ɶ ). A f + g is fulfilled for all λ ( ɶ ) ( ) ( ɶ ) old f 2. f + λ g λ, gɶ = λ f old gɶ f 3. f + λ g λ, gɶ = λ f λ old gɶ { } { } { } min min f gɶ =,min f gɶ = Business Computing and Operations Researh 652

46 7.4 Ford-Fulerson Algorithm This algorithm is a modified primal-dual solution proedure The DRP is diretly solved, however, that is why no Simplex proedure is neessary for this step On the other side, this has onsiderable onsequenes aording to the termination of the solution proedure This will be disussed thoroughly later Business Computing and Operations Researh 653

47 A redued networ 7.4. Definition: ( ) Assuming N = V,E,,s,t is an s - t-networ and f a feasible E = E E f b f f f s - t - flow. Then, we introdue, with ( ) ( ) { } f E = e = i, j e = i, j E f < and f ( ) ɶ ( ) { } ɶ b E = e = i, j e = j, i E f >. E f f f b denotes the set of forward ars while E defines ( V,E,,s,t ) the baward ars. Then, we denote as the orresponding redued networ. f f Business Computing and Operations Researh 654

48 Interpretation Forward ars are used by the urrent flow f, but they are not used up to apaity I.e., they are not saturated by now Baward ars are not used by the urrent flow f, but the inverted ar is used by flow f Consequently, these ars are used in opposite diretion by the urrent flow f Consequently, forward ars are andidates for augmenting the flow in the urrent diretion (sine they offer remaining apaities) baward ars are andidates for reduing the flow (sine the opposite diretion transfers something) Business Computing and Operations Researh 655

49 7.4.2 Lemma: Observation ( ) ( ) ( DRP) A path i,..., i with i =, i = n, and i, i E l l f indiates an optimal solution to. Business Computing and Operations Researh 656

50 Proof of Lemma p = ( i = s i = t) ( ) ( ) { } ( ) ( ) ( ) { } Based on the path,...,, we define as follows: if e = i, j = il, il E, for l,..., or if e = n, g = if e = i, j = il, il E, for l,..., otherwise Sine p is a path, eah visited node is reahed and left by ars one. If this is done aording to ar diretions, we use =, otherwise we have g =. Sine the and values in A are hanged g ( A g ) aordingly, we obtain in both ases for the respetive row i: =. In addition, it holds: { } { } = ( DRP) g g,i J = f = g,i J = f =. Thus, g i γ i δ is feasible. Sine g, it is also an optimal solution to. i Business Computing and Operations Researh 657

51 Conlusions Let us assume that suh a path between s and t annot be established in the redued networ. We define for this onstellation: ( ) ( ) { } l l f W = i V p = s = i,...,i = i : i,i E W = V \W and additionally... π i ( ) if i W if e = i, j i W j W =, γ =, if i W otherwise and finally if δ = ( ) ( ) e = i, j t,s i W j W otherwise Business Computing and Operations Researh 658

52 We obtain : ( ) = ( e ) W,W the ut we now f = f = ( i,j ) Sine all nodes of W ( W,W ) e E e = The s-t-ut ( i,j ) ( ) ( e = W,W ) and is therefore maximal. γ,i W j W. ( e ) are used up to apaity by flow = e =, γ = = e E ( e ) were not reahable, all ars bridging γ f. Consequently, In addition, f annot be augmented Business Computing and Operations Researh 659

53 Maximum augmentation The maximum augmentation δ that is possible for the urrent flow, is determined by δ min { } ars of path p f e is forward ar, = min min { } ars of path p f e is baward ar Business Computing and Operations Researh 66

54 Ford-Fulerson Algorithm In what follows, we introdue the desription provided by Papadimitriou and Steiglitz (982) p.23 Business Computing and Operations Researh 66

55 Ford-Fulerson Algorithm Input: Networ N=(s,t,V,E,) Output: Max-Flow f Set f=, E f =E; While an augmenting s-t-path with min apaity value δ > an be found in the redued networ E f : Set f = f +δ; Update redued networ E f (derease apaities in path diretion by value δ and inrease apaities in opposite diretion by value δ for all edges on the augmenting path) End while An augmenting path an be found with the labeling algorithm on the next slide. Business Computing and Operations Researh 662

56 Labeling Algorithm We try to label every node with one possible predeessor on a path from s until we reah t: LIST={s}; While LIST not empty and t not in LIST: San x: Remove x from LIST. Label not all labeled yet adjaent nodes to x in E f with x as predeessor and put them on LIST. End while If t is labeled, we an reate the augmenting path by onsidering the predeessors in the labels. Business Computing and Operations Researh 663

57 An example e, =4 2 e 3, 3 =5 e 4, 4 =3 4 e 7, 7 =4 6 e 2, 2 =3 e 5, 5 = e 8, 8 =3 3 e 6, 6 = 5 e 9, 9 =7 Business Computing and Operations Researh 664

58 . Iteration We ommene our searh with f= All labels are zero LIST={} san Updating LIST LIST={2,3}, and san 2 LIST={3,4,5}, and san 3 LIST={4,5}, and san 4 LIST={5,6} and stop sine 6=t is labeled already We have labeled node 6=t. Path is therefore Thus, we now an augment our urrent flow f by δ=min{4,5,4}=4 Business Computing and Operations Researh 665

59 Current flow Edge Current Flow Found path +4= = = =4 Business Computing and Operations Researh 666

60 Updated redued networ e, =4 2 e 3, 3 = e 3, 3 =4 e 4, 4 =3 4 e 7, 7 =4 6 e 2, 2 =3 e 5, 5 = e 8, 8 =3 3 e 6, 6 = 5 e 9, 9 =7 Business Computing and Operations Researh 667

61 2. Iteration We ommene our searh with f All labels are zero LIST={} san Updating LIST LIST={3}, and san 3 LIST={4,5}, and san 4 LIST={5,2}, and san 5 LIST={6} and stop sine 6=t is labeled already We have labeled node 6=t. Path is therefore Thus, we now an augment our urrent flow f by δ=min{3,,3}= Business Computing and Operations Researh 668

62 Current flow Edge Current Flow Found path 4 2 += = = 9 5 Business Computing and Operations Researh 669

63 Updated redued networ e, =4 2 e 4, 4 =3 e 3, 3 = e 3, 3 =4 4 e 7, 7 =4 e 2, 2 =2 e 5, 5 = e 8, 8 =2 6 e 2, 2 = e 3 e 6, 6 = 8, 8 = 5 e 9, 9 =7 Business Computing and Operations Researh 67

64 3. Iteration We ommene our searh with f All labels are zero LIST={} san Updating LIST LIST={3}, and san 3 LIST={,4}. Sine is labeled, LIST={4}, and san 4 LIST={2}, and san 2 LIST={,4,5} Sine,4 are labeled, LIST={5}, and san 5 LIST={6} and stop sine 6=t is labeled already We have labeled node 6=t. Path is therefore Thus, we now an augment our urrent flow f by δ=min{2,,4,3,2}= Business Computing and Operations Researh 67

65 Current flow Edge Current Flow Found path 4 2 +=2 3 4-=3-4 += 5 += =2 9 5+=6 Business Computing and Operations Researh 672

66 Updated redued networ e 3, 3 =2 2 4 e 3, 3 =3 e, =4 e 7, 7 =4 e 4, 4 =2 e 2, 2 = e e 8, 8 = 5, 5 = e 4, 4 = e 2, 2 =2 e 3 e 6, 6 = 8, 8 =2 5 e 9, 9 =7 6 Business Computing and Operations Researh 673

67 4. Iteration We ommene our searh with f All labels are zero LIST={} san Updating LIST LIST={3}, and san 3 LIST={}. Sine is labeled, LIST={}, and terminate Thus, we obtain the s-t ut W={,3} and W ={2,4,5,6} The ut has total osts =4++=6 Business Computing and Operations Researh 674

68 Maximal flow Edge Flow Business Computing and Operations Researh 675

69 Updated redued networ f =4 e, =4 2 e 4, 4 =3 f 3 =3 e 3, 3 =5 f 5 = 4 f 7 =4 e 7, 7 =4 f 2 =2 e 2, 2 =3 e 5, 5 = f 6 = f 4 = f 8 =2 e 8, 8 =3 6 3 e 6, 6 = e 9, 9 =7 5 f 9 =6 Business Computing and Operations Researh 676

70 Optimality Clearly, the optimality of the proedure depited above may be diretly derived from the Primal- Dual Algorithm design There are, however, some speifi interesting attributes oming along with the proedure of Ford and Fulerson that are worth mentioning In what follows, we briefly disuss or just mention them Business Computing and Operations Researh 677

71 Corretness of the proedure Lemma: When the Ford and Fulerson labeling algorithm terminates, it does so at optimal flow. Business Computing and Operations Researh 678

72 Proof of Lemma When the algorithm of Ford and Fulerson terminates, there are some nodes that are already labeled while others are still unlabeled. We define W and W as above Consequently, all ars that are running from W to W are saturated now Additionally, ars running in the opposite diretion have flow zero Therefore, by Theorem 7.2.3, the s-t-ut (W,W ) and flow f are optimal Business Computing and Operations Researh 679