FEASIBLE FLOWS IN MULTICOMMODITY GRAPHS. Holly Sue Zullo. B. S., Rensselaer Polytechnic Institute, M. S., University of Colorado at Denver, 1993

Transcription

1 FEASIBLE FLOWS IN MULTICOMMODITY GRAPHS by Holly Sue Zullo B. S., Rensselae Polytechnic Institute, 1991 M. S., Univesity of Coloado at Denve, 1993 A thesis submitted to the Faculty of the Gaduate School of the Univesity of Coloado at Denve in patial fulllment of the equiements fo the degee of Docto of Philosophy Applied Mathematics 1995

2 This thesis fo the Docto of Philosophy degee by Holly Sue Zullo has been appoved by Havey Geenbeg Jennife Ryan David Fishe J. Richad Lundgen Gay Kochenbege Date

3 Zullo, Holly Sue (Ph. D., Applied Mathematics) Feasible Flows in Multicommodity Gaphs Thesis diected by Pofesso Havey Geenbeg ABSTRACT This thesis establishes the minimal epesentation of the necessay conditions fo feasible supplies and demands fo a given multicommodity netwok. The fundamental theoem is an extension of the Wallace- Wets connectivity esult fo both diected and undiected gaphs. A system of absolute value inequalities is developed fo the undiected case, and special popeties of this system ae exploed. Additional esults include counting the numbe of nonedundant inequalities fo specic classes of gaphs. This abstact accuately epesents the content of the candidate's thesis. I ecommend its publication. Signed Havey Geenbeg iii

4 CONTENTS Chapte 1. Intoduction : : : : : : : : : : : : : : : : : : : : : : : : : 1 2. Technical Backgound : : : : : : : : : : : : : : : : : : : : Tems and Concepts : : : : : : : : : : : : : : : : : : The Liteatue : : : : : : : : : : : : : : : : : : : : : 6 3. The Wallace-Wets Theoem : : : : : : : : : : : : : : : : : Feasibility in Diected Multicommodity Gaphs : : : : : : Peliminaies : : : : : : : : : : : : : : : : : : : : : : The System : : : : : : : : : : : : : : : : : : : : : : : Redundancy Theoem : : : : : : : : : : : : : : : : : Feasibility in Undiected Multicommodity Gaphs : : : : The Absolute Value System : : : : : : : : : : : : : : Redundancy Theoem : : : : : : : : : : : : : : : : : Some Building Blocks : : : : : : : : : : : : : : : : : Avenues fo Futhe Reseach : : : : : : : : : : : : : : : : 56 Glossay : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 58 Bibliogaphy : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 59

5 1. Intoduction The set of feasible supplies and demands fo a netwok ow poblem can be descibed by a system of linea inequalities. Wallace and Wets have chaacteized the edundant inequalities in this system. The main theoem of this thesis is an extension of the Wallace-Wets Theoem to the case of undiected multicommodity gaphs. In ode to obtain the extension, a special system of inequalities is developed and seveal chaacteistics of this system ae discussed. Following the extension, we pove seveal theoems on its usefulness. In Chapte 2 we povide technical backgound on netwok ows. We discuss some dieences between single and multicommodity netwok ows, as well as the petinent liteatue on multicommodity ows. We also discuss the backgound liteatue fo the Wallace-Wets Theoem. In Chapte 3 we pesent the Wallace-Wets Theoem fo single commodity netwoks along with a detailed poof of the theoem. Chaptes 4 and 5 contain the main esults of the thesis. In Chapte 4 we show that the Wallace-Wets Theoem has a natual extension to diected multicommodity gaphs. In Chapte 5 we conside the case of undiected multicommodity gaphs, which is not as staightfowad. A system of inequalities which gives a necessay condition fo feasible supplies and demands fo undiected multicommodity gaphs is developed, and we exploe popeties of this system. Next we pesent ou extension of the Wallace-Wets Theoem. We also pesent seveal theoems elated

6 to using the extension and the benet ealized. In Chapte 6 we suggest avenues fo futhe eseach. This includes futhe wok elated to the system of inequalities developed in Chapte 5, as well as evaluating the eects of peclassication of nodes as supply, demand, and tansshipment nodes. 2

7 2. Technical Backgound 2.1 Tems and Concepts Let G = [V; E] be a nite gaph with node set V and edge set E. Fo the poblems we will conside in this thesis, thee is no loss in geneality to assume G does not contain any paallel edges. We theefoe denote an edge by its endpoints: (i; j). This is an unodeed pai in the context of undiected gaphs, but an odeed pai in the context of diected gaphs. We also assume, without loss in geneality, that G is connected. Fo any Y V, let Y denote the complement of Y : Y = fi 2 V : i 62 Y g. Then the associated cut is the edge set: < Y; Y >= f(i; j) 2 E : i 2 Y and j 2 Y g: Note that fo undiected gaphs, < Y; Y >=< Y; Y >; but that this is not tue fo diected gaphs. Fo each (i; j) 2 E thee is a capacity U ij which limits total ow acoss the edge. This extends natually to subsets of E: U(Y; Y ) P (i;j)2<y;y > U ij : Fo each Y V, we let G(Y ) denote the induced subgaph on the node set Y. Fo each i 2 V, we let the demand at node i be given by b i : That is, if b i > 0 then node i has a demand, if b i < 0 then node i has a supply, and if b i = 0 then node i has neithe demand no supply. In the context of multicommodity gaphs, s k and t k will denote the oigin and destination, espectively, of commodity k, k = 1; 2; : : : ; p: We will let q k epesent the demand fo commodity k. If the oigin and

8 destination of each commodity ae not known, then epesents the demand fo commodity k at node i. The wok in this thesis deals mainly with multicommodity gaphs. It is well known that many of the \nice" popeties associated with netwok ow poblems do not extend to the multicommodity case. A complicating matte in multicommodity ows is that ows in opposite diections on an edge do not cancel if they ae dieent commodities. This leads to a vaiety of poblems. In a single commodity netwok ow poblem, the maximum ow is equal to the minimum cut, and given intege capacities, the maximum ow is guaanteed to be intege. Neithe of these popeties extend to the geneal multicommodity case. Additionally, in single commodity netwok ow poblems, the condition that the capacity of any cut is geate than o equal to the demand acoss the cut, is a necessay as well as sucient condition fo feasibility of the netwok. That is, if q is the equied amount of ow to be sent though the netwok, the netwok is feasible if and only if q U(Y; Y ) 8Y V: The analogous multicommodity condition (qk : Y sepaates s k and t k ) U(Y; Y ) 8Y V is necessay fo undiected gaphs, but it is not sucient fo feasibility in geneal. Much wok has been done to detemine specic classes of gaphs fo which this condition is sucient, and we will examine seveal of the esults in the next section. We now pesent some examples of the above statements. 4

9 Example s s 2 = t 3 s 3 = t 4 d s 4 = t 2 t 1 Assume a capacity of 1 on each edge. A maximum ow is given by the following, whee x i is the ow vecto fo edge i. x a = x b = x c = x d = x e = x f = 1 3 ; 1 3 ; 1 3 ; ; 1 3 ; 1 3 ; ; 0; 1 3 ; ; 0; 1 3 ; ; 1 3 ; 0; ; 1 3 ; 0; 1 3 The ow is factional, even though the capacities ae intege. Also, the value of the maximum ow, which is the sum of the ows fo each commodity, is = 3: The notion of a cut in single commodity gaphs tanslates to a disconnecting set in multicommodity gaphs, that is, a set of edges whose emoval will disconnect the oigin and destination of each commodity. Fo moe on disconnecting sets, see [3]. A 5

10 minimum disconnecting set fo this gaph consists of the edges fa; c; d; fg. The capacity of this set is 4, which is not equal to the maximum ow. Example s 1 t 1 QA c A Q d A a Q Af A A A Q A Q Q A QA b s 2 = s 4 s 3 = t 2 t 3 = t 4 e Suppose the capacity of each edge is 1, and the demand fo each commodity is 1. That is, we wish to send one unit of commodity 1 fom s 1 to t 1, and so on. It can be veied that the cut condition, (qk : Y sepaates s k and t k ) U(Y; Y ); holds fo evey set Y. Fo example, if Y = fs 1 g, then < Y; Y >= fa; b; cg, so U(Y; Y ) = 3. The only commodity sepaated by Y is commodity 1, so the left-hand-side of the cut condition is 1. Thus the inequality holds, as could be shown fo evey Y. Howeve, it is not possible to satisfy all of the demands simultaneously, so the poblem is not feasible. 2.2 The Liteatue As discussed in the pevious section, a cental issue in the study of multicommodity netwok ow poblems is that the cut condition, (qk : Y sepaates s k and t k ) U(Y; Y ); 6

11 is not always a sucient condition fo feasibility. Hee we will pesent seveal of the esults which chaacteize gaphs fo which this condition is sucient fo feasibility. We efe the eade to the appopiate papes fo the poofs. As the esults ae vey dieent fo the undiected and diected cases, we will discuss them sepaately, beginning with the undiected case. Some of the pioneeing wok on this subject was done by T.C. Hu ([16],[17]). Hu poved that the cut condition is sucient fo feasibility if thee ae only two commodities. Additionally, he poved that in the two-commodity case, the maximum sum of the ows is equal to the minimum capacity of all cuts sepaating the oigins and destinations of both commodities (the minimum disconnecting set). He povided an algoithm fo constucting the ows. Finally, Hu showed that if the demands and the capacities ae all even integes, then a two-commodity ow will have the popety that the ows fo each commodity on each edge ae intege. This is commonly efeed to as a half-intege popety, and it aises in othe cases as well. Hu's esult is the only one that elies on a specic numbe of commodities, and it does not genealize to thee o moe commodities. All othe esults ely on the gaph having special chaacteistics. A few denitions ae in ode befoe we discuss the next esult. A plana gaph is a gaph that can be dawn in the plane so that no edges coss. Given a xed dawing of a plana gaph, we can identify vaious egions, which ae simply the aeas enclosed by the edges of the gaph. The innite egion is the entie egion outside the gaph, and the bounday of the innte egion consists of the nodes and edges that sepaate the innite 7

12 1 A B 4 3 C Figue 2.1: A Plana Gaph and Its Regions egion fom the \inteio" of the gaph. Fo example, in Figue 2.1, A is the egion enclosed by the edges (1,2), (2,3), and (1,3); B is the egion enclosed by the edges (1,3), (3,4), and (1,4); C is the innite egion. The bounday of the innite egion is 1, (1,2), 2, (2,3), 3, (3,4), 4, (4,1). Note that we could also daw the gaph with edge (1,3) on the othe side of node 2. This would still be a plana epesentation, but dieent egions would be dened. The bounday of the innite egion fo that dawing would be 1, (1,3), 3, (3,4), 4, (4,1). We ae now eady fo the theoem of Okamua and Seymou ([28]). Theoem 2.1 (Okamua and Seymou, 1981) If G is plana and can be dawn in the plane so that s 1 ; s 2 ; : : : ; s p ; t 1 ; t 2 ; : : : ; t p ae all on the bounday of the innite egion, then (qk : Y sepaates s k and t k ) U(Y; Y ) is a sucient condition fo feasibility. Futhe, if the demands and capacities ae all integes, then thee exists a set of feasible ows with the ow fo each commodity on each edge being intege. 8

13 That is, if a gaph can be dawn in the plane so that all the souces and sinks ae on the bounday of the innite egion, then the cut condition is sucient fo feasibility. Thee have been some extensions of this theoem which elax slightly the condition of all the souces and sinks being on the bounday of the innite egion. Okamua ([27]) poved that the cut condition is sucient if G is a plana gaph and can be dawn with one set of souces and sinks on the bounday of any egion, and the emainde of the souces and sinks on the bounday of the innite egion, with the souce and sink fo any one commodity on the same bounday. In the same pape, it is shown that the cut condition is also sucient if G is plana and can be dawn with the souces and sinks fo some commodities located on the bounday of the innite egion, and all othe commodities shae the same sink, also located on the bounday of the innite egion, but those souces may be located anywhee. Seymou ([39]) poved that the cut condition is sucient if G is a plana gaph and if the souce and sink fo each commodity may be joined by an edge without violating planaity. Fo a discussion of all of the above esults plus algoithms fo nding plana multicommodity ows, the eade is efeed to [26]. The nal class of undiected gaphs that we will pesent hee actually elies on the conguation of the souces and sinks of the commodities. We call this conguation the commodity gaph. Given a connected undiected netwok G = [V; E] with souces s 1 ; s 2 ; : : : ; s p and sinks t 1 ; t 2 ; : : : t p, let the commodity gaph, H = [T; U]; (T V ) be the undiected gaph whose edges coespond the the souce-sink pais of G. That is, e 2 U () e = (s k ; t k ): Note that U is not necessaily a subset of E. 9

14 Example Given the following gaph s 2 = s 3 = t 4 s 4 = @? t 1 we constuct the commodity gaph by dawing each node and then connecting s 1 to t 1, s 2 to t 2, and so on. The esulting gaph is A A AA A Kazanov ([19]) pesents a esult of Papenov egading the commodity gaph. Theoem 2.2 (Papenov, 1976) (1) If H is K 4 (the complete gaph on fou nodes), C 5 (the cycle on ve nodes), o a union of two stas, and if the cut condition is satised, then the poblem is feasible. (2) If H is a gaph which does not belong to the collection above, then thee exists a gaph G, capacities U, and demands q, such that the cut condition is satised and yet the poblem is not feasible. 10

15 In othe wods, the cut condition can be known to be sucient based only on the commodity gaph if and only if the commodity gaph is a complete gaph on fou nodes, a cycle on ve nodes, o a union of two stas. If the commodity gaph is othe than the gaphs mentioned, then it is possible to constuct a gaph which has that commodity gaph and fo which the cut condition is not sucient fo feasibility. We ae now eady to discuss some esults on feasibility in diected gaphs. As mentioned ealie, the diected case is vey dieent fom the undiected case, and it has eceived much less attention in the liteatue. Note that in the case of diected gaphs, the cut condition changes slightly to become (qk : s k 2 Y; t k 2 Y ) U(Y; Y ): It is known that Hu's esult fo two-commodity gaphs does not extend to the multicommodity case ([21]). Howeve, the cut condition is sucient fo feasibility if the netwok is plana with all the souces on the left side and all the sinks on the ight side (a tanspotation netwok) and if the souces and sinks appea in the same ode (see [29]). The main wok is by Nagamochi and Ibaaki, and we will give a bief oveview of thei esults. In seveal papes and a Ph.D. thesis ([22], [23], [24]), Nagamochi and Ibaaki studied diected multicommodity netwoks. They developed thee classes of netwoks, capacity balanced (CB), capacity semi-balanced (CS), and capacity semi-balanced unilateal (CU). In a CB netwok, the capacity is \balanced" at each node. That is, fo evey node, the capacity on outgoing acs plus the demand at the node is equal to the capacity on incoming acs plus the supply at the node. It is shown that the max-ow 11

16 min-cut theoem holds fo multicommodity netwoks in CB, and also that the ows will be intege if the capacities and demands ae intege. The class CS is shown to be a elaxation of class CB, and the same popeties shown fo CB netwoks also hold fo CS netwoks. Finally, it is shown that the popeties do not extend to CU netwoks. Fo a moe geneal suvey of multicommodity netwok ows, see eithe [1] o [21]. The foundations needed fo the Wallace-Wets Theoem come fom the wok done by Gale and Homan fo single-commodity netwok ow poblems. In [11], Gale gives a system of linea inequalities which ae necessay and sucient fo feasibility of a set of supplies and demands in a netwok. Homan ([15]) extends this esult to feasible ciculations. The inequalities ae those geneated by consideing all bipatitions of the vetices of the netwok. It is assumed that any node may be a supply node, demand node, o tansshipment node. The esulting system of inequalities is b i U(Y; Y ) fo all Y V. This is the system which Wallace and Wets wok with, and the inequalities will be efeed to as the Gale-Homan inequalities. 12

17 3. The Wallace-Wets Theoem In [43], [44], [45], Wallace and Wets examine the system of inequalities which descibe the feasible supplies and demands fo a netwok. The inequalities ae the ones given by Gale and Homan, and Wallace and Wets detemine the mimimum numbe of inequalities needed. The Wallace-Wets Theoem gives a chaacteization of the edundant inequalities based on a connectedness citeion. The Gale-Homan inequalities can be geneated fom all possible patitions of the vetices into two sets. Wallace and Wets pove that the esulting inequality is edundant if and only if the induced subgaph on at least one of the patitions is not connected. Thei theoem is the following. Theoem 3.1 [Wallace-Wets Theoem] Fo all nonempty Y V, the associated inequality, b i U(Y; Y ) is edundant in the cut system if and only if eithe G(Y ) o G( Y ) is not connected. It is impotant to note that this theoem applies to the case whee all supplies must be used and all demands must be met exactly. Thus, the cut system also includes the equation P i2v b i = 0, o supply equals demand. Fo moe on this and othe aspects of the Wallace-Wets Theoem, see [14].

18 We pesent a poof of the Wallace-Wets Theoem fo completeness and to claify seveal of thei steps. The suciency pat of the poof is done by constuction. We assume that eithe G(Y ) o G( Y ) is not connected, and we show how to constuct the dependency set fo the inequality coesponding to Y. Fo the necessity pat of the poof, we st pove a lemma showing that we may estict ou consideation of the ighthand-sides to an ac membeship function. We then show that the only sets which may be used in constucting a edundant inequality ae those whose cutset acs ae a subset of the cutset acs of the oiginal set. If both G(Y ) and G( Y ) ae connected, then thee ae no sets which have that chaacteistic. Thus, the oiginal inequality is not edundant. Lemma 3.2 Let Y l dene a patition of the vetices. If the inequality coesponding to Y l is edundant, then the acs in the union of the cutsets fo patitions coesponding to the inequalities in the dependency set must be exactly the acs in < Y l ; Y l > : Poof: Since the inequality coesponding to Y l is edundant, thee exist j ; 0, j = 1; 2; : : : ; 2 jv j ; such that b i = j b i? i 2 Y j6=l l i 2 Y j i2v b i j6=l j U(Y j ; Y j ) U(Y l ; Y l ): Let J be the set of j's such that j > 0: Fist, assume thee exists a 2< Y l ; Y l > such that a 62< Y j ; Y j > 8j 2 J. Let a = (; ) ( 2 Y l ; 2 Y l ). We will deive a contadiction. Fo any j 2 J if 2 Y j then 2 Y j (o else a 2< Y j ; Y j >. Thus, is in at least as many sets Y j as, so is in at most as many sets Y j as. But since 2 Y l and 2 Y l, we must have 14

19 in exactly one moe set Y j than in ode fo the left-hand-side of the inequality to wok out. So we have a contadiction. Thus, thee does not exist a 2< Y l ; Y l > such that a 62< Y j ; Y j > 8j 2 J. This tells us that the acs of the cutset fom Y l ae a subset of the union of all acs in the cutsets fom Y j fo j 2 J. What we have left to show is that thee is no ac in any set < Y j ; Y j >; j 2 J such that the ac is not in < Y l ; Y l >. Let a 2< Y l ; Y l >; a = (; ) whee 2 Y l ; 2 Y l. If the inequality fom Y l is edundant, then the following must be tue: j : 2 Y j j? = 1 j : 2 Y j j = whee the st equation comes fom setting b = 1 and all othe b i 's to zeo, and the second equation comes fom setting b = 1 and all othe b i 's to zeo. Fom the two equations above, we get that This can be ewitten as j? j = 1: j : 2 Y j j : 2 Y j j + j? j? j = 1: j : 2 Y j 2 Y j j : 2 Y j 2 Y j 2 Y j 2 Y j 2 Y j 2 Y j The second and fouth tems cancel. Since the thid tem is nonnegative, we have j:(;)2<y j ;Y j > j 1: So the capacity of each edge in < Y l ; Y l > is added in to P j6=l j U(Y j ; Y j ): Theefoe, we cannot have a 2< Y j ; Y j >; j 2 J; and a 62< 15

20 Y l ; Y l > because that would give us P j6=l j U(Y j ; Y j ) > U(Y l ; Y l ): We ae now eady to pove the Wallace-Wets Theoem. We estate the theoem fo convenience. Note that this theoem holds fo both the diected and undiected cases. Theoem 3.1 Fo all nonempty Y V, the associated inequality, b i U(Y; Y ) is edundant in the cut system if and only if eithe G(Y ) o G( Y ) is not connected. Poof: We begin with the \if" diection. Fist, suppose G(Y ) is not connected. Let Y = Y 1 [ Y 2 whee Y 1 \ Y 2 = ; and thee ae no edges between Y 1 and Y 2 (See Figue 3.1). Since thee ae no edges between Y 1 and Y 2, we know that < Y 1 ; Y 1 > [ < Y 2 ; Y 2 >=< Y; Y >. The inequality fom Y is The inequality fom Y 1 is b i U(Y; Y ) = U(Y 1 ; Y 1 ) + U(Y 2 ; Y 2 ): 1 b i = The inequality fom Y 2 is b i = 2 2 b i + 1 b i + b i = U(Y 1 ; Y 1 ): b i = U(Y 2 ; Y 2 ): Summing, we get b i + 2 b i + 1 b i + b i = b i + 2 b i U(Y; Y ): 16

21 '$ '$ &% &% Y 1 Y 2 Y Y Figue 3.1: Illustation of Y and Y Subtacting P i2v b i = 0 leaves us with b i U(Y; Y ): Theefoe, the inequality fom Y is edundant. Next, suppose G( Y ) is not connected. Let Y = Y 1 [ Y 2 whee Y 1 \ Y 2 = ; and thee ae no edges between Y 1 and Y 2. Thus, < Y; Y >=< Y 1 ; Y 1 > [ < Y 2 ; Y 2 >. Again the inequality fom Y is The inequality fom Y 1 is The inequality fom Y 2 is Summing, we get b i U(Y; Y ) = U( Y 1 ; Y 1 ) + U( Y 2 ; Y 2 ): 1 b i U( Y 1 ; Y 1 ): b i b i U( Y 2 ; Y 2 ): 2 b i = Theefoe, the inequality fom Y is edundant. b i U(Y; Y ): 17

22 We now pove the \only if" diection. We assume that G(Y l ) and G( Y l ) ae both connected, and we will show that the inequality coesponding to Y l cannot be edundant. Let l index the set that is the complement of Y l, that is, Y l = Y l. Let c j be the ac membeship vecto fo the cut dened by Y j (c j a = 1 () a 2< Y j ; Y j >). Also, let d j be the node membeship vecto fo the set Y j (d j i = 1 () i 2 Y j ). Let J l = f1; 2; : : : ; 2 jv j g n fl; lg: Let e be the vecto of ones. The tem 0 e will account fo being able to subtact multiples of the equation that states that total supply equals total demand. We want to show that the following system has no solution: c l = j c j + l c l j 2 J l d l = j d j 2 J l j + l d l? 0 e; 0 ; l ; j 0 8j 2 J l : Suppose the system is feasible, and let ( ; l ; ( j ; j 2 J l )) be a solution. Let J + l = fj 2 J l j j > 0g: That is, J + l indexes the inequalities which ae used in constucting the edundancy of inequality l. If k 62< Y l ; Y l >, then c l k = 0 and so 0 = j c j k + l c l k 8k 62< Y l ; Y l > : j 2 J + l This equation tells us that c j k = 0 if k 3< Y l; Y l > and if j 2 J + l. Witten anothe way, this says: k 3< Y l ; Y l >! k 3< Y j ; Y j > if 18

23 Y j H H H Y ~ j Yj Y l Y l Figue 3.2: Illustation of Y ~ j and Y j j 2 J + l If we negate the implication, we get that j 2 J + l implies k 2< Y j ; Y j >! k 2< Y l ; Y l >. This then is the same as saying j 2 J + l!< Y j ; Y j >< Y l ; Y l > : That is, we cannot use any sets whose cutset acs ae not a subset of the acs in < Y l ; Y l >. Fo all Y j V, we dene two new sets (see Figue 3.2): ~Y j = Y j \ Y l and Yj = Y j \ Y l : We will show that if Y j is such that eithe Y ~ j is a pope subset of Y l (; 6= Y ~ j 6= Y l ) o Y j is a pope subset of Y l (; 6= Y j 6= Y l ) then j 62 J + l. That is, then the inequality coesponding to Y j is not used in constucting the edundancy of inequality l. Fist we will wok with Y ~ j. If we wite the equation fo d l i (efeing back to the equation fo d l ) we have d l i = j j di + l d l i? 0 e i : j 2 J l 19

24 When i 2 Y l, d l i = 1, so we have 1 = j j di + 0? 0 j 2 J l = j 2 J l j d j i 8i 2 Y l : If q 2 J + l and ~ Yq is a pope subset of Y l, then eithe thee is an ac fom ~ Y q to Y l n ~ Y q, o thee is an ac fom Y l n ~ Y q to ~ Y q. Since < Y q Y q >< Y l ; Y l >, thee cannot be any acs fom ~ Yq to Y l n ~ Yq. Hence, thee must be an ac fom Y l n ~ Y q to ~ Y q. Let this ac be (i; i 0 ). Now conside any set Y j such that i 2 Y j and j 2 J + l : Since < Y j ; Y j >< Y l ; Y l >; we must also have i 0 2 Y j : Thus, any set Y j that contains i must also contain i 0. This means that the following must be tue: = j d j i < q + j d j i j d j i 0 = ; j 2 J + l j 2 J + l j 2 J + l which is a contadiction. Theefoe, thee is no q 2 J + l with ~ Y q a pope subset of Y l. The same agument is used to show that thee is not q 2 J + l with Y q a pope subset of Y l. Fo this case, note that i 2 Y l ) d l i = 0; so we have the equation 0 = l? 0 = j 2 J + l j d j i + l? 0 j d j i 8i 2 Y l : j 2 J + l If q 2 J + l and Yq is a pope subset of Y l ; then eithe thee is an ac fom Y q to Y l n Y q o thee is an ac fom Y l n Y q to Y q. Since 20

25 < Y q Y q >< Y l ; Y l >, thee cannot be any acs fom Y q to Y l n Y q. Hence, thee must be an ac fom Y l n Y q to Y q. Let this ac be (i; i 0 ). Now conside any set Y j such that i 2 Y j and j 2 J + l : Since < Y j ; Y j >< Y l ; Y l >; we must also have i 0 2 Y j : Thus, any set Y j that contains i must also contain i 0. This means that the following must be tue: l? 0 = j d j i < q + j d j i j d j i 0 = l? 0 ; j 2 J + l j 2 J + l j 2 J + l which is a contadiction. Theefoe, thee is no q 2 J + l with Y q a pope subset of Y l. Thus, the only index that could possibly belong to J + l is the one coesponding to V, say Y p = V. But then we would have = p 0? l = p 0 ; l ; p 0 which is not possible. Thus, J + l must be empty, but then we have d l = l d l? 0 e 0 ; l ; 0 which is also not possible. Theefoe, when G(Y ) and G( Y ) ae both connected, b i U(Y; Y ) is neve edundant. 21

26 4. Feasibility in Diected Multicommodity Gaphs 4.1 Peliminaies This thesis deals with linea multicommodity ows, ignoing intege estictions. We begin by noting that a multicommodity netwok with multiple souces and sinks fo each commodity can be tansfomed into a multicommodity netwok that has a single souce and sink fo each commodity. This tansfomation is analogous to the tansfomation fom multiple souces/sinks to a single souce/sink as given by Fod and Fulkeson ([9]). Theoem 4.1 A capacitated multicommodity netwok N = [V; E; U] with multiple souces and multiple sinks fo each commodity can be tansfomed into an equivalent multicommodity netwok N 0 = [V 0 ; E 0 ; U 0 ] that has a single souce and single sink fo each commodity. Poof: Let S k V and T k V be the indices fo the set of souces and sinks, espectively, fo commodity k. Let b k : V! < be such that = 0 fo i 62 S k [ T k ; < 0 fo i 2 S k; > 0 fo i 2 T k; and P = 0 fo each k. Constuct N 0 as follows. V 0 = V [ fo k ; D k g; and E 0 = E [ f(o k ; i : i 2 S k ); (i : i 2 T k ; D k )g: U 0 (O k ; i) =? fo i 2 S k ; U 0 (i; D k ) = b i fo i 2 T k ; and U 0 (e) = U(e) fo e 2 E: b 0 O k = P i 2 S k ; b0 D k = P i 2 T k ; b0 i = 0 fo i 2 V n (S k [ T k ).

27 Let x be any feasible ow in N. We will constuct a feasible ow x 0 in N 0. Let (x k e) 0 = x k e fo e 2 E; (x k O k ;i) 0 = U 0 (O k ; i) fo i 2 S k ; and (x k i;d k ) 0 = U 0 (i; D k ) fo i 2 T k : The capacity constaints ae satised fo all edges by denition. each node. We also need ow consevation at P i(x k i;j) 0? P i(x k j;i) 0 = P i x k i;j? P i x k j;i = 08k fo i 2 V n (S k [ T k ): P i 2 S k (x k O k ;i) 0 = P i 2 S k? =?b 0 O k 8k: P i 2 T k (x k i;d k ) 0 = P i 2 T k = b 0 D k 8k: P j(x k j;i) 0? P j(x k i;j) 0 = (x k O k ;i )0 + P j x k j;i? P j x k i;j =? + = 08k fo i 2 S k : P j(x k j;i) 0? P j(x k i;j) 0 = P j xk j;i? ( P j xk i;j + (x k i;d k ) 0 ) =? = 08k fo i 2 T k : Theefoe, x 0 is feasible in N 0 : Convesely, let x 0 be any feasible ow in N 0 : We will constuct a feasible ow x in N. Let x k e = (x k e) 0 fo e 2 E: Since U 0 (e) = U(e) fo e 2 E; P k xk e = P k(x k e )0 U 0 e = U e: So the capacity constaints ae satised. P i x k i;j? P i x k j;i = P i(x k i;j) 0? P i(x k j;i) 0 = 08k fo i 2 V n(s k [T k ): If i 2 S k then P j(x k j;i )0? P j(x k i;j )0 = (x k O k ; i )0 + P j xk j;i? P j xk i;j = 0 ) P j x k j;i? P j x k i;j = : If i 2 T k then P j(x k j;i) 0? P j(x k i;j) 0 = P j x k j;i? P j xk? i;j (xk ) 0 = 0 ) P i;d k j xk? P j;i j xk = i;j bk i : Theefoe, x is feasible in N: We illustate this theoem with the following example. Example The following gaph illustates the addition of supe-oigins and supe-destinations fo each commodity. 23

28 O 1 @ s 1 2 = s 1 1 = 1 t 2 1 s 2 = 2 1 t 1 = @ s 2 2 t 2?? 2 = 2??? t 1 2 = 1 D 2 D 1 Fo this example, node O 1 has a supply of 4 (s s1 2 ) and node O 2 has a supply of 6 (s s 2 2). Placing a capacity of 1 on edge (O 1 ; s 1 1) and a capacity of 3 on edge (O 1 ; s 1 2 ) will insue that nodes s1 1 and s1 2 will send the pope amounts of commodity 1. Similaly, we place a capacity of 2 on edge (O 2 ; s 2 1 ) and a capacity of 4 on edge (O2 ; s 2 2 ). Node D 2 has a demand of 6 (t t 2 2) and node D 1 has a demand of 4 (t t1 2 ). Placing a capacity of 4 on edge (t2 1 ; D2 ) and a capacity of 2 on edge (t 2 2 ; D2 ) will insue that nodes t 2 1 and t 2 2 will eceive the pope amounts of commodity 2. Similaly, we place a capacity of 3 on edge (t 1 1 ; D1 ) and a capacity of 1 on edge (t 1 2 ; D1 ). Thus, we see that any ow that is feasible in the oiginal netwok is feasible in the new netwok, and vice vesa. Note that this theoem holds whethe the gaph is diected o undiected. Fo the emainde of this thesis we will assume, without loss 24

29 of geneality, that each commodity has a single oigin and a single destination. 4.2 The System Wallace and Wets wok with a system of inqualities which dene feasibility fo a set of supplies and demands when the supply and demand nodes ae not known a pioi. We wish to develop an analogous system fo the multicommodity case, and then we will chaacteize the edundant inequalities in that system. If the souce and sink is known fo each commodity, then a necessay condition fo feasibility of a set of demands q is (qk : 1 k p; s k 2 Y; t k 2 Y ) U(Y; Y ) 8Y V: This condition is sucient fo the classes of gaphs identied by Nagamochi and Ibaaki (see Chapte 2). In ode to place this condition in the Wallace and Wets setting, we need to genealize this condition fo the case whee the supply and demand nodes ae not know a pioi. Looking at the system fo a single commodity, b i U(Y; Y ) 8Y V; one might guess that the system fo multicommodities can be obtained by simply summing ove each commodity. The esulting system is p k=1 U(Y; Y ) 8Y V: The following example shows that this is not enough. 25

30 Example g b 1 1 b 2 1 b 3 1 u 1 u 2 - g b 1 2 b 2 2 b 3 2 The inequalities ae: Y = f1g : b b2 2 + b3 2 u 1 Y = f2g : b b b 3 1 u 2 This system allows fo b 1 1 =?1; b 2 1 =?1; b 3 1 = 1; b 1 2 = 1; b 2 2 = 1; b 3 = 2?1; and u 1 = 1; u 2 = 1: Howeve, it is easily seen that this set of demands and capacities ae not feasible. Thus, we ty a stonge system. We actually need to sum ove all possible combinations of commodities. This esults in k2k U(Y; Y ) 8Y V; 8K; whee K is a nonempty set of commodities. This condition gives us the desied system. Theoem 4.2 The set of inequalities k2k U(Y; Y ) 8Y V; 8K is a necessay condition fo feasibility of b, the vecto of demands. Poof: We stat with the system of constaints fo a diected multicommodity ow poblem, x k ji? jj(j;i)2a jj(i;j)2a x k ij = 8i 26

31 p k=1 x k ij u ij x k ij 0 8i; j 8k; i; j and we want to show that this system implies the inequality k2k = U(Y; Y ): Recall that U(Y; Y ) = P (j;i):j2y; u ji : We begin by woking with the st constaint in the multicommodity ow poblem. Fist, we sum both sides of the equation ove i 2 Y to obtain = ( jj(j;i)2a x k ji? We next sum both sides ove k 2 K, obtaining k2k = k2k = ( k2k j2y = k2k j2y ( jj(j;i)2a j2y jj(i;j)2a x k ji? x k ij): jj(i;j)2a jj(i;j)2a x k? ji jj(j;i)2a k2k x k ji + x k ji? x k ij? = k2k Since x k ij 0, we now have j2y k2k x k ji? k2k k2k j2y x k ij : j2y x k ij): Now if we wok with the second constaint in the multicommodity ow j2y x k ji : poblem, again using the constaint that x k ij 0, we have k2k x k ji p k=1 x k ji u ji : x k ij x k ij) 27

32 This gives us u ji = U(Y; Y ): j2y k2k x k ji j2y This gives us ou desied inequality, k2k U(Y; Y ): The next lemma elates this system to the system of inequalities fo a diected multicommodity gaph when the souces and sinks ae known a pioi. Lemma 4.3 The set of inequalities k2k U(Y; Y ) 8Y V; 8K contains the set of inequalities (qk : 1 k p; s k 2 Y; t k 2 Y ) U(Y; Y ) 8Y V: Poof: We know that q k = b k t =?b k k s k and = 0 fo i 6= s k ; t k fo all k. We have a set of inequalities fo each set of commodities; in paticula, pick K 0 such that 8k 2 K 0, s k 2 Y and t k 2 Y. So ou system contains the inequalities k2k 0 U(Y; Y ) 8Y V: The left side of this inequality has fo i = t k and = 0 fo all othe i 2 Y. Fo i = t k ; = q k ; so the summation yields P k2k 0 q k: We now have a system of linea inequalities which seve as a necessay condition fo feasibility of multicommodity ows in a diected 28

33 gaph when the supply and demand nodes ae not known a pioi. In the next section, we will extend the Wallace-Wets Theoem to this system. 4.3 Redundancy Theoem The Wallace-Wets Theoem extends quite natually to the diected case. Using the system fom the pevious section, along with the equiement that supply equals demand fo each commodity, we ae able to chaacteize the edundancy in the system. This chaacteization is the same as the Wallace-Wets chaacteization fo the single commodity system with one exception. Fo ou system we need to sepaate the cases of U(Y; Y ) = 0 and U( Y; Y ) = 0. Theoem 4.4 [Redundancy Theoem fo Diected Gaphs] Let G be a connected, capacitated, diected, multicommodity gaph with p commodities, whee each commodity has one oigin and one destination. (1) If U(Y; Y ) > 0 and U( Y; Y ) > 0, then the inequality coesponding to the cut < Y; Y >; k2k U(Y; Y ) 8K; is edundant in the cut system if and only if eithe G(Y ) o G( Y ) is not connected. (2) If U(Y; Y ) = 0 (connectedness of the gaph then implies that U( Y; Y ) > 0), then all sets K with moe than one element yield edundant inequalities. The inequalities U(Y; Y ) 8k 29

34 ae edundant in the cut system if and only if eithe G(Y ) o G( Y ) is not connected. (3) If U( Y; Y ) = 0 (connectedness of the gaph then implies that U(Y; Y ) > 0), then all sets K yield edundant inequalities except fo K = f1; 2; : : : ; pg: The inequality p k=1 U(Y; Y ) is edundant in the cut system if and only if eithe G(Y ) o G( Y ) is not connected. Poof: (1) We will st pove the \if" diection. Assume that one of G(Y ) and G( Y ) is not connected. Case 1: G(Y ) is not connected. Let Y = Y 1 [ Y 2 whee Y 1 \ Y 2 = ; and thee ae no edges between Y 1 and Y 2. We examine the inequalities k2k U(Y; Y ) 8K: The inequalities fom Y 1 ae k2k 1 = b k + i k2k 2 k2k U(Y 1; Y 1 ) 8K: The inequalities fom Y 2 ae k2k 2 = + k2k 1 k2k U(Y 2 ; Y 2 ) 8K: Summing these inequalities gives +2 k2k k2k U(Y 1 ; Y 1 )+U(Y 2 ; Y 2 ) = U(Y; Y ) 8K: 30

35 Subtacting P P k2k i2v k2k = 0 gives U(Y; Y ) 8K: Thus, those inequalities ae edundant. Case 2: G( Y ) is not connected. Let Y = Y 1 [ Y 2 whee Y 1 \ Y 2 = ; and thee ae no edges between Y 1 and Y 2. We examine the inequalities k2k U(Y; Y ) 8K: The inequalities fom Y 1 ae k2k The inequalities fom Y 2 ae k2k 1 U( Y 1 ; Y 1 ) 8K: 2 U( Y 2 ; Y 2 ) 8K: Summing these two inequalities gives k2k ( ) = k2k U( Y 1; Y 1 )+U( Y 2 ; Y 2 ) = U(Y; Y ) 8K: Thus, the inequalities ae edundant. These poofs ae suciently geneal to hold fo U(Y; Y ) 0. Thus, we have also poved the \if" diection fo (2) as well. We now pove the \only if" diection. We assume that an inequality is edundant, and we will show that at least one of the coesponding gaphs is not connected. If k2k 0 q U(Y q ; Y q ) 31

36 is edundant, then it is a nonnegative combination of linea inequalities fom sets othe than Y q. To see this, assume that the above inequality is a nonnegative combination of linea inequalities in the cut system, some of which may be geneated by the set Y q. Clealy the only way the inequality can be a nonnegative combination of inequalities geneated only by the set Y q is if U(Y; Y ) = 0: This exception is teated in pat (2) of the theoem. So the dependency set must contain at least one inequality geneated by a set othe than Y q : Howeve, it may not contain any edges not in < Y q ; Y q > : We assume that G(Y q ) and G( Y q ) ae both connected, and we will deive a contadiction. Fist we will show that an inequality fom Y s, Y s \ Y q 6= ;; Y s \ Y q 6= Y q ; cannot be in the dependency set. Since G( Y q ) is connected, thee is an edge in < Y j ; Y q =Y j > o thee is an edge in < Y q =Y j ; Y j > : If the fome is tue, we ae done with this case. If the fome is false, then the latte is tue. Call the edge (; ): Sine and must be counted an equal numbe of times, thee must be some set, containing and not containing ; whose coesponding inequality is in the dependency set. But then this cutset contains the edge (; ); which is not possible. Thus, an inequality coesponding to Y s with Y s \ Y q 6= ;; Y s \ Y q 6= Y q ; cannot be in the dependency set. The same can be shown fo a set Y t with Y t \ Y q 6= ;; Y s \ Y q 6= Y q : The only emaining possibility is that an inequality coesponding to Y q is in the dependency set. This can only occu if U( Y q ; Y q ) = 0; 32

37 which is handled in pat (3) of the theoem. Thus, we have shown that the edundant inequality k2k 0 q U(Y q; Y q ) must be a nonnegative combination of linea inequalities fom sets othe than Y q : That is, 9 2 < 2jV j + such that k2k 0 q = s6=q s k2k 0 s = s k2k 0 s6=q s : Note that the above equation holds fo all b. In paticula, fo any choice of k, we can set b j i to be zeo fo all i and fo all j 6= k. Theefoe, q fo each k. This gives us that = s6=q s s q U(Y q; Y q ) is edundant in the cut system fo each commodity k. Thus, each inequality coesponding to Y q is edundant fo each commodity k. We can now apply the Wallace-Wets Theoem to conclude that eithe G(Y q ) o G( Y q ) is not connected. (2) Assume U(Y; Y ) = 0: We have the inequalities U(Y; Y ) = 0 8k: Summing ove k 2 K : jkj > 1 gives k2k 0 = U(Y; Y ) 8K : jkj > 1: 33

38 Thus, the inequalities k2k U(Y; Y ) 8K : jkj > 1 ae always edundant when U(Y; Y ) = 0. (3) Assume U( Y; Y ) = 0: Let K 0 be any pope subset of the commodities. We will show that the inequality is edundant. k2k 0 U(Y; Y ) Fom the set Y; we have the inequalities k2k In paticula, if K p inequality U( Y; Y ) = 0 8K: is the set of all commodities, we have the k2k p =K 0 0: Since P =? P ; this gives us? Adding this to the inequality k2k p =K 0 k2k p esults in the oiginal inequality. 0: U(Y; Y ) We note that when p = 1 (one commodity) this system educes to the system used by Wallace and Wets. The inequality fo the set Y becomes P b i U(Y; Y ). We poceed now to the undiected case, which is not as staightfowad as the diected case. 34

39 5. Feasibility in Undiected Multicommodity Gaphs 5.1 The Absolute Value System We will now wok with undiected multicommodity gaphs, meaning that the edges ae not oiented and ow may tavel in eithe diection on an edge. In paticula, ow fom dieent commodities may tavel in opposite diections on the same edge. In woking towads an extension of the Wallace-Wets Theoem to the undiected multicommodity case, we need to have a system of inequalities which is a necessay condition fo feasible demands (and which will be sucient fo the cases discussed in Chapte 2). The taditional system of cut inequalities, (qk : 1 k p; Y sepaates s k and t k ) U(Y; Y ) 8Y V; assumes that the oigin and destination nodes fo each commodity ae known a pioi. We need to develop a system in which the oigin and destination nodes do not need to be known a pioi. As a st attempt, one might ty genealizing the single commodity cut system by summing ove all commodities, as in the diected case. If we use the system fom the diected case, we have: k2k i2v = 0 8k U(Y; Y ) 8Y V; 8K: Howeve, this system is not estictive enough in the undiected case. Conside the following gaph.

40 b 1 1 b 2 1 u b 1 2 b 2 2 The above system would give ise to the following set of inequalities: b b 1 2 = 0 b b 2 2 = 0 Y = f1g : K = f1g : K = f2g : b 1 2 u b 2 2 u K = f1; 2g : b b2 2 u Y = f2g : K = f1g : K = f2g : b 1 1 u b 2 1 u K = f1; 2g : b b 2 1 u This system allows fo b 1 1 = 1; b 1 2 =?1; b 2 1 =?1; b 2 2 = 1; and u = 1. Howeve, we can easily see that this set of demands, along with the given capacity, is not feasible. In ode to satisfy the demands, we would have to send a total of 2 units of ow acoss the edge, when only 1 unit is allowed. The poblem is that at each node, the demand fo one commodity cancels with the supply fo the othe commodity. This poblem is alleviated by taking the absolute value of the sum of the b i 's fo each commodity. That is, we use the system i2v = 0 8k 36

41 p k=1 j Now the system fo the above gaph is: j U(Y; Y ) 8Y V: b b 1 2 = 0 b b2 2 = 0 Y = f1g : jb 1 2j + jb 2 2j u Y = f2g : jb 1 j + 1 jb2j 1 u We see that the set of 's that was feasible in the old system is not feasible in this system. In fact, this gives us the coect system fo evey multicommodity gaph, as the next theoem shows. Note that we do not need to sum ove k 2 K because the inequalities coesponding to sets K othe than the full set of commodities will be dominated by the inequality above. Fo example, fo the gaph in the above example, we would get Y = f1g : K = f1g : jb 1 2j u; which is dominated by the inequality obtained by summing ove all commodities. Theoem 5.1 The set of inequalities p k=1 j j U(Y; Y ) 8Y V is equivalent to the set of inequalities (qk : 1 k p; Y sepaates s k and t k ) U(Y; Y ) 8Y V: 37

42 Poof: Since q k = b k t k =?b k s k and = 0 fo i 6= s k ; t k fo all k, we will teat sepaately the cases when Y does and does not sepaate s k and t k. Case 1: Y does not sepaate s k and t k. Then s k and t k ae eithe both in Y o both in Y. If s k and t k ae both in Y, then j j = jb k s k + b k t k j = 0: If s k and t k ae both in Y, then = 0 8i 2 Y. Theefoe, if s k and t k ae not sepaated, then P = 0: Case 2: Y does sepaate s k and t k. Then eithe s k 2 Y o t k 2 Y, but not both. = 0 fo all othe i 2 Y. Then j j = b k t k = q k : Thus, the absolute value inequality is equivalent to limiting Y to sets that sepaate s k and t k, in which case the two inequalities ae the same. This system of inequalities, which we will efe to as the cut system, has many popeties which ae not common to a geneal system of absolute value inequalities. Fist, we know that evey inequality in the cut system contains absolute values. Futhe, fo a given inequality, the numbe of tems within each absolute value is the same, and it is equal to the cadinality of the set Y. So, fo example, we cannot have jx 1 j + jx 2 + x 3 j u in a cut system. Similaly, each inequality in the cut system has the same numbe of absolute value tems, which is equal to the numbe of commodities. That is, we cannot have both jx 1 j u and 38

43 jx 1 j+jx 2 j u in the same system. Additionally, a tem that appeas inside of one absolute value will not appea anywhee else in that inequality. That is, we cannot have jx 1 + x 2 j + jx 2 + x 3 j u in a cut system. Finally, we note that the coecients on all tems within the absolute values, as well as on the absolute value quantities themselves, ae 0-1, and the ight-hand side is always nonnegative. These obsevations lead us to some lemmas about the cut system. An absolute value inequality fom the cut system is equivalent to a system of 2 p linea inequalities. We can expess the jth linea inequality as p kj k=1 U(Y; Y ); whee kj 2 f?1; 1g: In the poof of the next lemma, we will simply wite U fo U(Y; Y ). To illustate the equivalent system of linea inequalities, conside the following absolute value inequality: jx 1 + x 2 j + jx 3 + x 4 j u: The coesponding system of linea inequalities is: x 1 + x 2 + x 3 + x 4 u?(x 1 + x 2 ) + x 3 + x 4 u x 1 + x 2? (x 3 + x 4 ) u?(x 1 + x 2 ) +?(x 3 + x 4 ) u: We will sometimes efe to this system as a family of linea inequalities. We next show that we cannot obtain an inequality in this system as a nonnegative combination of only othe inequalities in this system. 39

44 Lemma 5.2 The family of linea inequalities coesponding to a given absolute value inequality fom the cut system is intenally nonedundant. Poof: Conside the jth linea inequality coesponding to a given absolute value inequality: p k=1 kj U: Revese the sense of the inequality and make it stict: p k=1 kj > U: The system with this new inequality in place of the old one is feasible if and only if the old inequality is nonedundant. We will constuct a feasible solution to the evesed system: the sum by Then 8i 2 Y; = p k=1 kj 8 >< >: U if jy jp?1 kj = 1?U if jy jp?1 kj =?1 = jy j p jy j p? 1 U > U: Futhe, changing any kj fom 1 to -1 o fom -1 to 1 deceases U, making it equal to U. Thus, this b satises all of the jy jp?1 linea inequalities in this family, so the jth inequality is not implied by the othes. This lemma does not hold fo the linea system coesponding to a geneal absolute value inequality. The following example will illustate this. 40

45 Example Conside the inequality jx 1 j + jx 2 j + jx 1 + x 2 j u A patial enumeation of the coesponding linea inequalities gives: x 1 + x 2 + x 1 + x 2 u ) 2x 1 + 2x 2 u?x 1 + x 2 + x 1 + x 2 u ) 2x 2 u?x 1? x 2 + x 1 + x 2 u ) 0 u x 1 + x 2? x 1? x 2 u ) 0 u The last two inequalities ae copies of each othe. Thus, this system is not intenally nonedundant. We can also make some obsevations about an inequality which is known to be edundant. Lemma 5.3 A edundant linea inequality cannot be fomed as a nonnegative combination of linea inequalities which ae all fom one family, dieent fom the family of the edundant linea inequality. Poof: Let a iq be the node membeship function fo the set Y q : a iq = 1 i i 2 Y q : Let c eq be the edge membeship function fo the cut dened by Y q : c eq = 1 i e 2< Y q ; Y q > : We want to show that p cannot be fomed as kj k=1 i2v a iq U(Y q ; Y q ) = e2e c eq U e p k=1 k a it i2v U(Y t; Y t ) = e2e c et U e ; ( 0; = 1; j = 0): 41

46 Assume the two inequalities ae equal. Then c t P = c q : This gives us that P = 1 and c t = c q : If the netwok is connected then eithe Y t = Y q o Y t = Y q : Since this is not possible, the two inequalities cannot be equal. Lemma 5.4 If a linea inequality fom the cut system is edundant, then it is edundant fo each commodity. Poof: If p k=1 kj i2v a iq U(Y q ; Y q ) is edundant, then it is a nonnegative combination of linea inequalities fom sets othe than q. That is, 9 2 < 2p 2 jv j + such that p kj a iq k=1 i2v = p s s6=q k=1 k a is = p s k a is : i2v k=1 s6=q i2v Note that the above equation holds fo all b. In paticula, fo any choice of k, we can set b j i to be zeo fo all i and fo all j 6= k. Theefoe, kj a iq i2v Which gives us that = s k s6=q i2v a is fo each k: kj i2v a iq U(Y q ; Y q ) is edundant in the cut system fo commodity k. 5.2 Redundancy Theoem We have seen many chaacteistics of single commodity ows that do not extend to the case of multicommodity ows. The question which motivated this thesis was whethe the Wallace-Wets Theoem holds in the 42

47 multicommodity case. In this section we pove that using the absolute value system, the Wallace-Wets Theoem can be extended to undiected multicommodity gaphs. In what follows, we assume the cut system to be p k=1 j i2v j U(Y; Y ) = 0 (k = 1; : : : ; p): 8Y V Theoem 5.5 [Redundancy Theoem fo Undiected Gaphs] Let G be a connected, capacitated, undiected, multicommodity gaph with p commodities, whee each commodity has one oigin and one destination. (1) Only one of the pai of inequalities p k=1 p j j k=1 j U(Y; Y ) j U(Y; Y ) is eve necessay. That is, we need only conside the inequality fom Y o Y, but not both. (2) The inequality coesponding to the cut < Y; Y >, p j j U(Y; Y ); k=1 is edundant in the cut system if and only if eithe G(Y ) o G( Y ) is not connected (Y 6= ;; Y 6= ;). 43

48 Poof: (1) Since P i2v = 0 (k = 1; : : : ; p); we have that P =? P : Theefoe, p j j = p j? j = p j k=1 k=1 k=1 So only one of each pai of inequalities is needed. (2) We will st pove the \if" diection. If G(Y ) and G( Y ) ae not both connected, we may assume, without loss of geneality, that G( Y ) is not connected. Let Y = Y 1 [ Y 2 whee Y 1 \ Y 2 = ; j: and thee ae no edges between Y 1 and Y 2. We will show that the inequality fom Y is dominated by the sum of the inequalities fom Y 1 and Y 2. The inequality fom Y is p j j = k=1 k=1 p j + 1 The inequality fom Y 1 is p k=1 The inequality fom Y 2 is p k=1 2 j U(Y; Y ) = U(Y; Y 1 )+U(Y; Y 2 ): j 1 j U(Y 1 ; Y 1 ): j 2 j U(Y 2 ; Y 2 ): Summing the inequalities fom Y 1 and Y 2 ; we get p k=1 (j 1 j + j 2 j) U(Y 1 ; Y 1 ) + U(Y 2 ; Y 2 ): Howeve, since thee ae no edges between Y 1 and Y 2 we get that U(Y 1 ; Y 1 ) + U(Y 2 ; Y 2 ) = U(Y 1 ; Y ) + U(Y 2 ; Y ): Thus p j j k=1 k=1 p (j j + j j) U(Y; Y ):

49 We next pove the \only if" diection. If the qth inequality p k=1 k j bi j U(Y q ; Y q ) i 2 Y q is edundant, then some linea inequality (of the 2 p coesponding to it) is edundant. Assume it is the jth linea inequality. By Lemma 5.4, if the jth inequality is edundant, then it is edundant fo some single commodity. At this point, we can apply the Wallace and Wets Theoem (Theoem 3.1) to this inequality, and we conclude that eithe G(Y q ) o G( Y q ) is not connected. We note that when p = 1 (one commodity), the system of absolute value inequalities educes to the same system of linea inequalities that is used in the Wallace-Wets Theoem. We get that the inequality fom set Y is j b i j U(Y; Y ): This is equivalent to the following two linea inequalities, b i U(Y; Y )? b i = b i U(Y; Y ); which ae the inequalities that Wallace and Wets would geneate fom the sets Y and Y. Thus, if p = 1, ou theoem is equivalent to the Wallace-Wets Theoem. 45