Projections of the capacitated network loading problem

` European Journal of Operational Research 122 (2000) 534±560

www.elsevier.com/locate/orms

Projections of the capacitated network loading problem Prakash Mirchandani

*

Katz Graduate School of Business, University of Pittsburgh, Pittsburgh, PA 15260, USA Received 6 August 1998; accepted 15 December 1998

Abstract Consider an undirected network and a set of commodities with speci®ed demands between various pairs of nodes of the network. Given two types of capacitated facilities that can be installed (loaded) for arc dependent costs, we have to determine the integer number of facilities to load on each arc in order to send the required ¯ow of all commodities at minimum total cost. We present a natural mixed-integer programming formulation of the problem and then consider its single commodity and multicommodity versions. We develop ``equivalent'' formulations in a lower-dimensional space by projecting out the ¯ow variables and study the polyhedral properties of the corresponding projection cones. Our results strengthen an existing result for multicommodity ¯ow problems. We also characterize several classes of facet de®ning inequalities for this lower-dimensional polyhedron, and conclude by identifying some open problems and future research directions. Ó 2000 Elsevier Science B.V. All rights reserved. Keywords: Integer programming; Capacitated network design problem; Multicommodity ¯ow; Facets; Extreme rays

1. Introduction Projection techniques are becoming increasingly important in the study of combinatorial polyhedra. Given some polyhedron in Rp  Rq , we can use these techniques to construct new polyhedra in lower- or higher-dimensional spaces. For example, suppose W is an arbitrary subset of Rq and we de®ne FY ˆ f…f; y† 2 Rp  Rq j Af ‡ By 6 b; f P 0; y 2 Wg by an m ´ p matrix A, an m ´ q matrix B and an m vector b. Assume that FY is nonempty. The projection of FY into the lower-dimensional subspace of y 2 Rq variables is de®ned as Y ˆ fy 2 Rq j there exists f 2 Rp for which …f; y† 2 FY g:

*

Tel.: +1-412-648-1652; fax: +1-412-648-1693. E-mail address: [email protected] (P. Mirchandani).

0377-2217/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 2 2 1 7 ( 9 9 ) 0 0 0 8 3 - 1

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We can correspondingly project FY into a higher-dimensional space. Projecting a polyhedron into a different space is potentially very powerful. First, Lovasz and Schrijver (1989), Sherali and Adams (1990) and Balas et al. (1993) develop a projection based procedure to solve integer programming problems. Second, Balas and Pulleyblank (1983, 1989) have used this methodology for facet identi®cation in the context of the perfectly matchable subgraph polytopes of bipartite and general graphs. (See also, for example, Ball et al., 1987; Vande Vate, 1989; Goemans, 1994.) Lastly, Barany et al. (1984) have used projection techniques to analytically study the ``equivalence'' of di€erent formulations for the uncapacitated economic lot-sizing model. Wolsey (1988) and Pochet and Wolsey (1988), respectively, have studied more general versions of the problem, i.e., those including start-up costs and backlogging. See Pulleyblank (1989) and Wolsey (1989) for a survey of recent work on projection. This paper studies projections and reformulations of a version of the capacitated network design problem that we call the network loading problem. In the network loading problem, we are given a connected, undirected network G(N, A), where N is the set of nodes and A is the set of arcs. We are also given two types of facilities that can be installed (loaded) on the arcs of the network at speci®ed costs. The capacities of these facilities, which we refer to as low capacity (LC) and high capacity (HC) facilities, equal to one and C, respectively. The facilities are undirected, i.e., loading a facility of capacity equal to one on an arc allows us to send a total bi-directional ¯ow of 1 unit on this arc. The cost structure of these facilities exhibits scale economies: loading C LC facilities is more expensive than loading a single HC facility although the capacities of both these alternatives is the same. The decision problem is to load enough facilities on the arcs of the network and send desired amounts of ¯ow between given origin±destination pairs at minimum total cost. In the multicommodity case, we are given multiple origins and/or multiple destinations that de®ne the commodities. The single commodity version considers the special case when a ¯ow of d must be sent from a (single) origin node O to a (single) destination node D. This paper is organized as follows. We begin by introducing the notation and a natural formulation for the multicommodity version of the problem in Section 2. This formulation, in the space of the ¯ow and the design variables, consists of the ¯ow conservation, capacity and cutset constraints. In Section 3, we develop a tight formulation of the single commodity version of the problem in the space of the design variables by projecting out the ¯ow variables. Section 4 studies the multicommodity case of the problem. We develop sucient conditions on the capacities that guarantee the existence of a multicommodity ¯ow in a capacitated network by reformulating the original problem in the subspace of design variables. Our conditions strengthen those existing in literature. Section 5 identi®es several classes of facets for this lower-dimensional polyhedron. Most of our results apply even when more than two facilities are available. Section 6 concludes the paper with some open questions and future research directions. 2. Notation and formulation For the network G…N ; A†, let xij (yij ) denote the number of LC (HC) facilities of capacity 1 (capacity C) that are loaded on arc fi; jg each at a cost of aij (bij ). For two nonempty sets S  N and T  N n S, we say that arc fi; jg belongs to {S, T} if nodes i and j belong to di€erent sets S and T. Let X X xij and YS;T ˆ yij XS;T ˆ fi;jg2fS;T g

fi;jg2fS;T g

respectively denote the aggregate number of LC and HC facilities loaded across fS; T g. Let …x; y† denote the design variable vector of size 2 |A|, indexed by the arcs of the network, i.e., …x; y† ˆ …x1;2 ; x1;3 ; . . . ; xjN jÿ1;jN j ; y1;2 ; . . . ; yjN jÿ1;jN j †

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in a complete network. In the above de®nition, we have used commas as separators in the subscripts for clarity. Let K denote the set of commodities with the origin of commodity k denoted by O(k), its destination by D(k) and its demand by dk . Given S  N and T  N n S, the aggregate demand, DS;T , between S and T equals the sum of the commodity demands for which the origin lies in S and the destination lies in T or viceversa. Let fijk denote the ¯ow of commodity k on arc fi; jg from i to j and fjik denote its ¯ow in the opposite direction. The network loading problem may be formulated as follows. Formulation FY(K): minimize

X

…aij xij ‡ bij yij †

…2:1a†

fi;jg2A

subject to: 8 > < ÿdk X X k k fji ÿ fij 6 dk > : j2N j2N 0 X k2K

 fijk ‡ fjik 6 xij ‡ Cyij

XS;T ‡ rYS;T P rdDS;T =Ce

if i ˆ O…k†; if i ˆ D…k†;

for all i 2 N ;

for all k 2 K;

…2:1b†

otherwise; for all fi; jg 2 A;

…2:1c†

for all S  N ; T ˆ N n S;

…2:1d†

0 6 xij ; yij 6 L and integer for all fi; jg 2 A;

…2:1e†

fijk ; fjik P 0 for all fi; jg 2 A;

…2:1f†

for all k 2 K:

In this formulation, the objective function (2.1a) minimizes the total cost incurred in loading facilities on the arcs of the network. Constraints (2.1b) are the ¯ow conservation constraints written in an (equivalent) inequality form instead of the usual equality form. Constraints (2.1c) are the capacity constraints and constraints (2.1d) are the cutset constraints. In these constraints, r equals DS;T mod(C); we adopt the convention that r equals C if DS;T is a multiple of C. Magnanti et al. (1995) show that the cutset constraints are facet de®ning under fairly mild conditions. (See also Mirchandani (1989) and Magnanti and Mirchandani (1993) and Magnanti et al. (1995) for polyhedral results of related models.) Constraints (2.1e) and (2.1f) are nonnegativity and integrality constraints with L a suciently large integer. The single commodity specialization of the above formulation is FY({1}) and we refer to it as FY(1) for convenience. In this formulation, we drop the superscript k for the ¯ow variables and the subscript k for the demand variables.

3. The single commodity case This section considers the single commodity version of the network loading problem. Using the max¯ow min-cut theorem, we can project the polyhedron corresponding to FY(1) into a lower-dimensional space. Let fS; T ˆ N n Sg denote an O±D cutset if the origin node, O and the destination node, D, of the commodity belong to di€erent sets S and T. Consider the following formulation in the subspace of …x; y† variables.

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Formulation Y(1): X …aij xij ‡ bij yij †; minimize

537

…3:2a†

fi;jg2A

subject to: XS;T ‡ CYS;T P d  XS;T ‡ rYS;T P r

for all O±D cutsets fS; T g; d C

…3:2b†

 for all O±D cutsets fS; T g;

0 6 xij ; yij 6 L and integer for all fi; jg 2 A:

…3:2c† …3:2d†

Notice that (i) we have included the cutset inequalities (2.1d) for O±D cutsets only since the inequalities corresponding to non O±D cutsets are redundant and (ii) projecting out the ¯ow variables increases the number of constraints exponentially. Proposition 3.1. The polyhedron corresponding to formulation Y(1) is the projection of the polyhedron corresponding to formulation FY(1) into the subspace of design variables. Proof. Suppose …^ x; ^ y† is a feasible solution to Y(1). Then the inequality X^ S;T ‡ C Y^ S;T P d for all O±D cutsets implies that the minimum capacity of an O±D cutset with x^ij ‡ C y^ij as the arc capacities is at least d. Consequently, we can ®nd a feasible ¯ow with an O±D ¯ow value of d, and thus a feasible solution to FY(1). h We now provide an alternate polyhedral proof of this proposition. This proof, although long, sets the stage for (and is useful in) the study of multicommodity case. We ®rst describe some preliminary polyhedral results for completeness. Let Conv [Y(1)] denote the convex hull of feasible solutions to the formulation Y(1). Proposition 3.2. Conv [Y(1)] is full-dimensional. Proof. Let (x, y)1 be some point belonging to Conv [Y(1)] satisfying (xij )1 , (yij )1 < L for all {i, j}. Since the choice of L is arbitrary, such a point exists. Now consider the points …x2 ; y2 † ˆ …x1 ‡ e1; y1 ‡ e1† and …x3 ; y3 † ˆ …x1 ‡ 2e1; y1 ‡ 2e1†, where e is a suciently small positive real number and 1 is a vector of ones of size |A|. The points (x2 , y2 ) and (x3 , y3 ) belong to Conv [Y(1)]. We will show that (x2 , y2 ) is an interior point. Assume not. Thus, there exists a valid inequality dx ‡ cy P c0 , with …d; c; c0 † 6ˆ …0†, for Conv [Y(1)] that is satis®ed at equality by (x2 , y2 ). Let ei be a unit vector of size |A| with a one in the ith position. Since (x1 , y1 ) belongs to Conv [Y(1)], so do the points (x2 ÿ eei , y2 ) for all i. Now, expressions dx2 ‡ cy2 ˆ c0 and d…x2 ÿ eei † ‡ cy2 P c0 , for all i, imply that all components of d are nonpositive. Repeating this argument for the y variable, we conclude that all components of c are also nonpositive. Similarly, since (x3 , y3 ) belongs to Conv [Y(1)], so do points (x2 + eei , y), for all i. Thus, all components of d must be nonnegative. Repeating this argument for the y variable, we conclude that all components of c are also nonnegative. Consequently, (d, c, c0 ) ˆ (0). This contradiction implies that (x, y)2 is an interior point, and thus, Conv [Y(1)] is full-dimensional. h The second method for projecting FY(1) into the subspace of the design variables is based on the following interpretation of Farkas' Lemma used by (Balas and Pulleyblank, 1983, 1989) in their study of the perfectly matchable subgraph polytopes for bipartite and general graphs.

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Theorem 3.3. Let Y be the projection of FY as de®ned in the introductory section of this paper and W ˆ fv 2 Rm j vA P 0; v P 0g: If v1 ; v2 ; . . . ; vs are the extreme rays of W, then Y ˆ fy 2 Rq j …vk B†y 6 vk b for all k ˆ 1; 2; . . . ; s; y 2 Wg: In this theorem, W is a pointed-polyhedral cone characterized by a ®nite number of extreme rays. We denote a ray by a point, and say that two rays, u and v, are equivalent if u ˆ av for some scalar a > 0. To project out the ¯ow variables from FY(1) for the single commodity case, we ®rst identify the cone corresponding to W. This cone is W …1† ˆ fa 2 RjN j ; b 2 RjAj j ai ÿ aj ‡ bij P 0; aj ÿ ai ‡ bij P 0 for all fi; jg 2 A; a; b P 0g:

…3:3†

W(1) is clearly polyhedral, and since it lies in the nonnegative orthant, it is also pointed. Proposition 3.4 describes some polyhedral properties of this cone. Proposition 3.5 shows that every extreme ray of W(1) must be of one of three di€erent forms. Proposition 3.4. Let W(1) be the polyhedral cone de®ned by (3.3). Then 1. W(1) is full-dimensional. 2. The following inequalities de®ne facets of W(1): (a) ai ÿ aj + bij P 0 for all {i, j}, (b) aj ÿ ai + bij P 0 for all {i, j} and (c) ai P 0 for all i. Proof. See Appendix A.

h

Proposition 3.5. Let (a, b) be a ray of W(1). Then (a, b) is an extreme ray of W(1) if and only if it is equivalent to one of the following forms. 1. Arc form: bi  j  ˆ 1

for some fi ; j g 2 A;

a; b ˆ 0

otherwise:

2. Node form: ai ˆ 1 for all i 2 N ; bij ˆ 0 for all fi; jg 2 A: 3. Cutset form: ai ˆ 1

for all i 2 S and 0 otherwise;

bij ˆ 1 for all fi; jg 2 fS; T g and 0 otherwise; for some nonempty sets S  N , T ˆ N n S and for which the subgraph induced by S is connected. Proof. Suciency. Suppose a ray, say …a; b†, is contained in W(1) but is not an extreme ray. By de®nition, 1 2 1 2 there exist two rays …a; b† and …a; b† that satisfy (i) Condition A: …a; b† and …a; b† belong to W(1), (ii) 1 2 1 Condition B: …a; b† and …a; b† are not equivalent to …a; b† and (iii) Condition C: …a; b† ˆ 1=2…a; b† + 1/2 2 …a; b† . (The multipliers 1/2 can be replaced by any positive scalars in Condition C.)

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2

First, suppose that …a; b† has the arc form. Since …a; b† and …a; b† satisfy Condition A, they are non1 2 negative. Condition C implies (ai )1 ˆ (ai )2 ˆ 0 for all i 2 N and …bij † ˆ …bij † ˆ 0 for all fi; jg 6ˆ fi ; j g. 1 2 Thus, both (a, b) and (a, b) are equivalent to (a, b), a contradiction. Next, assume that …a; b† has the node form. Conditions A and C imply that …bij †1 ˆ …bij †2 ˆ 0 for all fi; jg 2 A. Since …ai †1 ÿ …aj †1 P 0 and …aj †1 ÿ …ai †1 P 0 for all fi; jg 2 A, and the network is connected, all components of (a)1 are equal, i.e., …ai †1 ˆ d1 > 0 for all i 2 N . Similarly, …ai †2 ˆ d2 > 0 for all i 2 N . Thus, both (a, b)1 and (a, b)2 are again equivalent to (a, b), a contradiction. 1 Finally, suppose that (a, b) has the cutset form. Since (a, b)1 and (a, b)2 are nonnegative, …ai † ˆ 2 1 2 …ai † ˆ 0 for all i 2 T and …bij † ˆ …bij † ˆ 0 for all fi; jg 2 fS; Sg or fi; jg 2 fT ; T g. Because the subgraph 1 2 1 1 2 induced by S is connected (and since …bij † ˆ …bij † ˆ 0 for all fi; jg 2 fS; Sg†, …ai † ˆ …aj † and …ai † ˆ 2 1 2 …aj † for all i,j 2 S. Thus, for some d1 ; d2 P 0; …ai † ˆ d1 and …ai † ˆ d2 for all i 2 S. Now, consider arc fi ; j g 2 fS; T g with node i 2 S. The de®nition of W(1) and the fact that …aj †1 ˆ …aj †2 ˆ 0 for all j 2 T imply …bi j †1 ÿ …ai †1 P 0

and

…bi j †1 ‡ …ai †1 P 0

…bi j †2 ÿ …ai †2 P 0

and

…bi j †2 ‡ …ai †2 P 0:

and

1

2

Since both …ai † and …ai † are nonnegative, the above inequalities reduce to 1

1

…bi j † P …ai † ˆ d1

and

2

2

…bi j † P …ai † ˆ d2 :

Since (a, b) is of the cutset form, ai ˆ 1. Further, Condition C implies 1=2…d1 ‡ d2 † ˆ 1. Assume, 1 1 2 without loss of generality, that …bi j † > d1 . Then, bi j ˆ 1=2‰…bi j † ‡ …bi j † Š > 1=2…d1 ‡ d2 † ˆ 1, a 1 2 contradiction. Consequently, …bi j † ˆ d1 and …bij † ˆ d2 . Since arc fi ; j g was chosen arbitrarily, …bij †1 ˆ d1 and …bij †2 ˆ d2 for all fi; jg 2 fS; T g. Furthermore, both d1 and d2 > 0, else (a, b)1 or (a, b)2 does not de®ne a ray. Thus, (a, b)1 and (a, b)2 are (positive) scalar multiples of (a, b), which contradicts Condition B. Necessity. Let (a, b) be an extreme ray of W(1). Let P ˆ ffi; jg 2 A: bij > 0g and Q ˆ fi 2 N : ai > 0g. If P ˆ /, then ai ˆ d P 0 for all i 2 N . If d ˆ 0, (a, b) ˆ 0 and therefore, it cannot be a ray. If d > 0, we obtain a ray that is equivalent to a node form ray. Next, assume that P ¹ /. If Q ˆ /, we show that |P| ˆ 1 (i.e., (a, b) is equivalent to an arc form ray). Suppose |P| > 1. Let bi1 j1 ; bi2 j2 ; . . . ; bijP j jjP j > 0. De®ne 1

…ai † ˆ 0 1  …a; b† : 2bij …bij †1 ˆ 0

for all i 2 N ; if fi; jg ˆ fi1 ; j1 g; otherwise;

and 2

…ai † ˆ 0 2  …a; b† : 2bij 2 …bij † ˆ 0

for all i 2 N ; if fi; jg ˆ fik ; jk g; k ˆ 2; 3; . . . ; jP j; otherwise:

But, then (a, b), (a, b)1 and (a, b)2 satisfy Conditions A, B and C, contradicting the assumption that (a, b) is extreme. Now, assume that P¹/ and Q ˆ N and de®ne a ˆ minfai : i 2 Qg > 0. Let

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P. Mirchandani / European Journal of Operational Research 122 (2000) 534±560



1

1

…a; b† :

…ai † ˆ

a 0

…bij †1 ˆ 0 and 2

…a; b† :

for all fi; jg 

2

…ai † ˆ

if i 2 Q; otherwise;

2ai ÿ a 0

2

…bij † ˆ 2bij 1

if i 2 Q; otherwise; for all fi; jg:

2

Since (a, b), (a, b) and (a, b) satisfy Conditions A, B and C, (a, b) cannot be extreme. Lastly, consider the case when P¹/ and Q¹/, Q  N . We proceed as follows. Claim 1. bij > 0 if fi; jg 2 fQ; N n Qg. Let i ; j 2 fQ; N n Qg with i 2 Q. Since ai > 0 and aj ˆ 0, inequalities (3.3) imply that bi j ˆ ai > 0: Claim 2. bij ˆ 0 if fi; jg 2 fN n Q; N n Qg: Assume not. Thus, for some nodes i and j 2 N n Q, ai ˆ 0; aj ˆ 0 but bi j > 0. Consider the following two nonequivalent rays that belong to W(1): …ai †1 ˆ 0  …a; b†1 : 2bij 1 …bij † ˆ 0

for all i 2 N ; if fi; jg ˆ fi ; j g; otherwise;

and …ai †2 ˆ 2ai  …a; b†2 : 0 2 …bij † ˆ 2bij

for all i 2 N ; if fi; jg ˆ fi ; j g; otherwise:

Again, …a; b† ˆ 1=2‰…a; b†1 ‡ …a; b†2 Š, contradicting the assumption that (a, b) is extreme. Claim 3. The subgraph induced by the nodes in Q is connected. Assume not. Let Q1 and Q2 ˆ Q\Q1 induce two subgraphs that are not connected to each other in Q. De®ne the following nonequivalent rays belonging to W(1):  …2ai † if i 2 Q1 ; 1 …ai † ˆ 0 otherwise; 1  …a; b† : ÿ
1 2bij if fi; jg 2 fN n Q; Q1 g or fi; jg 2 fQ1 ; Q1 g; bij ˆ 0 otherwise; and



…2ai † if i 2 Q2 ; 0 otherwise; 2  …a; b† : ÿ
2 2bij if fi; jg 2 fN n Q; Q2 g or fi; jg 2 fQ2 ; Q2 g; bij ˆ 0 otherwise: 2

…ai † ˆ

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We again have (a, b) ˆ 1/2 [(a, b)1 + (a, b)2 ], contradicting the hypothesis that (a, b) is extreme. Claim 4: If i 2 Q, then ai ˆ d > 0; if fi; jg 2 fQ; Qg, then bij ˆ 0; and if fi; jg 2 fN n Q; Qg; then bij ˆ d. Assume not and de®ne, as earlier, a ˆ minfai : i 2 Qg > 0. Let ( a if i 2 Q; 1 …ai † ˆ 0 otherwise; 1 …a; b† : ( ÿ
1 a if fi; jg 2 fN n Q; Qg; bij ˆ 0 otherwise; and 2

…ai † ˆ



2ai ÿ a

if i 2 Q;

0 otherwise; 8 2 …a; b† : > 2bij ÿ a if fi; jg 2 fN n Q; Qg; ÿ
2 < if fi; jg 2 fQ; Qg; bij ˆ 2bij > : 0 otherwise: If any of the conditions in Claim 4 is not true, then (a, b), (a, b)1 and (a, b)2 satisfy Conditions A, B and C, contradicting the assumption that (a, b) is extreme. Claims 1±4 show that (a, b) is equivalent to a cutset form ray with S ˆ Q and T ˆ N\Q. h We use these extreme ray forms to apply Theorem 3.3 and project FY(1) into the space of (x, y) variables. Suppose the extreme ray is described by the arc form. Then we obtain xij ‡ Cyij P 0 for all fi; jg 2 A, which is clearly redundant since it is a linear combination of the nonnegativity constraints. If the extreme ray is of the node form, we obtain a vacuous constraint 0y P 0. Now, suppose that the extreme ray is of the cutset form. If nodes O and D are contained in S for some S  N , then we obtain XS;T ‡ CYS;T P 0, which is redundant. By symmetry, if O and D are contained in T, then the corresponding inequality is also redundant. If O 2 T and D 2 S, then we obtain XS;T ‡ CYS;T P ÿ d, which is also redundant. Lastly, if O 2 S and D 2 T , then we obtain XS;T ‡ CYS;T P d: Consequently, we have generated all the required inequalities of Y(1) except those for which the subgraph induced by S is not connected. But it is easy to see that these inequalities are implied by the inequalities obtained above. We have thus completed the second proof of Proposition 3.1. 4. The multicommodity case We now turn to the multicommodity case of the network loading problem and project out the ¯ow variables from FY(K). Viewed from another standpoint, suppose the capacity of arc {i, j} equals cij and we solve the corresponding multicommodity ¯ow problem. Now, even if the aggregate capacity of every cutset exceeds the total demand across the cutset, a feasible ¯ow meeting the ¯ow conservation and capacity constraints might not exist (Hu, 1966; Seymour, 1980). Iri (1971) has proved that there exists a ¯ow satisfying the ¯ow conservation constraints for all the commodities and capacity constraints for all the arcs if and only if

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X

cij bij P

fi;jg2A

X k2K

dk lenk …b† for any bij P 0; fi; jg 2 A:

…4:1†

In this expression, lenk (b) denotes the length of the shortest path from O(k) to D(k) using bij as the arc lengths. Using Theorem 3.3, we derive a stronger result from a network loading viewpoint, i.e., when the arc capacities are variables. Since the ¯ow conservation constraints (2.1b) are separable, the cone W(1) generalizes to the multicommodity case and is given by W …K† ˆ fa 2 RjN jjKj ; b 2 RjAj jaki ÿ akj ‡ bij P 0; akj ÿ aki ‡ bij P 0 for all k 2 K;

for all fi; jg 2 A;

a; b P 0g:

…4:2†

Using the argument of Proposition 3.4, we can show Proposition 4.1. Let W(K) be the polyhedral cone de®ned by (4.2). Then 1. W(k) is full-dimensional. 2. The following inequalities de®ne facets of W(k): (a) aki ÿ akj ‡ bij P 0; for all fi; jg; for all k, (b) akj ÿ aki ‡ bij P 0; for all fi; jg; for all k and for all i, for all k. (c) aki P 0 Proposition 4.2 characterizes the necessary conditions that must be satis®ed by any extreme ray of W(K) using arguments that are generalizations of the necessity proof for the single commodity case. These conditions will help us in projecting out the ¯ow variables and in strengthening inequality (4.1). To avoid some technicalities, we will assume that the network is complete in the sequel. For each k 2 K; define z…k† ˆ fi 2 N : aki ˆ 0g: Proposition 4.2. (a, b) is an extreme ray of W(K) only if it is equivalent to one of the following forms. (i) Arc form: bi  j  ˆ 1

for some …i j † 2 A;

…a; b† ˆ …0†

otherwise:

(ii) Node form: 

aki ˆ 1

for all i 2 N ; for some k  2 K;

…a; b† ˆ …0†

otherwise:

(iii) Shortest path form: b 2 RjAj ‡ : …a† bij ‡ bjl P bil jjKj : a 2 RjN ‡ …b† aki ˆ 0 …c† aki 6 lenk …b†

for all triplets i; j; l 2 N ; for each k 2 K; for some i 2 N : for all i; k; where lenk …b† denotes the shortestpath length from any node in z…k† to i with bij as the arc lengths:

Proof. Let (a, b) be an extreme ray of W(K). De®ne the following index sets:

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543

P ˆ ffi; jg 2 A: bij > 0g; Qk ˆ fi 2 N : aki > 0g

for each k 2 K; and

R ˆ fk 2 K: there exists an i 2 N satisfying aki > 0g: First, suppose P ˆ /. Then, |R| ˆ 0 implies that (a, b) ˆ (0), and therefore …a; b† cannot be a ray. Next, suppose that |R| > 1, and without loss of generality, assume that commodity 1 2 R. Since P ˆ /, aki ˆ dk > 0 for all i 2 N ; k 2 R and Qk ˆ N for all k 2 R. De®ne ( ÿ
1 a1i ˆ 2d1 for all i 2 N ; 1 …a; b† : 1 …a; b† ˆ 0 otherwise; and 2

…a; b† :

( ÿ
2 aki ˆ 2dk 2

…a; b† ˆ 0

for all i 2 N ;

for all k 2 R n f1g;

otherwise:

1

2

Since …a; b† ˆ 1=2‰…a; b† ‡ …a; b† Š and Conditions A, B and C are satis®ed, we get a contradiction to the assumption that …a; b† is extreme. Thus, |R| ˆ 1, and therefore, …a; b† must be of the node form. Next, suppose that P 6ˆ U. If Qk ˆ / for all k 2 K, then the argument of Proposition 3.5 proves that …a; b† is of the arc form. Finally, consider the case when P 6ˆ /; Qk 6ˆ / for some k. Consequently, jRj P 1. If |R| ˆ 1, then the situation is similar to the one in Proposition 3.5. For jRj P 2, we proceed as follows. Claim 1: aki ˆ 0 for some node i, for all k 2 R. Assume that this equality does not hold for commodity 1. Let a ˆ min…a1i : i 2 N † > 0 and construct the following two rays to obtain a contradiction. 1

…a; b† : and 2

…a; b† :

( ÿ
1 a1i ˆ a1i ÿ a 1

…a; b† ˆ …a; b† ( ÿ
2 a1i ˆ a1i ‡ a 2

…a; b† ˆ …a; b†

for all i 2 N ; otherwise

for all i 2 N ; otherwise:

Thus condition (b) of the shortest path extreme ray form holds. k Claim 2: For a speci®ed b 2 RjAj ‡ ; ai is not greater than the shortest path length from any node in z(k) to node i for all k 2 R. using bij as the arc lengths.

Follows from linear programming duality upon observing that with bij ®xed, the constraints of W(K) are simply the dual constraints of the shortest path problem for each k. Thus condition (c) of the shortest path extreme ray form holds.   De®ne an arc {i, j} to be tight with respect to commodities in H  K if bij ˆ maxk2K jaki ÿ akj j ˆ jahj ÿ ahj j for all h 2 H . jN jxjKj , every arc is tight with respect to some commodity. Claim 3: Given a 2 R‡

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ÿ
Assume arc f1; 2g is not tight. Then clearly b12 > 0. Moreover, for d ˆ b12 ÿ maxk2K jak1 ÿ ak2 j > 0, let ( 1 …b12 † ˆ b12 ÿ d; 1 …a; b† : 1 …a; b† ˆ …a; b† otherwise; and

( 2

…a; b† :

2

…b12 † ˆ b12 ‡ d; 2

…a; b† ˆ …a; b†

otherwise;

de®ne two nonequivalent extreme rays satisfying Conditions A, B and C. Consequently, a contradiction ensues. In particular, Claim 3 shows that condition (a) of the shortest path form holds. For, if bil > bij + bjl then arc fi; lg will not lie on the shortest path for any commodity and therefore, will not be tight with respect to any commodity. Claims 1±3 prove that the extreme ray must have the shortest path form. h We now project FY(K) into the subspace of (x, y) variables. Note that the conditions that we impose on the b's are tighter than those proposed by Iri (1971). As a consequence, we replace the in®nite set of inequalities introduced by Iri by a ®nite set that correspond to the extreme rays of the polyhedral cone W(K). It also follows that we restrict ourselves to only those nonnegative b's that satisfy the triangle inequality. To project FY(K), observe that as in the single commodity case, the constraints corresponding to the arc and node form extreme rays are redundant and vacuous, respectively. The constraints corresponding to the shortest path form extreme rays are  X  Xÿ
xij ‡ Cyij bij P dk akO…k† ÿ akD…k† for all …a; b† 2 W …K†: k2K

fi;jg2A

For a given b such that …a; b† 2 W(K), we obtain a nonredundant constraint when akO…k† ˆ lenk …b† and akD…k† ˆ 0 for all k. This constraint is at least as strong as the other constraints generated by extreme rays of the shortest path form. We have thus proved Theorem 4.3. The feasible region of FY(K) projected into the subspace of …x; y† variables is described by X Xÿ
xij ‡ Cyij bij P dk lenk …b†; for all b such that …a; b† 2 W …K†; …4:3a† k2K

fi;jg2A

 XS;T ‡ r YS;T P r

 DS;T ; C

0 6 xij ; yij 6 L and integer for all fi; jg 2 A:

…4:3b† …4:3c†

For the single commodity case, the cutset form of the extreme ray (Proposition 3.5) is a special case of the shortest path form and is obtained by setting bij equal to one for all arcs across the fS; T g cutset and 0 otherwise. The corresponding constraint (3.2b) is a special case of (4.3a) as well. Both the cutset form extreme ray and constraint (3.2b) also have their multicommodity analogs. Thus, we have generated a large number of constraints (4.3a) for values of b between 0 and some large upper bound. Section 5 identi®es several conditions on b for which constraints (4.3a) are facet de®ning.

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5. New families of extreme rays and facets We now focus our attention on the formulation for the network loading problem in the lower-dimensional space of x and y variables, and identify conditions on b for which inequality (4.3a) is facet de®ning. To do so, we will ®rst identify extreme rays of cone W(K) for two special networks, and show that the corresponding inequalities are facet de®ning. We will then generalize these results. Observe that if the aggregate capacity across any cutset exceeds the demand across the cutset, then there exists a feasible multicommodity ¯ow that satis®es the ¯ow conservation and the capacity constraints if the demand network has a (i) two-star topology (Papernov, 1976) and (ii) k4 topology (Papernov, 1976; Seymour, 1980). (See Schrijver, 1982 for other comparable topologies.) Thus, in these cases, the only facet de®ning inequalities arising from inequality (4.3a) are the aggregate capacity-demand inequalities XS;T + CYS;T P DS;T . (We obtain these inequalities if the multicommodity analog of the cutset form of extreme ray is used for projection.) In the following discussion, we identify new classes of extreme rays and ®rst consider a 5-node 4commodity network. Note that a 4-node 4-commodity network is a special case of the two-star topology, and any smaller network cannot have four distinct origin±destination pairs. Thus, we need at least ®ve nodes if we want to study a meaningful case of the 4-commodity problem. We next consider a 6-node 3commodity network, again the smallest interesting case for a 3-commodity problem. As mentioned earlier, our analysis assumes that the underlying networks are undirected and complete. 5.1. 1-2 extreme rays This subsection describes a class of extreme rays of W(K) with coecients equaling 1 or 2. Instead of (wordily) characterizing this extreme ray in text form, we depict the ray graphically. Fig. 1 describes the 1-2 extreme ray for the 5-node 4-commodity network. The origin±destination pairs for each commodity are tabulated at the bottom of Fig. 1. The numbers on the arcs represent the corresponding b values and the numbers on the nodes represent the a values ± the ®rst number corresponds to the ®rst commodity, the second number to second commodity and so forth. The motivation for the indicated choice of b values is to admit sucient degeneracy in the shortest paths between the origin±destination pairs of nodes. These alternative shortest paths allow us to derive conditions relating the b values to each other. Proposition 5.1. Let G(N, A) be the 5-node 4-commodity network shown in Fig. 1 and let W(K) be the corresponding cone. Then, the 1-2 ray depicted graphically in Fig. 1 is an extreme ray of W(K). Proof. Let (a, b) denote the ray described in Fig. 1. If it is not extreme, then two rays, say, (a, b)1 and (a, b)2 must satisfy the Conditions A, B and C introduced in Proposition 3.5. Observe that nonnegativity of (a, b)1 1 2 and (a, b)2 implies that …akD…k† † ˆ …akD…k† † ˆ 0 for all k. Now, consider arc {1, 2}. Feasibility of (a, b)1 and 1 1 2 2 (a, b)2 implies that …b12 † P …a12 † and …b12 † P …a12 † . Now, if either of these inequalities holds as a strict 1 2 1 2 1 1 inequality, then 2 ˆ b12 ˆ 0:5……b12 † ‡ …b12 † † > 0:5……a12 † ‡ …a12 † † ˆ a12 ˆ 2. Thus, …b12 † ˆ …a12 † and 2 2 …b12 † ˆ …a12 † . Proceeding in a similar fashion, we obtain the (a, b)1 relationships shown in Fig. 2. 1 1 Now, consider arc {1,2} and …a12 † . Since …a12 † cannot be larger than any other path length, 1 1 1 2 2 …b12 † 6 …b13 † ‡ …b23 † . Similarly, …b12 † 6 …b13 † ‡ …b23 †2 . Suppose the ®rst inequality holds as a strict inequality. Then, 2 ˆ b12 ˆ 0:5……b12 †1 ‡ …b12 †2 † < 0:5……b13 †1 ‡ …b23 †1 ‡ …b13 †2 ‡ …b23 †2 † ˆ b13 ‡ b23 ˆ 2. This contradiction, and similar reasoning for the other arcs implies the following equations: 1 1 1 1 1 1 1 (i) …b12 † ˆ …b13 † ‡ …b23 † ˆ …b14 † ‡ …b24 † ˆ …b15 † ‡ …b25 † ; 1 1 1 1 1 (ii) …b34 † ˆ …b13 † ‡ …b14 † ˆ …b23 † ‡ …b24 † ,

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Fig. 1. Graphical representation of a 1-2 ray for the 5-node 4-commodity case.

Fig. 2. …a†1 variables ®xed on the basis of …b†1 variables. 1

1

1

1

1

(iii) …b35 † ˆ …b13 † ‡ …b15 † ˆ …b23 † ‡ …b25 † , 1 1 1 1 1 (iv) …b45 † ˆ …b14 † ‡ …b15 † ˆ …b24 † ‡ …b25 † : 1 1 1 1 1 1 Eliminating variables, we obtain …b13 † ˆ …b14 † ˆ …b15 † ˆ …b23 † ˆ …b24 † ˆ …b25 † . Substituting these 1 1 1 1 1 values into (i)±(iv) shows that …b12 † ˆ …b34 † ˆ …b35 † ˆ …b45 † ˆ 2…b13 † , and consequently, (a, b)1 is equivalent to (a, b). Similarly, (a, b)2 is equivalent to (a, b), which must, therefore, be extreme. h

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547

Fig. 3. …a; b† values for a 5-component subdivision of a network.

We can ``lift'' these extreme rays for a larger network subdivided into ®ve components Si that correspond to nodes i, for i ˆ 1; 2; . . . ; 5. (See Fig. 3.) Each component consists of either a single node or a set of nodes. Classify the commodities of the network into the following ®ve categories. 1. Category 1. All commodities whose origin nodes belong to S2 and destination nodes belong to S1 . Since the network is undirected, we can assume that no commodity has its origin node in S1 and destination node in S2 . A similar remark applies to the following categories also. 2. Category 2. All commodities whose origin nodes belong to S4 and destination nodes belong to S3 . 3. Category 3. All commodities whose origin nodes belong to S5 and destination nodes belong to S3 . 4. Category 4. All commodities whose origin nodes belong to S5 and destination nodes belong to S4 . 5. Category 5. All other commodities. Further, let the b values for an arc whose both endpoints belong to the same component equal zero, and the b values for an arc whose endpoints lie in two di€erent components be as shown in Fig. 3. For example, bij equals 0 for all fi; jg 2 fS1 ; S1 g and bij equals two for all fi; jg 2 fS1 ; S2 g. Let the numbers on the components in Fig. 3 represent the a values ± the ®rst number corresponds to all commodities in the ®rst category, the second number to all commodities in the second category, and so forth. Extending our arguments used in Proposition 5.1, we can easily establish that the (a, b) ray de®ned graphically in Fig. 3 is an extreme ray. 5.2. 1-2-3 extreme rays We now describe sucient conditions for characterizing extreme rays of W(K) for situations with three commodities in a six node network. Each node is either the origin or the destination of some commodity. Again, we will describe the 1-2-3 ray in a graphical manner in Fig. 4 for the three origin±destination pairs shown. Proposition 5.2. Let G(N, A) be the 6-node 3-commodity network shown in Fig. 4 and let W(K) denote the corresponding cone. Then, the 1-2-3 ray described graphically in Fig. 4 is an extreme ray of W(K). Proof. Let (a, b) represent the 1-2-3 ray described in Fig. 4, and let (a, b)1 and (a, b)2 satisfy Conditions A, B and C. First, we observe that the arcs incident to the destination nodes are tight with respect to (b)1 . Thus, as with the 1-2 extreme ray, we can ®x the (a)1 coecients as shown in Fig. 5.

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Fig. 4. Graphical representation of a 1-2-3 ray of the 6-node 3-commodity case. 1

1

1

We now derive relationships between the various (b)1 values. First, we note that …b14 † ˆ …b12 † ‡ …b24 † , otherwise, as in the proof of Proposition 5.1, we have a contradiction to Condition C. This argument applied to the other arcs proves that 1

1

1

1

1

1

1

1

1

1

1

1

1

…i† …b14 † ˆ …b12 † ‡ …b24 † ˆ …b13 † ‡ …b34 † ˆ …b15 † ‡ …b45 † ˆ …b16 † ‡ …b46 †

1

…ii† …b13 † ˆ …b12 † ‡ …b23 † ˆ …b15 † ‡ …b35 † ; …iii† …b16 †1 ˆ …b12 †1 ‡ …b26 †1 ˆ …b15 †1 ‡ …b56 †1 ; …iv† …b24 †1 ˆ …b23 †1 ‡ …b34 †1 ˆ …b26 †1 ‡ …b46 †1 ; 1

1

1

1

1

1

1

1

1

1

1

1

1

1

…v† …b25 † ˆ …b23 † ‡ …b35 † ˆ …b26 † ‡ …b56 † ˆ …b12 † ‡ …b15 † ; …vi† …b36 † ˆ …b23 † ‡ …b26 † ˆ …b35 † ‡ …b56 † ˆ …b34 † ‡ …b46 † : 1

1

1

1

1

1

1

1

Eliminating variables, we obtain …b12 † ˆ …b15 † ˆ …b23 † ˆ …b26 † ˆ …b34 † ˆ …b35 † ˆ …b46 † ˆ …b56 † . Sub1 1 1 1 1 1 1 stituting these coecients in (i)±(vi), we obtain …b13 † ˆ …b16 † ˆ …b24 † ˆ …b25 † ˆ …b36 † ˆ …b45 † ˆ 2…b12 † 1 1 and …b14 † ˆ 3…b12 † . Consequently, we have proved that the 1-2-3 ray described in Fig. 4 is an extreme ray. h

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Fig. 5. Relationships between coecient values fo …a; b†1 .

These extreme rays are asymmetric with respect to the commodities. Thus, with the commodities ®xed, we can generate six such extreme rays. Furthermore, we can generalize Proposition 5.2 to a larger network with six components as we had done for Proposition 5.1. In general, extreme rays do not de®ne facets of the projected polyhedra. However, as with the 1-2 extreme ray discussed earlier, the 1-2-3 extreme rays also generate facet de®ning inequalities for the network loading problem in the space of x and y variables. We discuss these facets in the following discussion. 5.3. New families of facets To motivate this discussion, let us ®rst generate the constraint corresponding to the 1-2 extreme ray developed for the 5-node 4-commodity case. This constraint is 2x12 ‡ x13 ‡ x14 ‡ x15 ‡ x23 ‡ x24 ‡ x25 ‡ 2x34 ‡ 2x35 ‡ 2x45 ‡ C …2y12 ‡ y13 ‡ y14 ‡ y15 ‡ y23 ‡ y24 ‡ y25 ‡ 2y34 ‡ 2y35 ‡ 2y45 † P 2  …d1 ‡ d2 ‡ d3 ‡ d4 †:

…5:1†

Similarly, generating the inequality corresponding to the 1-2-3 ray, we obtain the following constraint for the 6-node 3-commodity problem: x12 ‡ 2x13 ‡ 3x14 ‡ x15 ‡ 2x16 ‡ x23 ‡ 2x24 ‡ 2x25 ‡ x26 ‡ x34 ‡ x35 ‡ 2x36 ‡ 2x45 ‡ x46 ‡ x56 ‡ C …y12 ‡ 2y13 ‡ 3y14 ‡ y15 ‡ 2y16 ‡ y23 ‡ 2y24 ‡ 2y25 ‡ y26 ‡ y34 ‡ y35 ‡ 2y36 ‡ 2y45 ‡ y46 ‡ y56 † P 3d1 ‡ 2  …d2 ‡ d3 †:

…5:2†

In fact, inequalities (5.1) and (5.2) de®ne facets of the respective lower-dimensional polyhedra in the x and y space if all the commodity demands are C or more. We ®rst prove that inequality (5.1) de®nes a facet and

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then show how it can be generalized to networks having a larger number of nodes, or a higher number of commodities. Next, we will discuss inequality (5.2) and its generalizations. Notation. In the sequel, we will use the following notation for denoting the facet inequalities.

The ®rst group of numbers denotes the set from which the coecients of the x variables are chosen. (Zero is trivially included in all coecient sets.) The number in the second section denotes the number of components in the network and that in the third section denotes the number of fundamental aggregate commodities (to be de®ned later). A component can be either an original node of the network or a set of nodes. The demand and design variables across the components are aggregated in the same fashion as we had aggregated them in de®ning the cutset inequalities (2.1d). Finally, the letter in the last section is either C or r and implies that the coecient of yij equals either C or r times the coecient of xij . To illustrate this notation with a concrete example, we would represent the cutset inequalities (2.1d) as 1/2/1/r. The coecients of all x variables in the inequality are either 0 or 1, and the coecients of the y variables are either 0 or r. The inequality has two components (S and T), and it models one fundamental aggregate commodity across these components. Similarly, the aggregate capacity±demand inequalities, XS;T + CYS;T P DS;T , would be denoted as 1/2/1/C. 5.4. The class of 1-2 facets Suppose we are given a network, a subdivision of its nodes into p components, say, S1 ; S2 ; . . . ; Sp ; and a partition of these components into two parts. (In the sequel, we will use the term subdivision to denote a mutually exclusive and collectively exhaustive collection of components; and partition to denote a grouping of components.) The 1-2 facet inequalities would have either of the following forms: X bij …xij ‡ Cyij † P b0 ; or …i† fi;jg2A

…ii†

X

bij …xij ‡ ryij † P b0 :

…5:3†

fi;jg2A

In these inequalities, the right hand side b0 is some function (to be de®ned momentarily) of the commodity demands and the other b coecients are …i† buv ˆ 0 …ii† buv ˆ 2

if fu; vg 2 fSi ; Si g for some i; if fu; vg 2 fSi ; Sj g; i 6ˆ j; and Si ; Sj lie on the same side of the partition;

…iii† buv ˆ 1

if fu; vg 2 fSi ; Sj g; i 6ˆ j; and Si ; Sj lie on opposite side of the partition:

As an illustration, suppose subdivide a larger, complete network into ®ve components: S1 ±S5 . Take a partition with components S1 and S2 on one side, and the remaining three components on the other. Next,

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Fig. 6. A 5-component subdivision and the corresponding b values. For arcs contained in the same component, the b values are O.

assign the b values as described above. Fig. 6 illustrates this graphically. Note that the b values refer to all arcs across the corresponding pair of components. Let every pair of components on the same side of the partition de®ne a fundamental component pair. For example, S1 and S2 de®ne a fundamental component pair. Similarly, S3 and S4 de®ne another such pair. Recollect that the aggregate demand between two components equals the sum of commodity demands for which the origin lies in one component and the destination lies in the other. Each aggregate demand between a fundamental component pair de®nes a fundamental aggregate commodity. We say that a network subdivision is demand satis®able with respect to a particular partition if the aggregate demand between every fundamental component pair is no less than C, that is, the demand for every fundamantal aggregate commodity is no less than C. The concept of demand satis®ability is important in determining when a 1-2 inequality is facet de®ning. 5.4.1. 1-2/5/4/C facet inequalities Consider the 5-component subdivision of a network shown in Fig. 6. Proposition 5.3 shows that the 1-2/ 5/4/C inequality is facet de®ning for the convex hull of feasible solutions to the network loading problem. As a corollary, it follows that inequality (5.1) is facet de®ning if dk P C for all k in the 5-node 4-commodity network of Fig. 1. Note that if dk < C, for some k, then inequality (5.1) may be weak. In particular, if dk ˆ 0, for some k, then we have a two-star or simpler topology and hence (5.1) is not a facet de®ning inequality. Proposition 5.3. Consider a 5-component subdivision of a complete network and let Conv [G(N, A)] denote the convex hull of feasible solutions to the corresponding network loading problem. If the network subdivision is demand satis®able, then inequality (5.4) de®nes a facet of Conv [G(N, A)]. X fi;jg2A

bij …xij ‡ Cyij † P

X

lenk …b†dk :

…5:4†

k2K

In this inequality, b's equal {FUNC {0}} 0, 1 or 2 as depicted in Fig. 6. Proof. The validity of this inequality follows from Theorem 3.3 and our remarks following Proposition 5.1. Next note that inequality (5.4) is not an improper face of Conv ‰G…N ; A†Š since some feasible solutions satisfy it as a strict inequality. Assume, with c and h of appropriate dimension, that

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Xÿ


cij xij ‡ hij yij P c0

…†

fi;jg2A

de®nes a face of Conv [G(N, A)] that contains the points satisfying inequality (5.4) at equality. We will show that each coecient cij and hij equals the corresponding coecient in inequality (5.4). First, we generate a feasible solution satisfying inequality (5.4) at equality. For each commodity k that does not belong to any of the (four) fundamental aggregate commodities, install dk LC facilities on arc fO…k†; D…k†g. Assume without loss of generality, that node i belongs to Si , for i ˆ 1; 2; . . . ; 5. Now, consider the (original) commodities that belong to the fundamental aggregate commodity de®ned by S1 and S2 . To satisfy the demand of any such commodity k, route it through nodes 1 and 2, installing dk LC facilities on arcs fD…k†; 1g and {2, O(k)} for each such k; and a total of DS1 S2 LC facilities on arc f1; 2g. Route the commodities that belong to the remaining fundamental aggregate commodities in a similar fashion. Augment the above solution by installing an additional LC facility on an arc fu; vg 2 fSi ; Si g for any i in 1; 2; . . . ; 5. Since the new solution continues to satisfy inequality (5.4) at equality cuv ˆ 0. Furthermore, since the choice of arc fu; vg was arbitrary, cuv ˆ 0 for all arcs fu; vg that are contained in the same component. Similarly, huv ˆ 0 for all arcs fu; vg that are contained in the same component. Next, suppose that commodity h belongs to the fundamental aggregate commodity de®ned by S1 and S2 . We reroute one unit of commodity h via node 3 by installing one LC facility on each of the arcs f1; 3g and f2; 3g and send the remaining (dh ÿ 1) units directly from node 1 to node 2 as before. This solution satis®es inequality (5.4) at equality. By hypothesis, it must also satisfy ( ) at equality. This observation implies that c12 ˆ c13 + c23 . In a similar fashion, we obtain the equations (i)±(iv) of Proposition 5.1, only now the variables are the c's. Upon solving this system of equations, we obtain the following relationship between the c's, i.e., c13 ˆ c14 ˆ c15 ˆ c23 ˆ c24 ˆ c25 ˆ c

and

c12 ˆ c34 ˆ c35 ˆ c45 ˆ 2c:

Since the nodes 1; 2; . . . ; 5 were arbitrarily chosen nodes from S1 ; S2 ; . . . ; S5 , respectively, the above relationship implies …i† cuv ˆ c

if fu; vg 2 fSi ; Sj g; i ˆ 1 or 2; and j ˆ 3; 4; or 5; and

…ii† cuv ˆ 2c

if fu; vg 2 fSi ; Sj g; i ˆ 1 and j ˆ 2; i ˆ 3 and j ˆ 4; i ˆ 3 and j ˆ 5; or i ˆ 4 and j ˆ 5:

To prove that hij ˆ Ccij for each arc fi; jg, it is sucient to note that since the network subdivision is demand satis®able (and, therefore, the demand for each fundamental aggregate commodity is no less than C), we can replace C LC facilities by a single HC on each arc successively. We can now scale the resulting inequality to show that inequality ( ) is equivalent to inequality (5.4). Thus, (5.4) de®nes a facet of Conv ‰G…N ; A†Š. h Since in the above proof, we require that the demands for the four distinguished aggregate commodities (i.e., those between components S1 and S2 , S3 and S4 , S3 and S5 , and S4 and S5 ) be no less than C, we have referred to these commodities as the fundamental aggregate commodities. Furthermore, notice that we can modify the above proof so that it continues to work if each of the ®ve components de®nes a connected subnetwork, and we do not necessarily require the original network to be complete. 5.4.2. 1-2/6/6/C facet inequalities We can lift inequality (5.4) for network subdivisions with a higher number of components. To keep the exposition simple, we will work with single-node components instead of larger ones; the same ideas easily carry through to the more general case. Consider a 6-node 6-commodity network as shown in Fig. 7. The numbers on the arcs represent b values with missing numbers indicating a b value equal to 1. Notice that we could have obtained this network by splitting node 2 of the network described in Fig. 2 into nodes 2 and 6.

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Fig. 7. 1-2/6/6/C facet inequality for the 6-node 6-commodity case.

Proposition 5.4. Let G(N, A) and the components bij of the vector b be given as shown in Fig. 7. If the network is demand satis®able, then X fi;jg2A

bij …xij ‡ Cyij † P

X

lenk …b†dk

…5:5†

k2K

de®nes a facet of Conv [G(6, 6)], the convex hull of feasible solutions to the corresponding network loading problem. Proof. Let ( ) represent an inequality of Conv ‰G…6; 6†Š that contains all feasible points that satisfy inequality (5.5) at equality. First, we set xi6 ˆ di6 for all i, and repeat the proof of Proposition 5.3 to obtain the relationship between all the c's except for those arcs that are incident to node 6. Consider the solution with xij ˆ dij for pairs of nodes i and j. The solution obtained by reducing x16 by 1, and increasing x13 and x36 by 1 satis®es inequality (5.5) at equality. Similarly, the solution obtained by reducing x16 by 1 and increasing x14 and x46 by 1 satis®es inequality (5.5) at equality. These observations imply that c36 ˆ c46 since we have already proved that c13 ˆ c14 ˆ c. Analogously, c36 ˆ c56 . But c34 ˆ c36 + c46 implies c36 ˆ c46 ˆ c. The relationships between the remaining c's follow. The fact that hij ˆ Ccij for all {i, j} follows from arguments similar to those used for the proof of Proposition 5.3. h We can repeat this process of splitting and lifting to obtain facet de®ning inequalities of a subdivision with a larger number of components. In the proof of Proposition 5.4, we ®rst set the design variables for all arcs incident to node 6, and derived a relationship between the c and h coecients for the remaining arcs. Then, we considered pairs of arcs emanating from node 6 and showed that these arcs must have equal c and h coecients. This approach is the essential idea of the proof for the next proposition. Again, to keep the exposition simple, we consider single-node components. Proposition 5.5. Let G(N, A) be a complete network and let S and T be a node partition with |S|, |T| P 3. De®ne b as follows:

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 bij ˆ

2 if fi; jg 2 fS; Sg or fT ; T g; 1 otherwise:

Assume the network is demand satis®able. Then inequality X X bij …xij ‡ Cyij † P lenk …b†dk fi;jg2A

…5:6†

…5:7†

k2K

de®nes a facet of the convex hull of feasible solutions to the corresponding network loading problem. Proof. We have shown in Proposition 5.4 that inequality (5.7) de®nes a facet if |S| ˆ |T| ˆ 3, and will now use induction to show that it de®nes a facet for any ®nite cardinality of S and T greater than 3. Let ( ) represent an arbitrary inequality that is satis®ed at equality by all feasible solutions (x, y) that satisfy (5.7) at equality. Consider subsets U  S and V  T of cardinalities u and v, respectively with u, v P 3. We will show that if cij ˆ 2d if fi; jg 2 fU ; U g

and

cij ˆ d if fi; jg 2 fU ; V g

…5:8a†

and hij ˆ 2Cd if fi; jg 2 fU ; U g

and

hij ˆ Cd if fi; jg 2 fU ; V g

…5:8b†

is valid for |U| ˆ u, then this relationship is also valid for |U| ˆ u + 1 6 |S|. Since, by symmetry, we can repeat the same argument for V, this will complete the proof. Assume that node 1 2 U and nodes 3 and 4 2 V. For all commodities, set xO…k†;D…k† ˆ dk . This solution satis®es (5.7) at equality. Now, consider an arbitrary node h 2 S n U . Since the network is demand satis®able, d1h P C. Reduce x1h by one unit and increase x13 and x3h by one unit each. This solution also satis®es (5.7) at equality, and consequently, c1h ˆ c13 ‡ c3h . Similarly, c1h ˆ c14 ‡ c4h . Since c13 ˆ c14 , we obtain that c3h ˆ c4h . But c34 ˆ c3h ‡ c4h . Thus, c3h ˆ c4h ˆ d and c1h ˆ 2d. Moreover, as in the proof of Proposition 5.3, we can show that h3h ˆ h4h ˆ Cd and h1h ˆ 2Cd. Now, since nodes 1 2 U and 3, 4 2 V were chosen arbitrarily, relationship (5.8) holds if we replace U by U [ fhg. Thus, ( ) is a multiple of (5.7) implying that (5.7) is a facet. h If G has n nodes, n even, and in Proposition 5.5 we select S and T of equal cardinality, we need m ˆ n=2 …n=2 ÿ 1† commodities for the network to be demand satis®able. Thus we have obtained a 1-2/n/m/C facet inequality in Proposition 5.5. These facet inequalities are generalizations of the aggregate capacity±demand constraints (3.2b) which are de®ned across a cutset. We will now show that by imposing additional restrictions on the demand values, we can use integrality arguments to generalize the cutset inequality (2.1d), (3.2c) as well. We will discuss this result for the case when we have only 5 components ± S1 ; S2 ; . . . ; S5 and four fundamental aggregate commodities. (Refer to Fig. 6.) Moreover, for this result, it is necessary to split the HC variables into two parts: on arcs with bij ˆ 1 we denote the variable as uij , and on arcs with bij ˆ 2 we denote the variable as vij . This rede®nition of variables is needed only to prove that inequality (5.9) is a valid inequality. 5.4.3. 1-2/5/4/r facet inequalities Proposition 5.6. Let G…N ; A† be a complete network with components Si for 1 6 i 6 5, divided into two partitions fS1 ; S2 g and fS3 ; S4 ; S5 g. Let the demand for one of the fundamental aggregate commodities equal lC ‡ r; l 2 Z‡ and the demands for the remaining three fundamental aggregate commodities be multiples of C.

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Ifr 6 C=2, and the network subdivision is demand satis®able, then inequality (5.9) de®nes a facet of the convex hull of feasible solutions to the corresponding network loading problem.    X 2D ‡1 : …5:9† bij …xij ‡ r yij † P r C fi;jg2A In this expression, D ˆ DS1 S2 ‡ DS3 S4 ‡ DS3 S5 ‡ DS4 S5 and the b values are as de®ned in Fig. 6. Proof. We ®rst show that inequality (5.9)Pis valid. Let us ®rst rede®ne the y variables into the u and v less than d2D e ‡ 1. Then inequality (5.9) is clearly variables as mentioned above. Suppose 2 fi;jg vij is noP C 2D 2D e is odd, and thus 2 v ˆ d e ÿ s, with s P 1. So, now we want to valid. Otherwise, r 6 C/2 implies d ij fi;jg C C P P ruij † P r…s ‡ 1†. If, fi;jg uij is no less than (s + 1), then inequality (5.9) is valid. prove that fi;jg …bij xij ‡P Otherwise, suppose that fi;jg uij equals (s + 1 ÿ l) with l P 1. With this assumption, we want to show that X bij xij P rl: fi;jg2A

IfPl ˆ 1, then this inequality is true by cutset arguments and we will prove this when s ˆ 1. Since ˆ d2DS1 S2 =Ce ÿ 1 or (without loss of generality) VS3 S4 ˆ DS3 S4 =C ÿ 1 2 fi;jg vij ˆ d2D=Ce ÿ 1, either 2VSP 1 S2 If 2VS1 S2 ˆ d2Ds1 S2 =Ce, and because fi;jg uij equals 1 (when s ˆ l ˆ 1), XS1 ;N nS1 ‡ XS2 ; N n S2 P r. On the other D hand, if VS3 S4 ˆ SC3 S4 ÿ 1, then XS3 ;N nS3 ‡ XS4 ;N nS4 P C > r. (A similar argument applies when s P 1.) So we assume that l P 2. We know that Xÿ ÿ

bij xij ‡ C uij ‡ 2vij P 2D fi;jg2A

is valid. Substituting the values of X bij xij P 2r ‡ C…l ÿ 2†

P fi;jg

vij and

P fi;jg

uij , we get the inequality

fi;jg2A

P which implies fi;jg2A bij xij P rl, and thus (5.9) for all l P 2. The facet proof is along the lines of the proof of Proposition 5.3 with some changes that are required because the coecients are no longer 0, 1 or C (see Mirchandani, 1989). We skip the details of this proof. h 5.5. The class of 1-2-3 facet inequalities We now describe facet inequalities that have their x variable coecients drawn from the set {0, 1, 2, 3}. As earlier, to simplify the exposition, we will focus our attention on single-node components with the implicit observation that all these results can be extended to the more general case. 5.5.1. 1-2-3/6/3/C facet inequalities We now discuss inequality (5.2) depicted graphically in Fig. 4. These inequalities de®ne facets and can be generalized to a larger network as we had done for the 1-2/5/4/C facet inequalities. Recollect that Fig. 4 contains three fundamental commodities: between nodes 1 and 4; 2 and 5; and, 3 and 6. Proposition 5.7. Let G(N, A) as shown in Fig. 4 be given, and let Conv ‰G…6; 3†Š denote the convex hull of feasible solutions to the corresponding network loading problem. If dk P C for all k, then inequality …5:2† de®nes a facet inequality for Conv ‰G…6; 3†Š.

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We omit the proof of this proposition. As in the 1-2 facet case, inequality (5.2) remains a facet if the number of commodities increases beyond 3; or, if the number of nodes in the network is more than six and we subdivide the network into six components. For inequality (5.2) to be a facet, we only need the condition that the network subdivision is demand satis®able. For the purpose of using the concept of demand satis®ability, we would have a 3partition in this case. Nodes 1 and 4 constitute the ®rst part of the partition, nodes 2 and 5 the second one, and nodes 3 and 6 the last one. 5.5.2. Generalizations We use the idea of splitting nodes to generate facets on larger networks. We will consider only node 5, though we could split the other nodes of the network in a similar fashion. Suppose we want to split node 5 into a set U of p (p P 2) nodes. We assign the b values for the arcs incident on the new nodes as follows. 1: If both nodes i and j belong to U ; then bij ˆ 2:

…5:10a†

2: If i 2 U and j 62 U ; then bij ˆ b5j :

…5:10b†

3: If neither node i nor node j belongs to U ; then the value of bij remains the same as in inequality …5:2†:

…5:10c†

Fig. 8 describes the network obtained after we have split node 5 into nodes 50 and 7 and illustrates a 1-23/7/5/C inequality. ‡ 2=C inequality With the b values as de®ned in (5.10), consider the 1-2-3/p+5/p…p‡1† 2 X X bij …xij ‡ Cyij † P lenk …b†dk : …5:11† fi;jg2A

k2K

We claim that inequality (5.11) de®nes a facet for this larger network of (p + 5) nodes if demands of the following commodities is no less than C: (i) between nodes 1 and 4; (ii) between nodes 3 and 6; (iii) between every pair of nodes in the set ff2g [ U g. These conditions correspond to demand satis®ability with respect to the 3-partition: (i) nodes 1 and 4, (ii) nodes 3 and 6 and (iii) nodes ff2g [ U g.

Fig. 8. b values obtained after splitting node 5. bij ˆ 1 for arcs without any label.

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Assume that the face described by inequality (5.11) is contained in the facet described by ( ). We will show inequality (5.11) de®nes a facet by showing that the coecients c and h for any arc in ( ) must equal the corresponding coecient in inequality (5.11). To see that this is true, choose an arbitrary node from U, say 50 , and ®x xij at dij for all i 2 U f50 g. Now, repeat the proof of Proposition 5.7. The proof applies directly because by setting xij for all iU n f50 g as indicated, we have e€ectively eliminated these variables from inequality (5.11). This procedure shows that c12 ˆ c16 ˆ c23 ˆ c26 ˆ c34 ˆ c350 ˆ c46 ˆ c50 6 ˆ c; c150 ˆ c24 ˆ c250 ˆ c36 ˆ c45 ˆ c47 ˆ 2c;

and

c14 ˆ 3c:

Next, consider an arbitrary node, say node h belonging to node U n 50 . Perturb the solution used above (that satis®es inequality (5.11) at equality) by reducing x2h by one unit, and increasing x23 and x3h by one unit. This solution continues to satisfy inequality (5.11) at equality; thus, c2h ˆ c23 ‡ c3h . Similarly, c2h ˆ c12 ‡ c1h . Since c12 ˆ c23 , we obtain c1h ˆ c3h . Now, reduce x14 by one unit, and increase x1h , x3h and x34 by one unit each. Thus, we deduce that c14 ˆ c1h ‡ c3h ‡ c34 , which implies that c1h ˆ c3h ˆ c and c2h ˆ 2c. Similarly, c6h ˆ c, and c4h ˆ c50 h ˆ c. Since, we chose node h arbitrarily, the same argument applies for any any node belonging to U\{50 }. Consequently, cij ˆ 2c for any i and j 2 U\{50 }. Lastly, because the network is demand satis®able, by arguing as in Proposition 5.3, we obtain hij ˆ Ccij . Consequently, we have shown that ( ) is equivalent to inequality (5.11). h In a similar fashion, we can split the nodes 1, 3, 4 and 6. We de®ne the coecients b for the arcs incident to the new nodes as in (5.10a), (5.10b) and (5.10c) and if there exists a commodity demand between every pair of these new nodes whose value is at least C, then inequality (5.11) applied to this larger network de®nes a facet. 6. Conclusion This paper discusses reformulations of the network loading problem. First, we study the single commodity model and although we can reformulate this problem by applying max-¯ow min-cut based arguments, we also present a polyhedral approach that serves as a motivation for the multicommodity case. Using this approach, we construct a formulation in the subspace of design variables and strengthen Iri's result for the general multicommodity case. This research raises a number of issues for further work. Suppose the aggregate capacity across every cutset exceeds the total demand across the cutset. Then we can ®nd a feasible ¯ow that satis®es the ¯ow conservation and capacity constraints for several important topologies, for example, (i) two-star topology (Papernov, 1976), (ii) k4 topology (Papernov, 1976; Seymour, 1980), (iii) C5 topology (Lomonosov, 1978), (iv) planar graph topology (Okamura and Seymour, 1981; Seymour, 1981). Di€erent approaches have been used to prove these results. Projection techniques might yield a unifying methodology for developing all these currently known ``feasibility'' results for the multicommodity ¯ow problem. Moreover, this investigation might also identify other con®gurations for which there exists a feasible multicommodity ¯ow if and only if the capacity of every cutset is no less than the total demand across the cutset. In particular, suppose the demand network has a k5 topology and the network loading problem is de®ned with only the x variables. For this case, we conjecture that if all the 1-2/5/10/0 and the 1/2/1/0 inequalities are satis®ed, then there exists a feasible multicommodity ¯ow. This paper raises two other related research questions: (i) can the necessary conditions of Proposition 4.2 be augmented to yield-even for networks with a special structure-suciency conditions for a ray to be extreme and (ii) what are the conditions on b for which inequality (4.3a) generates a facet of the underlying

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polyhedron? We identify several such families of facets for the multicommodity problem in the space of x and y variables. A more exhaustive characterization of the classes of facets, and of the corresponding extreme rays of W(K), remains an open, unsolved research problem. Finally, we observe that even for small networks of bounded size, the number of facets of each family that we identify is enormously high. For, example, for a 25 node network, the number of possible 1-2/5/4/r facet inequalities is more than 2.4 ´ 1015 (Stirling's number of the second kind (Abramowitz and Stegun, 1970)). Implementing these facets and determining their e€ectiveness in a computational setting, therefore, remains a challenge. Appendix A Proposition 3.4. Let W(1) be the polyhedral cone de®ned by (3.8). Then 1. W(1) is full-dimensional. 2. The following inequalities de®ne facets of W(1): (a) ai ÿ aj ‡ bij P 0 for all fi; jg 2 A, (b) aj ÿ ai ‡ bij P 0 for all fi; jg 2 A, and for all i 2 N . (c) ai P 0 Proof. 1. Construct the following feasible solutions: (a) For each fi ; j g 2 A, de®ne  bi j ˆ 1; …a; b†: …a; b† ˆ …0† otherwise: (b) For each i 2 N , de®ne 8 < ai ˆ 1; for all fi ; jg 2 A; …a; b†: bi j ˆ 1 : …a; b† ˆ …0† otherwise: We thus have |A| + |N| linearly independent solutions, completing the proof. 2(a). For simplicity of notation, let us consider arc {1, 2}. Construct the following solutions satisfying a1 ÿ a2 ‡ b12 P 0 at equality. (aa) For each fi ; j g 6ˆ f1; 2g, de®ne  bi j ˆ 1 …a; b†: …a; b† ˆ …0† otherwise: (bb) For each S equaling one of the sets {2}, {2,3}, {2,4}; . . . ; or {2,n}, de®ne  ai ˆ 1 for all i 2 S; and 0 otherwise; …a; b†: bij ˆ 1 for all fi; jg 2 fS; N n Sg; and 0 otherwise: (cc) De®ne  ai ˆ 1 …a; b†: bij ˆ 0

for all i 2 N ; for all fi; jg 2 A:

We thus have a total of …jAj ÿ 1† ‡ …jN j ÿ 1† ‡ 1 ˆ …jAj ‡ jN j ÿ 1† solutions. The linear independence of these solutions can be veri®ed easily from the following matrix, which is constructed assuming that the arcs shown in the variable column exist in the network.

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Since arc {1, 2} was chosen arbitrarily, the proof is complete. 2(b) Proof similar to the proof of part 2(a). 2(c) Suppose i ˆ 1. Then replace the last solution in 2(a) by b12 ˆ 1, (a, b) ˆ (0) otherwise, to obtain the requisite number of linearly independent solutions.

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