ripples, complements, and substitutes in singly constrained monotropic parametric network flow...

Ripples, Complements, and Substitutes in Singly Constrained Monotropic Parametric Network Flow Problems *

Antoine Gautier

Facult6 des Sciences de I’Administration, Universitb Laval, Quebec City, Canada

Frieda Granot

Faculty of Commerce and Business Administration, The University of British Columbia, Vancouver, British Columbia, Canada

We extend the qualitative theory of sensitivity analysis for minimum-cost flow problems developed by Granot and Veinott to minimum-cost flow problems with one additional linear constraint. Two natural extensions of the “less dependent on” partial ordering of the arcs are presented. One is decidable in linear time, whereas the other yields more information but is NP-complete in general. The Ripple Theorem gives upper bounds on the absolute value of optimal-flow variations as a function of variations in the problem parameters. The theory of substitutes and complements presents necessary and sufficient conditions for optimal-flow changes to consistently have the same (or the opposite) sign in two given arcs. The Monotonicity Theory links the changes in the value of the parameters to the change in the optimal arc-flows, and bounds on the rates of changes are discussed. The departure from the pure network structure is shown to have a profound effect on computational issues. Indeed, the complexity of determining substitutes and complements, although linear for the unconstrained (no additional constraint) case, is shown to be NP-complete in general for the constrained case. However, for all intractable problems, families of cases arise from easily recognizable graph structures which can be computed in linear time. 0 1994 John Wiley & Sons, lnc.

1. INTRODUCTION

Many real-world problems can naturally be formulated as Network Flow Problems in which flows are subject to one additional linear constraint. Examples of such problems include inventory-production models [ 1 11, asset allocation management [ 91, telecommunication satellites [ 22 J , bicriterion decision analysis [ 21, newspaper distri- bution [ 151. matrix scaling [17] , as well as critical sub-

* This research was partially supported by the Natural Sciences and Engineering Research Council of Canada Grants 5-83998 and OGPOIZ 1627 and by the Government of Canada through the Entraide Universitaire Mondiale du Canada.

routines to solve minimum convex-cost dynamic network flow problems [ 16 1.

The special case where no additional constraints are present, which we refer to as the unconstrained case, has received considerable attention in the literature. Efficient solution techniques for problems with linear objective functions are too numerous to be surveyed here. For the case where the objective function is nonlinear, one can refer to, e.g., [4. 7, 14, 211. The majority of these algorithms exploit the particular structure of the constraint matrix and the combinatorial properties of network flows and circulations. Unfortunately, the additional linear constraint partially destroys the attractive computational features of pure network flow problems. Even though, for linear objective functions, specialized solution methods

NETWORKS, VOl 24 (1994) 285-296 c 1994 John Wiley & Sons, Inc CCC 0028-3045/94/050285-12

285

286 GAUTIER AND GRANOT

may be used, see, e.g., [ 3,5, 121, for nonlinear arc-additive costs, one has to resort to solving a monotropic program (see, e.g., [21]). Moreover, many of the problems discussed above are very large scale problems, and their solution, expensive. It would thus be desirable to study changes in the optimal solution as well as changes in the optimal value resulting from changes in problem data. In addition to the theoretical interest of such a study, it would help bypass the necessity of resolving such problems whenever a change in the data occurs.

The aim of this paper was to develop a qualitative sensitivity analysis for network flow problems with one additional linear constraint and with convex, arc-additive cost function. Qualitative sensitivity analysis and the study of substitutes and complements (terms defined in Section 4) for the unconstrained case was the objective of [ 131. Here, however, the presence of a single additional linear constraint increases the complexity of solving the problem and partially abolishes qualitative sensitivity analysis results developed for unconstrained problems. For an analysis of substitutes and complements in linear models using the requirement space, the reader is referred to the work ofprovan [19].

In our case, the cost function is, for most results, required to be convex in the arc-flow and may reflect real costs and upper and/or lower bounds. The sensitivity analysis is performed by modifying the parameter associated with one or several given arcs. Any variation of the cost structure can be represented in this fashion. Our results are concerned with the “reactions” of the optimal flows to such changes. For applications, such parameter variations may, e.g., reflect changes in upper and/or lower bounds, additive or multiplicative increases in cost, pricing alternatives, technological advances, or discounts or even simulate the deletion/addition of an arc. In cases where the parameter change results in multiple optima, our results are valid for at least one of the possible optimal flows.

The paper is organized as follows: Notation and definitions are introduced in Section 2, in particular, the key concepts of elementary vectors and dependent subgraphs. Section 3 focuses on bounds on the magnitude of change of the optimal flow in two given arcs, the main results being given in the Ripple Theorem (Theorem 1 ) . To that end, we define two partial orders on the set of arcs: the less-dependent-on relation and the quantitatively less-dependent-on relation, as well as a coefficient M j k that is associated with each pair of arcs ( ej, ek) . Computational issues associated with the partial orders and the coefficients k t j k are discussed. In Section 4, we introduce the notions of complement and substitute pairs of arcs and present a result on the direction of changes in the optimal arc-flow for such pairs of arcs. Computational complexity is again addressed. The Monotonicity Theory presented in Section 5 relates and compares the rate of change in the optimal flow of a given arc to that of another arc’s parameter when

the two arcs are either complements or substitutes, a result stated in the Monotone Optimal-Flow Selection Theorem (Theorem 1 1 ) . The Smoothing Theorem (Theorem 13) provides an upper bound on the maximum arc-flow change over the network in terms of L I-norm of the corresponding parameter variation. Finally, Theorem 14 shows the subadditivity property of the minimum cost. Although the paper is self-contained, we emphasize throughout similarities and differences with the case of unconstrained monotropic network flow problems discussed in [ 131.

2. NOTATION, DEFINITIONS, AND PRELIMINARIES

Let G be a finite directed graph with a set of nodes N = { 1, 2, . . . , n } and a set of arcs E = { e l , e2, . . . , em} (possibly with parallel arcs). With each arc ej we associate a flow xj E 8, a parameter tj in a lattice (for background on lattices and lattice programming, see, e.g., [ 6, 25]), a +00 or real-valued cost bj( xi, t j ) , and a real number aj. The parameter vector t = ( t i ) lies in the lattice product T = XejEETJ. We discussed in this paper the following ( minimum-cost ) Parametric Singly Constrained Mono- tropic Network Flow Problem:

m def P( t ) : min { b ( x , t ) = 2 bj( xi, t,) : Ax = a , ax = a0 } ,

j = 1

where A is the node-arc incidence matrix of G, the demand vector a verifies Z p I a, = 0, and a. E W is the right-hand side of the additional constraint. Dropping the additional constraint yields the Unconstrained monotropic network flow Problem [ 2 11.

We denote by X ( t ) the set of optimal flows of P( t ) . The usual nonnegativity and upper/lower bound constraint ( s), if any, are naturally incorporated into the cost- function by setting b,( x,, t,) to + 00 for all forbidden values

A circulation x verifies Ax = 0, and C = { x E W” : Ax = 0, ax = 0 } is the linear subspace of constrained circulations. The terms path and cycle refer both to node/ arc sequences (with no repeated nodes) and to { 0, 1, - 1 } m-vectors corresponding to one unit of flow going through these arcs. A cycle c is neutral if ac = 0 and active otherwise. By extension, a subgraph of G is active if it contains active cycles and neutral otherwise.

A graph is connected iff there exists a path between any two nodes and biconnected if every pair of arcs is contained in a cycle [ 26, p. 347 ] (by extension, we say that a graph with only one arc is biconnected). Biconnected components of a graph (maximal biconnected subgraphs) have at most one node in common with one another (see, e.g.,

of x,.

MONOTROPIC PARAMETRIC NETWORK FLOW PROBLEMS 287

[ 23, pp. 90-921). In the study of flows, it can be assumed without loss of generality that biconnected components have no common nodes with one another (modulo a simple linear time transformation, see [13]). The decomposition of a graph into biconnected components is unique [ 261 and can be achieved in linear time, see, e.g., [ 1, pp.

The remainder of this section is adapted from [ 101. We define an active constrained graph as a rim if it is composed of several neutral biconnected subgraphs, its rim components arranged in a “cycle,” each having a unique node (called a bolt) in common with each of its two neighbors. A one-rim is the degenerate case with one rim component obtained by condensing all but one rim component of a rim into a single node, the bolt, so as to result in an active subgraph (see Fig. 1 ). Rims and one- rims are biconnected and have the property that all active cycles c go through all the bolts and have the same value for I (YC I . Active cycles that verify ac > 0 (resp., < 0) are called positive (resp., negative) rim cycles. Rims and one- rims can be recognized in O( rn + n) [ 101.

The support of a vector x E !Xm is the subset { i = 1, . . . , m : x, # 0 } . An elementary vector of a linear subspace K of !Xm is a vector f E K whose support is minimal among the supports of nonzero vectors of K. Any vector of K can be expressed as a conformal sum (two vectors u and v are conformal if u,v, 2 0 for all i) of elementary vectors of K [ 20, pp. 108-109, Harmonious Superposition Theorem; 2 11. Let H be the set of elementary vectors of C , or elementary circulations, and define H, = {f€ H : 5 # 0 } . The elementary circulations take one of the two

forms ( i ) f = c and c is a neutral cycle and (ii) f i s a weighted sum of active cycles, i.e.,

179-1871.

where cI and c2 are active cycles, the union of their supports does not contain any neutral cycle, and cI and c2 share at most a single node or a path [ lo] (see example 1).

H gives rise to the following relation between arcs: Two arcs ei and ej in G are dependent [ 101 iff there exists an elementary vector f = (fk) E H for which f;f; # 0. The dependence relation is an equivalence relation, and G is

Neutral Biconnected Subgraph

said to be dependent if every pair of its arcs is dependent. The decomposition of G into its dependent components (maximal dependent subgraphs) is unique and leads to the following decomposition of P( t ).

Dependent Components

The dependent components are obtained from the biconnected components as follows: Each neutral biconnected component is a dependent component [type (I)]. If the number of active biconnected components is at least 2, then the subgraph composed of all of the active biconnected components forms one dependent component [type (11)] . If a unique biconnected component is active and is neither a rim nor a one-rim, then it is also a dependent component [type (III)], whereas if it is a rim (resp., one-rim), then each of its (resp., its unique) rim component( s) is a dependent component [type (IV)] . G may be altered so that its dependent components have no nodes in common.

Problem Decomposition

P( t ) can be decomposed into independent subproblems on the dependent components so that X( t ) is the Cartesian product of the solution sets of the subproblems. At most, one of the resulting subproblems is constrained [type (11) or (III)] , the remaining subproblems being unconstrained [type ( I ) or (IV)]. Both the graph decomposition and the problem decomposition can be achieved in O( m t n) .

3. THE RIPPLE THEOREM FOR CONSTRAINED NETWORKS

We study in this section the effect of a change in the parameter of a given arc on the flow throughout the network. We show that the magnitude of the change in any arc depends on how “dependent” the arc is with the arc whose parameter is being changed. Clearly, in view of the above problem decomposition, changes in an arc’s parameter may not affect the set of optimal flows in arcs belonging to other dependent components. Therefore, we may assume in the following that G is dependent. Further, if G

Fig. 1. Special constrained graphs.


is of type ( I ) or type (IV), then the associated subproblem is an unconstrained one and the qualitative theory developed in [ 131 for such problems can be applied. Thus, we further assume in the following that G is either of type (11) or type (III), i.e., that G is composed of either several or of a unique active biconnected component (s). We start by introducing two partial-orders on the set of arcs E.

3.1. d- and q-Order Relations

For e l , el, ek E E , say that el is less dependent on el than ek is and write el 5," ek if

forall f EH,J;J#O*fkPO

(for all f E H,,f; P 0 *fk # 0 ) .

If el I," ek and ek I," e, , then el and ek are said to be d- equally dependent on e, and we write el =," ek. Clearly, el = ," ek if for all elementary vectors f E HI either = fk = 0 orf;fk # 0. Further, say that el is quantitatively less dependent on el than e k is and write el I," ek iff

or, equivalently, if [J; I s Ifk I for all f E HI. If e, 5," ek and ek I," e l , we say that e, and ek are q-equally dependent on e, and write el =I ek. Clearly, e, =," e k iff for all f E H,, If; I = lfkl . It is easy to see that both relations I," and

I," are partial-orders; hence, =," and =," are equivalence relations. The sets &," (resp., &,") of d- (resp., q-) equivalence classes (induced by the relation =,"( resp., =,")) are partially ordered by I," (resp., I,"). We denote by &,"(k) (resp., &,"(k)) the unique element of &," (resp., 6,") containing ek.

Comparisons with the Unconstrained Case

The d- and q-relations constitute genuine extensions of Granot and Veinott's [ 131 less-biconnected-to relation. However, the characterization of the equally biconnected- to relation =, given in [ 13, Theorem 41 is no longer valid for constrained networks (two arcs el and ek are equally biconnected to a third arc el iff e, = ek or deleting e, and ek leaves G disconnected). One can easily construct examples of graphs in which deleting two dependent arcs will leave the graph connected. Also, by contrast with the unconstrained case where the greatest element in the partial order s, always contains arc e,, here there are examples for which there exists an arc ek such that el <," e k . In particular, this may happen if the constraint is of the form X, + a k x k = CYO and I CYk I < 1. Moreover, the d- and q- relations are no longer equivalent. Clearly, for el , e,, and ek E E, if el 5," e, (resp., el =," ek), then, also, e, I," ek (resp., el =," ek) . This implies that &,"( i ) E &,"( i ) for all

ei E E , but the reverse inclusion may not be true, as is shown in the following example.

Example 1. Consider thegraph given in Figure 2 with the additional constraint 2x4 + x5 = 3. For any circulation x E C , 2x4 + x5 = 0. Further, one can verifv that H = { f I ,

f 2 , f 3 , f 4 } > , where

f ' = ( e l + e4 + e3) - 2(e2 + e6 + e5)

= ( 1, -2, 1, 1, -2, -2, 0)

f ' = ( e l + e2 + e6 - e7 + e3)

= (1, 1, 1 ,0 ,0 , 1, -1)

f = (e4 + e7 + e5) - 3(e2 + e6 + e5)

= (0 , -3, 0, 1, -2, -3, 1)

f 4 = 2(e4 + e7 + e5) - 3(el + e4 + e3)

= (-3, 0, -3, -1, 2,0, 2 ) .

Now, the two elementary vectors f ' and f that contain el and e4 (resp., e5) also contain e5 (resp., e4) and it follows from the definitions that e4 = f e5. However, one can easily show that e4 #': e5.

3.2. The Ripple Theorem

Theorem 1: The Ripple Theorem. Let P( t ) be dejined on G. Suppose that t and t' d@er only in one component tj, that bk(. , tk) is convex for each ek E E\&,d( j ) , x is any element of X ( t ) and X ( t ' ) is nonempty. Then, for some x' E X ( t ' )

(R. l ) x f - x is a sum of conformal elementary circulations

(R.2) Ripple Bounds. For any two arcs ei and ek in E, whose supports contain arc ej.

i fe i I: ek, then

(R.3) I x i - x k l does not increase the less dependent qualitatively ek is on ej.

Proof (R. 1 ) follows by using Rockafellar's Harmo- nious Superposition Theorem described at the beginning of Section 2 and the convexity of the cost functions as was done in [ 13 1. The proof is included for completion. For an element x' ofX( t f ) , X ' - x is a sum of conformal elementary circulations. Let y be the sum of the conformal


elementary circulations whose support contains e, and z the sum of the remaining elementary circulations. Then, y and z are conformal and z k = 0 for all k E &,d( j ) since, by definition, any elementary support that does not contain eJ does not contain ek either whenever ek =," e,. By convexity.

b(.u + y, t ' ) + b ( x + z , t ' ) I b ( x , t ' ) + b(x', t ' ) . ( 3 )

Moreover, since only t, is changed and z, = 0,

b ( x + Z , t ' ) - b ( x , t ' ) = b ( x + Z , t ) - b ( x , t ) . (4 )

Combining ( 3 ) and (4) , one obtains b ( x + y , t ' ) + b ( x + z , t ) 5 b ( x , t ) + b(x ' , t ' ) . Hence, x t y E X ( t ' ) and x+ zEX(t),becausex'EX(t')andxEX(t). Therefore, x + J' is optimal for t' and (R. 1 ) holds.

To show ( R.2), assume that el I ," ek . From ( R. 1 ) , we can write

x' - x = zafa where za 2 0 for all a . ( 5 ) foe Hj

cunformol

By the conformality of the sum,

Moreover, e, 5," ek implies that & = 0 for all elementary vectorsf, E H,\Hk and, by the definition of Mkl, I& I I MkI \ f a k I for d l f , E Hk. Thus, (R.2) fOllOWS Since

1.4 - x, I = C zalfa, I /h HJ

con/ormal

To prove (R.3), we need to show that for any two arcs e, , CL E E such that el I; ek, we have 1 x: - x, I I I xi - x k I . By hypothesis, Ifo, I I lhk I for all fa E H,; thus, (6 ) yields

Constraint: 2 x4 + x5 =3

Demand: a=O e4 e5

el e2 Fig. 2. Example.

The bounds obtained in the Ripple Theorem are sharp as is demonstrated below.

Example: Sharp bounds in the Ripple Theorem. Consider again the graph given in Figure 2 and specijjy the demand a = 0 and the costs

The assumptions of Theorem 1 are satisfied, and for t = ( O , O , . . . , O ) a n d t ' = ( l , O , . . . , O),it iseasytoverIfy that X ( t ) = { x } and X ( t ' ) = { x'}, where x = ( 0 , 0, 0, 1, 1 , 0, 1) andx ' = ( 1 , -2, 1, 2, -1, -2, I ) (this since zero is a lower bound on the objective value for any value of the parameter, and one can check that both these solutions achieve an objective value of zero and are unique in that respect). From ( 2 ) and the above expression of H , MI* = max{ 1-2/11, 1 1 / 1 1 , 10/-31) = 2. Thus, 1x5 - x2 I = MI* I x', - x i I and the bound given in ( 2 ) is tight.

Constant Changes in q-equivalence Classes

In the unconstrained case, Granot and Veinott [ 131 showed that, under the assumptions of the Ripple Theo- rem, the magnitude of the change in optimal arc-flow remains constant over all arcs in a given equivalence class for the less-biconnected-to relation. This remains valid for the q-relation (since if e, =," ek, then by definition, Ifn, 1 = Idk 1 for allf, E H,, and thus by (6) , I x: - x, I

= I x i - xk I ) but not for the d-relation. This can be seen in the above example for which it was shown that e4 =;' e5, whereas Ixh -x41 = 1 f 2 = Ix; - x51.

Notice that under the assumptions of Theorem 1 it follows that ( i ) if e, 5," ek and x i = xh, then x: = x, , and (i i) since e, <ye, for all i , then I xi - x, I 5 MJ, Ix; - -r/i for all i .

3.3. Computational Issues

We establish in Theorems 2, 3, and 4 results regarding the complexity of verifying some of the hypotheses of the


Ripple Theorem. We then provide examples in which some of the hard problems may be solved easily.

Theorem 2. For e;, ej, and ek in G, ei 1; ek ifdeleting ek leaves e; and ej independent.

The proof is a straightforward application of the definition. Theorem 2provides a linear time algorithm to verifv whether three arcs e i , ej, and ek satisfy ei sf ek, for it suffices to decompose G\ { e k } (obtained by deleting ek from G) into dependent components in O(m + n ) , and check whether or not e; and ej are in the same dependent component [lo].

Theorem 3. The problem [MI of computing the ripple bound h f l k in (2) for any given pair of arcs ( ej, ek) is NP- complete.

Proof We reduce [MI to the NP-complete (see [ 8, p. 2 131 ) problem of the Maximum Weight Cycle [ MWC] : max { I ac I : c cycle of G } as follows. Assume that there exists an algorithm A, which, given G and two arcs ej, ek € E , evaluates the coefficient h f j k and runs in at most P( s) elementary operations, where P( s) is some function of s (s is the number of bits required in the encoding of the data which is a measure of the size of the input). We use A to solve the [ MWC] in the following manner:

The input to the [MWC] is composed by a graph G and arc-weights that we denote by (ai ) E !TIm. Form the graph G* from G by adding the nodes n + 1, n + 2, and thearcsem+l = ( n + 1, n + 2)and em+2 = ( n + 2, n + 1 ) . The constraint is a* x = 0 with a* = ( a I , . . . , a,,,, - 1, 0). Choose ek in G (See Fig. 3) . Now, since no cycle goes through both ek and em+l, any elementary vector containing those two arcs has to be a weighted sum of two active disjoint cycles, i.e.,

where c is an active cycle of G for which ck = 1 and& is determined by ( 1 ), resulting in

fk = -a*(e,,,+l + em+2)/a*c 2 l / a * ~ = l / a ~ . ( 8 )

Therefore, for G*, we can write

where H: is the set of elementary vectors of the circulation space of G* that contain f?k. Thus, A will, when applied to G * and a particular pair of arcs ( e k , e,,, + ) , determine the cycle of maximum weight that contains arc ek. By repeating the above for k = 1, . . . , m and then selecting the largest of the m optimal values, one solves the [ MWC] in O( m P ( s)), completing the reduction and the proof..

Tractable Cases of Ripple Bounds

We present here examples where the value of Mjk can easily be computed. For convenience, we assume without loss of generality that all elementary vectors f encountered with J # 0 are scaled to J = 1.

( i ) Arcs in rims. Consider first the case where G has several biconnected components, and the biconnected component(s) that contains ej and ek is(are) a rim(s). Notice now that a biconnected subgraph, say GI, with a unique arc ei that verifies a; # 0, is a rim whose bolts are the end-nodes of ei and whose rim components are { ei } and the biconnected components of GI \ { ei } . Such subgraphs naturally fall into the category described here.

(i-a) If at least one of the biconnected components of G, say GI, is a rim, then for every ej, ek E G I , the elementary vectors that contain both ej and ek are either neutral cycles of GI or active cycles of GI paired with active cycles not in GI. All such elementary vectors verify 1 J 1 = lfkl, and, thus, M]k - 1.

(i-b) If at least two of the biconnected components of

-

n+ 1

n+2

Fig. 3. The extended graph G *.


r - - - - -

Fig. 4. ei 59 e,,.

G, say GI and G2, are rims, then for every el E GI and ek E G 2 , any elementary vector that contains both el and ek is of the form .f = c ' Sfkc ' , where c ' and c2 are. respectively, active cycles of GI and G2. c,! = c i = 1 and.f;, = -ac1 /ac2 as in ( 1 ) . Therefore, Ifk/Jl = I ( -ac l /acz ) / l I = Iac1/ac21 is independent of the choice of c 1 and c2, and MJk is equal to that ratio.

(i i) The support of a on some biconnected component of G is a pair of arcs. Suppose now that the entries of a on a given biconnected component GI are all zero except for two, say a] and ak. Given that an elementary vector ,/.is a circulation, it verifies that alJ + a k f k + CGiG, a,$ = 0. If f i s composed of two cycles. one in G , and the other one not in G I , then Ih-1 = 141 = 1. If, however, both cycles are in G I , then CC\C, a,f; = 0, aJJ + a k f k

= 0. and l.fi/.Jl = lal /akl . Thus, MIL = max( I ,

Theorem 4. The problem [ QLD] ofdeciding whether fhree given urcs e l , el, and ek verifi el 17 ek is NP-comp1ete.t

Proof We reduce [ QLD] to the NP-complete (see [ 8, p. 2 131) problem of the Maximum Weight Path [ MWP] : Given u vector of weights ( w, ) E !H'", u specific arc el = ( nz , n l ) with w1 = 0, and u real number X 2 0, d o all paths p in G \ { el } ,from n l to n2 verij.? wp I A? Assume without loss of generality that X I 2' (this since w can be encoded with less than .Y bits) and consider the graph G* i n the proof of Theorem 3 with

I a , / f f h I ) .

NT = -2 ' / (2 ' - A ) ,

= w1/(2' - A ) for i = 2 , . . . , m ,

aZil = -1. and a$+? = 0.

Now " e l I %+? em + I " iff I 1 / a*c I I 1 for any active cycle

t We are thankful to Richard Anstee for suggesting improvements to an earlier version of the proof.

c of G* that contains e l . This is trivially equivalent to I a*c I 2 1 for all cycles c of G with c1 = 1 . Since such a cycle is of the form c = el + p, where p is a path from nl to n2, el 1%+2 holds iff 1(2s - ~ p ) / ( 2 ~ - A ) l 2 1 for all paths p from n l to n2. Now, since I wc I I 2 ', then ( 2 s - ~ p ) / ( 2 ~ - A ) 2 I and, thus, wp I X for all paths p from n l to n2, solving the [ MWP]. Since the reduction is linear, the proof follows.

Tractable Cases of the q-Relation

Recall that el 1; ek =+ el I," e k ; thus, cases for which el I," ek does not hold (see Theorem 2) can be ruled out. Two situations where the q-relation can easily be decided positively require the following result:

Lemma 5. Given two arcs el, ek, everv acrive cycle that contains el also contains eh if the biconnecred componenr of G\ { e k } that contains el is neutral.

Proof: First, every active cycle of G that contains e, also contains eh iff e, does not belong to any active cycle in G \ { ek} or. equivalently, if el does not belong to any active cycles in the biconnected component of G\ { e k } in which it lies. The result follows from Lemma 7 in [ lo] , which states that a biconnected graph with at least two arcs is active iff every arc belongs to an active cycle.

G has at least two biconnected components, one of which, say G I , is a rim. Then, for any e l , ek E GI and el that does not belong to the rim component of GI that contains e l , any elementary vector that contains both el and el is of the form given in ( 1 ) where c ' is an active cycle of GI that contains el and c2 is some active cycle of G \ G l (if el E G I , then c ' contains el; otherwise, c2 does). Conversely, any active cycle c' in GI that contains el can be completed into an elementary vector that contains e, as well. Clearly, such an elementary vector contains ek iff c 1 does, and, thus, e, 5:' eh iff every active cycle of GI that contains el also contains ek. From Lemma 5 , this holds iff the biconnected component of


Constraint: 2 x1 + x7+ 2 x8 = 4

Demand: a=O

Fig. 5. Example.

G \ { ek} that contains e, is neutral. The possible configurations are given in Figure 4 where ( i ) e, may belong to GI or (ii) to G\ GI . Since it follows from the above that I f ; I = I c f I = l f k l = I c L l = 1, then e, 5," ek holds whenever el I," ek does. G has exactly two biconnected components, say GI and G2, that are rims. Then, for every e, in G I , e, and ek in G2, any elementary vector that contains both e, and e, is of the form given in ( l ) , where c1 and c2 are, respectively, active cycles of GI and G2. Therefore, el I," ek iff every active cycle of G2 that contains e, also contains ek. The two possible configurations are given by Lemma 5. [See the subgraph GI in Fig. 4(i) and (ii).] For e, 5; ek to hold, we require the additional condition that I f ; I I Ifk I wheneverJ # 0, which from the above is equivalent to I ex2 I I I acl I.

Two arcs ei and e, in G are said to be complements (resp., substitutes) iff for all elementary vectors f E H , f ; J 2 0 (resp., I 0 ) .

Example. In the graph given in Figure 2, a simple explo- ration of H yields that el and e7 are substitutes, el and e3 are complements, while el and e2 are neither.

Although it was shown in Section 3 that optimal arc- flow changes are not constant across d-equivalence classes, one can predict the comparative direction of changes as follows:

Lemma 6: Variations in pairs of complements and substitutes. Under the assumptions of the Ripple Theorem, iftwo arcs ei and ek are complements (resp., substitutes) and one at least belongs to &f( j ) , then

4. COMPLEMENTS AND SUBSTITUTES ( x : - xi ) (xL - xk) 2 0 (resp., I 0 ) .

In this section, we define the concepts of complements and substitutes for constrained networks. These notions are crucial to the development of the Monotonicity The- ory in Section 5. The definition of substitutes and complements as given in [ 131 for unconstrained problems only involved the topology of the graph. Indeed, two arcs are said to be complements (resp., substitutes) therein if every cycle in the network orients them in the same (resp., in opposite) direction( s) . Here, the side constraint a x = a0 comes into play through the elementary vectors, as seen in the definition below. Although we present in Lemma 6 some preliminary results on comparative flow changes between two arcs that are substitutes or complements, the central role played by substitutes and complements in predicting flow changes in a given arc resulting from changes in the parameter of another arc is described later in Theorem 11. We further show that, in contrast with the unconstrained case where one can determine whether two given arcs are substitutes, complements, or neither in linear time, for the constrained case, the problem is, in general, NP-complete. We then point out particular classes of instances of problems where a fast solution is provided.

Proof: If ( x : - xi )( xj, - xk) = 0, the proof is complete. Otherwise, suppose that ek E &jd(j) and notice that H' = Hj n H, = Hj fl Hk whenever ei = jd ek. For the optimal flow x' E X ( t ' ) defined in Theorem 1 by ( 5 ) , we obtain

def

where all the terms in both sums are nonzero and z, > 0 for all fa E H'. By conformality, all the terms in the first sum are of the same sign, and so are the terms in the second sum. Now, since e, and ej are complements ( resp., substitutes), these signs are the same (resp., opposite) and the proof follows. H

Remarks. In view of the above two results, it is natural to ask whether the property for a pair of arcs to be complements or substitutes is transitive. This is, in general, not true, as was shown in [ 13 ] for the unconstrained case


and, thus, for the constrained case as well.$ Still, in the unconstrained case, every pair of arcs in an equivalence class of the less-biconnected-to relation defined therein is either complement or substitutes. This is not valid for constrained networks. For instance, in the graph given in Figure 5 ,Hz = I f ” } . where,P = (el + e4 t e8 t e5) - 4(e2 + e7). and since .fl =.fi = I , then e l = f e g , el =; e8.

However, el and e8 are neither complements nor substitutes since f” andf’ = (el - e6 - e8 + e3) orient them in different directions. Finally, one might ask whether the substitutes/complements relation in the unconstrained graph is informative on the substitutes/complements relation in the constrained one. In general, it is not, as can easily be seen by taking two arcs, say e l , e,, which are, say, complements in the unconstrained version of a graph, and considering the possible constraints x, - x, = a. ( e l , e, are complements) and x, + x, = a. ( e l , e, are substitutes).

Lemma 7: Conditional Transitivity of Complements and Substitutes. Let e, , e,, and ek be arcs in G.for which [ i] el and e, are complements, [ ii] e, and ek are complements (resp., substitutes), and [ iii] eh I: el .$ Then, e, and ek ure complements ( resp., sirbstitirtes) .I1

Proqf Let f E H , and suppose that .t;X # 0. By [ iii], ./; # 0, and thus by [i] and [ ii],.hh > 0 andf;X > 0 (resp., < 0 ) . Therefore. sign(.f;ji) = s ign( . f zJ fk ) = sign(1;J) X sign(.{h) = 1 (resp.. - I ), completing the proof. rn

Theorem 8: The Complexity of Determining Substitutes and Complements. The problem ojdetrrmining whether a given pair o f arcs ure substitutes, complements, or neither is NP-complete.

Proof We sketch a reduction to the [ MWC] by build- ing G* as in the proof of Theorem 3 and replacing a: by ah - X for one chosen k . Then, ek and are “neither substitutes nor complements” iff there exists a pair of elementary vectors of G*. sayf” andf.- such that fLt l - = I . f ‘ f > 0, and f i < 0. From (7) and ( 8 ) , there exists two cycles c+ and c - with e l = ck = 1 and

-

f + = + ern+?) + ( l/a*cf)c+ and

f’- = (em+l + ernt2) + ( l / a * c - ) c - ,

$ Counter examples for the unconstrained case are given in [13]. For an example specific to constrained networks. refer to Figure 2, where ( e2, e4) are substitutes, (e4, e,) are complements, but ( e2. e , ) are neither.

5 In particular. [ i i i ] holds if e, and ek are in the same d-equivalence class for =,”.

I( If e, and e, are substitutes, then. trivially. a similar result holds, in which the conclusions are reversed.

cue+ > X and ac- < A. If. however, e!, and are substitutes (resp.. complements), then only c- (resp., c + ) exists. Repeating for k = 1, . . . . m solves the [MWC] and completes the proof. a

Tractable Cases

There are, however, cases in which one can easily determine if two arcs are substitutes or complements. More- over, the question of whether a given instance of the above problem is one of these cases can be answered in linear time by well-known graph algorithms.

Cut Arcs

For example, if two arcs el and e, form a cut (i.e., if removing them augments the number of connected components by at least one, but removing only one does not), then, clearly, ,4x = 0 implies one of ( i ) x, - x, = 0 or (ii ) x, + x, = 0. Thus, either.{ - 6 = 0 or /; + 6 = 0 holds for all elementary vectors .f = ( J ), and the two arcs e, and e, are ( i ) complements or ( i i ) substitutes.

Arcs in a Rim

Suppose that G is of type (11) and that one of its biconnected components, say G I , is a rim. Then, any elementary vector with arcs in G I is either a neutral cycle of GI or a pair of active cycles, one in GI and the other in G\G, . Thus, two arcs e, and e, in G I are complements (resp., substitutes) iff any cycle c of G , verifies c! c, 2 0 (resp., I 0) , i.e., if the two arcs are complements (resp., substitutes) in the sense defined in [ 131. This can be done in linear time. In particular, it is a consequence of the topology of rims that two arcs which belong to different rim components and each are incident to a bolt (not necessarily the same bolt) are conformal.

5. THE MONOTONICITY THEORY

The main result of this section provides conditions under which an optimal flow in one arc is monotone in the parameter associated with another arc. Bounds on the rate of change of the flow are established, namely, we show how the rate of change of the flow is bounded by a function of the rate of change of the parameter vector. We start by stating the extensions of two lemmas from Granot and Veinott [ 131 that are needed in the proof of our main result, the Monotone Optimal-Flow Selection Theorem for constrained networks.

Lemma 9: Ascending Optimal-Arc-Flow Multifunction. If To is a .series ofparameter vectors that differ onl,v in one


component j E E, and b, is subadditive* *, then the jth

If b( - , t ) is strictly convex, then X ( t ) = { x ( t ) } is a singleton and one can interpret the conclusion of Lemma 9 as stating that xj( t ) is nondecreasing in tj. To prove the monotonicity result for the case where not all arc costs are strictly convex, we define for a parameter t E [ 0, 11 the following perturbed problem:

P( t , t ) : min{ b ( x , t , t ) = b ( x , t )

projection o f X ( t ) , is ascending?? in t on To.

def

+ 2 tie-xl : A x = d , ax = a o } . e,EE

Let X ( t , t ) be the set of optimal solutions of P( t , t ) , and note that since b ( x , t , 0) = b ( x , t ) thenX(t, 0 ) = X ( t ) . A function f from 'iRn into itself is said to have an iterated limit 1 as x + y in Rfl, written Zlimx-,f( x ) = g ; if for some order of taking limits, 1irnxfln * limx2+n lim,,,, f ( x ) = 1.

Lemma 10: Iterated Optimal Constrained Flow. Zft E T, bj( * , t j) is convex and lower semicontinuous$$ for each ej € E, and X( t ) is nonempty and bounded, there is a unique x ( t , E ) E X ( t , € ) for each t % 0 (i.e., t i > Ofor all i) and

H Given an ordering of the arcs, the function t - x ( t ,

0 ) is called an Iterated Optimal-Constrained-Flow Selec- tion. It may depend on the ordering of the arcs and, thus, on the order of taking limits, as it does on the type of perturbation applied to the cost function [the perturbation b ( x , t , t ) proposed above is clearly not the only way to make the costs strictly convex].

Theorem 11: Monotone Optimal-Flow Selection. Suppose that bj( * , t j ) is convex and lower semicontinuous for each t, E TJ, b j ( - , - ) is subadditive for ej E E, and X ( t ) is nonempty and bounded for each t E T. Then, there exists an iterated optimal-constrained-flow selection t - x( t ) with the Ripple Property and for which x j ( t ) is nondecreasing (resp., nonincreasing) in tk whenever ej and ek are complements (resp., substitutes).

Proof: Similar to the proof of Theorem 10 in [ 13 1. Assume first that b ( x , t ) is strictly convex in x , in which case X ( t ) = { x ( t ) } is a singleton, and x ( t , 0) = x ( t ) . If t' is obtained from t by increasing tk to t i , then by Lemma

Zlim,+ox( t , t ) exists and is in X ( t ) .

* * Recall that a function f o n a lattice T is subadditive if for any two parameters t and t ' , f ( t A t') + f ( t V t ' ) s f ( t ) + f ( t ' ) , where f A t' and t V f ' are, respectively, the meet (coordinatewise minimum) and join (coordinatewise maximum) o f t and t'.

tt A series S1, . . . , S, of subsets of W is ascending iff for all x E S,, x' E S,,, with p < p', min(x, x') E S ( p ) and max(x, x') E S ( p ' ) .

$$ Recall that a real-valued convex function fon W" is lower semicontinuous iff all its level sets { x : f( x) I a }, a € %, are closed.

9, x k ( t ' ) - x k ( t ) 2 0, and since ei and ej are complements (resp., substitutes), then, by Lemma 6 , xi( t ' ) - xi( t ) 2 0 (resp., S O ) , completing the proof. In the case where the arc-costs are not all strictly convex, the perturbed costs b ( x , t , t ) are, and thus from Lemma 10, the perturbed problem possesses a unique optimal solution x( t , t ) for each t E T and t % 0. Now, clearly from the above, the result holds for x ( t , t ) , t E T . By taking the limit, the result also holds for x( t ) = x( t , 0 ) = Zlim,+ox( t , E ) , which by Lemma 10 will have the desired properties, and the proof is complete.@

5.1. The Smoothing Theorem

The Smoothing Theorem (Theorem 13) presents qualitative results regarding the rate of change of the flows compared with changes in the parameters and is an extension of Theorem 1 5 in [ 1 3 ]. We start with two preliminary definitions:

U { +co } is given by g#(a , a') = g(a' - a , a') . The function g is said to be doubly subadditive if both g and g# are subadditive. For examples of doubly subadditive functions, see, e.g., [13, p. 4891.

Two parameters t and t' in T are said to be monotonically step-connected if there is a finite sequence t = t o , t o , . . . , tk = t' such that t' and tr+' differ at most in one component for all r and t ' is coordinatewise monotone in r. Lemma 12 states that, under certain circumstances, a modification of an arc's parameter will not produce an optimal-flow change in this arc larger in magnitude than the variation of the parameter.

The dual g# of a function g from 'iR2 onto

Lemma 12. If P is a chain whose elements difler only in their jth entry, E %, by is subadditive and bi ( - , t,) is convex for all t E P , and all i Z j , then the multifunction t - t, - r j X ( t ) is ascending on To.

Theorem 13: Smoothing Theorem. Suppose that Ti E 3, bi( - , t ) is convex and lower semicontinuous for each t E Ti, that bi is doubly subadditive for all ei E E, and that X ( t ) is nonempty and bounded for each t E T. Then, there exists an iterated optimal-constrained-flow selection x( - ) with the Ripple property and such that x i ( t ) and Mjitj - xi ( t ) (resp., -Mjitj - xi ( t ) ) are nondecreasing (resp., nonincreasing) in tj whenever ei and ej are complements (resp., substitutes). Moreover, Ilx(t') - x( t ) l l , s Mllt' - tll for all monotonically step-connected t and t' in T, whereM= max{ l J /A[ : e,, e j € E , f E H j } .

Proof Suppose that two arcs ei and ej are complements and that t - x( t ) is an iterated optimal selection, then,

§§ Notice that if the costs are strictly convex, then X ( t) is the singleton { x ( t ) } , x(t) = x(t , 0), and thus the notation is consistent.


by Theorem 1 1, xi ( t ) is nondecreasing in t j . Moreover, by Lemma 12, ti - xi( t ) is nondecreasing in t j . Now, let t E T and let t‘ be obtained from t by augmenting its j th component. From Theorem 1 and Lemma 6,

0 I x i ( t ’ ) - x i ( t ) I Mji(xj(t’) - x j ( t ) ) -

and, thus, by the choice o f t and t‘, the quantity M,,x,( t ) - x, ( t ) is also nondecreasing in t,. Consequently, M,,t, - x , ( t ) =M,,(t,-x,(t))+(M,,x,(t)-x,(t))isthesum of two quantities that are nondecreasing in tl and, thus, nondecreasing in t,. Now, if e, and e, are substitutes instead of complements, then by inverting the direction of el and replacing x, ( t ) by -x , ( t ) , the arcs become complements, the above conclusions hold, and the proof of the first part follows. To show the second part, suppose that t and t’ differ in only one component, sayj. Then, from Theorem 1 and the definition of M , we have for all e, E E

by Theorem 9. Next, consider the case where t and t‘ are monotonically step-connected. Then, there exists a sequence t o = t , t ’ , . . . , t k = t’ such that t’ - t‘-’ has only one nonzero component for each 1 and t‘is coordinatewise monotone in 1 , and it follows from (9) that

completing the proof.

5.2. Subadditivity of Minimum Cost in the Parameters

In this section, we extend Theorem 17 in [ 131 and present a result concerning the variations of the minimum cost b ( t ) = C F l b,(x,(t) , t i ) , where x ( r ) is any optimal flow for P( t ) . For any subset D of E , denote by xD the subvector of x whose entries are indexed by D, and let Y , Z be the partition of E for which x ; < x y and x5 2 xz.

Theorem 14. Let C be a subset of E such that the arcs in C are all pairwise complements, and assume that XejEE\CTJ = { tE\C} , bj( - , ti) is convex for each ej E E\ C, T is a sublattice, bc is subadditive and b( - ) > -a on T, then b( . ) is subadditive on T.

Proof First notice that it suffices to show that, for any two feasible (not necessarily optimal ) constrained flows x and x‘ and a pair of parameters ( t , t ’ ) , b( t A t ’ ) + b( t V t ’ ) 5 b ( x , t ) + b(x’, t ’ ) . If either of the last two terms is + co , then the equality holds trivially; thus, in the following, we assume that all four terms are finite. Now, x’ - x is a sum of conformal elementary circulations, and if y is the sum of the conformal elementary circulations whose support contains an arc in Y and z the sum of the remaining elementary circulations, then y and z are conformal, y y = x’,. - x y and z y = 0. We now claim that yz = 0, and thus that zz = x> - xz . Indeed, suppose that yj # 0 for some ej E 2, then one of the elementary vectors fthat constitute y verifiesf;J # 0 for some ei E Y. Since the arcs in Care complements, f ; J > 0, contradicting the fact that f is conformal with x ’ - x. Therefore, xc A xk = xc + yc and xc V xk = xc + zc. It now follows from our assumptions on the arc-costs that b(t A t ’ ) + b ( t v t’) I b ( x + y , t A t ’ ) + b ( x + Z , t v t ’ ) I b ( x , t ) + b(x’, t ’ ) .

The above result can be further extended to include substitutes by noticing that two arcs e, and ej are substitutes iff they are complements in the graph obtained by reversing e, , replacing ai by -ai and bi( -, - ) by - b j ( * , a ) .

The authors appreciate constructive suggestions made by the referees.

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Received January 23, 1992 Accepted March 9, 1994

ripples, complements, and substitutes in singly constrained monotropic parametric network flow...

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