Three-Partition Flow Cover Inequalities for Constant Capacity Fixed-Charge Network Flow Problems

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1 Three-Partton Flow Cover Inequaltes for Constant Capaty Fxed-Charge Networ Flow Problems Alper Atamtür and Andrés Gómez Department of Industral Engneerng and Operatons Researh, Unversty of Calforna, Bereley, Calforna 9470 Smge Küçüyavuz Department of Integrated Systems Engneerng, Oho State Unversty, Columbus, Oho 430 Flow over nequaltes are among the most effetve vald nequaltes for apatated fxed-harge networ flow problems. These vald nequaltes are based on mplatons for the flow quantty on the ut ars of a two-parttonng of the networ, dependng on whether some of the ut ars are open or losed. As the mplatons are only on the ut ars, flow over nequaltes an be obtaned by ollapsng a subset of nodes nto a sngle node. In ths artle, we derve new vald nequaltes for the apatated fxed-harge networ flow problem by explotng addtonal nformaton from the networ. In partular, the new nequaltes are based on a three parttonng of the nodes. The new three-partton flow over nequaltes nlude the flow over nequaltes as a speal ase. We dsuss the onstant apaty ase and gve a polynomal separaton algorthm for the nequaltes. Fnally, we report omputatonal results wth the new nequaltes for networs wth dfferent haratersts. 06 Wley Perodals, In. NETWORKS, Vol. 67(4), Keywords: nteger programmng; lftng; superaddtvty; fxedharge networ flow; three-partton; faets. INTRODUCTION Many logsts, supply han, and teleommunatons problems are modeled as apatated fxed-harge networ flow problems (CFNF). A CFNF s defned on a dreted graph, wth gven supply or demand on the nodes of the graph, Reeved February 05; aepted February 06 Correspondene to: A. Atamtür; e-mal: atamtur@bereley.edu Contrat grant sponsor: Natonal Sene Foundaton (A. A.); Contrat grant number: Contrat grant sponsor: Offe of the Assstant Seretary of Defense for Researh and Engneerng (A. A.); FA Contrat grant sponsor: Natonal Sene Foundaton (S. K.); Contrat grant number: DOI 0.00/net.677 Publshed onlne Marh 06 n Wley Onlne Lbrary (wleyonlnelbrary.om). 06 Wley Perodals, In. and apaty, fxed and varable osts of flow on the ars of the graph. The problem s to hoose a subset of the ars and route the flow on the hosen ars whle satsfyng the supply, demand, and apaty onstrants, so that the sum of fxed and varable osts s mnmzed. Gven a dgraph G = (V, A), let y a be the flow on ar a A and let x a = f the ar a s used, 0 otherwse. Then, the CFNF problem an be formulated as mn a A(p a y a + q a x a ) s.t. a δ() + y a a δ() y a = d, V () y a a x a, a A () x B n, y R n +, where δ() + s the set of nomng ars to V, δ() s the set of outgong ars, V d = 0, and p a and q a are the varable and fxed ost of flow on ar a A, respetvely. The flow onservaton onstrants () ensure that the demand s met at node (when d > 0) and supply s not exeeded (when d < 0), and the upper bound onstrants () ensure that the ar apaty a, a A, s not exeeded. Almost all prevous wor on vald nequaltes for CFNF s based on ut-set relaxatons. Gven a subgraph, a ut refers to the ars that have one end n the subgraph, the other outsde. A ut set refers to the relaxaton of the feasble set of CFNF gven by the onstrants for the ut ars and the aggregaton of the supply/demand onstrants for the nodes n the subgraph defnng the ut. Smply, vald nequaltes based on a ut set are mplatons for the flow quantty on the ut ars dependng on whether some of these ars are open or losed. The flow over nequaltes (e.g., [5, 0,, 3]) and the flow pa nequaltes desrbed n Atamtür [] belong to ths ategory of nequaltes. Addtonally, Gu et al. [7] show how to strengthen the exstng nequaltes usng superaddtve lftng. These ut-set nequaltes are based on NETWORKS 06 DOI 0.00/net

2 a two-parttonng of the networ and gnore the networ struture n eah of the parttons, essentally ollapsng the orrespondng subgraphs nto a sngle node. Vald nequaltes that onsder strutures other than a twoparttonng of the networ have reeved less attenton n the lterature. Path nequaltes are studed n Atamtür and Küçüyavuz [3] and Van Roy and Wolsey []. The submodular nequaltes gven n Wolsey [5] are very general, but they are not gven n losed form sne the oeffents of the nequaltes are expressed as solutons to optmzaton problems. Vald nequaltes based on a three-parttonng of the networ have been studed n the onstant apaty networ desgn problem (e.g., [, 4, 8, 9]), but they have not been onsdered for CFNF problems. In ths wor, we onsder three-partton relaxatons for the onstant apaty CFNF (.e., a = for all a A). We present new vald nequaltes for ths problem, whh nlude as a speal ase the flow over nequaltes. We dsuss ther separaton and mplementaton as uttng planes to solve the onstant apaty CFNF, and we present omputatonal experments n networs wth dfferent haratersts. The artle s organzed as follows. In Seton, we revew the flow over nequaltes for the onstant apaty CFNF. In Seton 3, we study the three-partton relaxaton and provde vald nequaltes for ths problem. In Seton 4, we dsuss the mplementaton and separaton problem for the three-partton flow over nequaltes. In Seton 5, we gve omputatonal results, and n Seton 6 we onlude the paper. Throughout the artle, for a vetor v R n, let v(s) = S v, + = max {0, } and = mn {,0}.. FLOW COVER INEQUALITIES Gven dsjont sets N + and N, let N := N + N, n := N and > 0, let X := {(x, y) B n R n + : y(n+ ) y(n ) = d, y j x j, j N}, and let X be the relaxaton where the equalty s replaed wth an nequalty. The set X arses n CFNF when several nodes are aggregated nto one (.e., addng the orrespondng flow onservaton onstrants), mpltly defnng a twoparttonng of the networ. For a subset of vertes V V, d = d(v ) s the aggregate demand on the vertes of V, N + = { (, j) E : V \ V, j V } s the set of nomng ars to V, N = { (, j) E : V, j V \ V } s the set of outgong ars from V and s the (ommon) apaty of the ars. Fgure depts a typal two-parttonng of a networ. Padberg et al. [0] studed the onvex hull of X and X. A set S + N + s alled a flow over f λ = (S + ) d > 0. Moreover, when λ<, we say that the flow over s mnmal. Note that λ = d d n a mnmal flow over. Gven a flow over S +, the flow over nequalty (y j + ρ( x j )) mn{y j, ( ρ)x j } d, (3) j N FIG.. Two-parttonng of the networ. where ρ = ( λ) +, s vald for X and X. Moreover, the lfted flow over nequalty (y j + ρ( x j )) mn{y j, ( ρ)x j } j N max{y j ρx j,0} d (4) j N + \S + s also vald for X and X. Moreover, nequaltes (4) together wth the flow onservaton and bound onstrants ompletely desrbe the onvex hull of X. When applyng nequaltes (3) (4) to an aggregaton of many nodes ollapsed nto one, only the ars between the two parttons are onsdered n the nequalty, and the nternal networ struture of eah of the parttons s not taen nto aount n the nequalty. Note that optmzng a lnear funton over X s easy. Proposton. There s an O(n) algorthm to solve the optmzaton problem mn {py + qx : (x, y) X}. (5) Proof. By omplementng varables n N, that s, defnng x = x and ȳ = y, N, we wrte the equvalent problem p(n ) + q(n ) + mn (p y + q x ) (x,y) B n R n N + (p ȳ + q x ) N s.t. y(n + ) +ȳ(n ) = d + (N ) 0 y x, N + x ȳ, N. Observe that f q < 0 for N +, then x = n any optmal soluton, and f q < 0 for N, then x = 0 n any optmal soluton. Therefore, we assume wthout loss of generalty that q 0. Denote by ξ the ontrbuton to the objetve funton of ar N f t s at full apaty,.e., ξ = p + q for N + and ξ =p q for N. Assume N s sorted n nondereasng order of ξ s, and let N 0 denote the frst d+(n ) elements of N n ths order. 300 NETWORKS 06 DOI 0.00/net

3 If λ = 0, then N 0 gves the optmal soluton. Otherwse, note that there exsts an optmal soluton where there s only a sngle ndex wth orrespondng flow equal to λ, where λ = d d, and There are two possbltes: d+(n ) other ars are at full apaty. / N 0 : Then set all ars n N 0 at full apaty, and let be the best hoe between argmn N + \N 0 ( λ)p + q and argmn N \N 0 ( λ)p. N 0 : Then set the frst d+(n ) elements of N at full apaty exept for, whh s the best hoe between argmn N + N 0 λp and argmn N N 0 λp + q. Note that N need not be sorted; we only requre the frst ars (n any order), whh an be done n O(n) tme d+(n ) wth quselet. Therefore, the omplexty of the algorthm s O(n). We now show how to update an optmal soluton after a small hange n d. We wll use ths result later n Seton 3. Remar. Suppose we have an optmal soluton to problem (5), and the demand s hanged by ±. We an ompute a new optmal soluton from the prevous one by dong exatly one of the followng: Add one ar at full apaty. Remove one ar at full apaty. Complete the flow on the nonsaturated ar, and nrease the flowto λ n one of the ars wth no flow. Mae the flow on the nonsaturated ar 0, and derease the flow to n λ one of the saturated ars. If we eep two sorted lsts, one wth the ontrbutons of the ars at full apaty and one wth the ontrbutons of the ars at partal apaty, then these operatons are done n O() tme. 3. THREE-PARTITION ANALYSIS 3.. Prelmnares In ths seton, we study the onstant apaty threepartton polytope T. Fgure shows a graphal representaton of T, where node 0 (mplt) s a supply node and nodes and are demand nodes, and eah node represents a partton n the orgnal graph. Let N +, =,, be the set of ars gong from node 0 to node, let N, =,, be the set of ars gong from node to node 0, and let N and N be the sets of ars gong from node to node, and from node to node, respetvely. Defne N + = N + N+, N = N N, N = N+ N N N and n = N. Fnally, let d, =,, be the demand at node, and let d = d + d. T s defned as the onvex hull of y(n + ) y(n ) y(n ) + y(n ) = d (6) y(n + ) y(n ) + y(n ) y(n ) = d (7) FIG.. y j x j, j N x B n, y R n +. Three-partton graph. Moreover let T be the relaxaton of T where equaltes n (6) and (7) are replaed by nequaltes. Proposton. For any networ, after aggregaton, we may assume wthout loss of generalty that d 0 and d 0 n equaltes (6)-(7). Proof. Suppose that n a three-parttonng, there are two supply nodes and one demand node. In other words, d 0, d 0, and d 0 =d d. Lettng N + = N, N = N +, =, and N = N, N = N, and d =d, =,, we get an equvalent model wth a sngle supply node and two demand nodes. By Proposton, we assume throughout the rest of the artle that d 0 and d 0. Proposton 3. There s an O(n log n) algorthm to solve the optmzaton problem mn {py + qx : (x, y) T}. Proof. There exsts an optmal soluton n whh at most two ars have flow strtly between 0 and. In partular, there exsts a par of nodes suh that the flow between them s a multple of. Keep sorted lsts for N + N, N+ N and N N as desrbed n Remar. Suppose y(n ) y(n ) =, where = N,...,0,..., N. Set = 0, and the problem deomposes nto two sngle node onstant apaty fxed-harge networ flow problems, solvable n O(n) tme. Then, hangng one unt at a tme, we an ompute optmal solutons for all possble values of, eah one n O() tme (Remar ). Choose the one that results n the best objetve funton value. The ases when y(n + ) y(n ) = and y(n+ ) y(n ) = are handled smlarly n O(n) tme. The omplexty s then gven by sortng, whh s done n O(n log n) tme. NETWORKS 06 DOI 0.00/net 30

4 The exstene of polynomal tme algorthm for optmzaton over onv(t) mples that the separaton problem over onv(t) s also polynomal tme solvable [6]. Ths motvates an effort to desrbe strong nequaltes for T. We now haraterze strong vald nequaltes for T and T. 3.. Vald Inequaltes Defnton (Three-partton flow over). For S + N +, S + N+ and S N, we say that the set S = S + S+ S s a three-partton flow over f. λ := (S + ) d > 0.. λ := (S + S ) d > λ := (S + S+ ) d > 0. Furthermore, we say that the three-partton flow over S s mnmal f 4. λ < and λ<. FIG. 3. Type three-partton example. Condtons,, and 3 mply that the ars n S are suffent to satsfy the demand n nodes and. Condton 4 mples that f any ar s removed from S, then the demand an no longer be met. Note that f the over s mnmal, then λ = d d d. d and λ = Gven a mnmal three-partton flow over S = (S +, S+, S ), onsder the three-partton flow over nequaltes =, (y j + ρ ( x j )) mn{y j, ( ρ )x j } =, j N + \S + =, j N max{y j ρ x j,0} { } mn yj, (ρ ρ )x j j N \S j S (ρ ρ )( x j ) max{0, y j + (ρ ρ )x j } d, (8) j N ( λ, λ + (λ λ ) + )) (Type three-partton flow over nequaltes) where (ρ, ρ ) = ((λ λ) +, λ + (λ λ) + ) (Type three-partton flow over nequaltes). Remar. Note that when λ λ, the type threepartton flow over nequaltes redue to the lfted flow over nequaltes for both nodes ollapsed. Moreover, when λ λ, the type three-partton flow over nequaltes are equvalent to the lfted flow over nequaltes for node after addng the flow onservaton onstrant for node. In Seton 3.3., we gve ondtons under whh nequaltes (8) strtly domnate the flow over nequaltes. Before provng the valdty of nequaltes (8), we provde examples of both types of three-partton flow over nequaltes. FIG. 4. Type three-partton example. Example (Type three-partton flow over nequaltes). Consder the networ depted n Fgure 3, and assume that the ar apaty s 0. Note that λ = 0 5 = and λ = 0 0 = 8, and sne λ > λ the type three-partton flow over nequaltes redue to the flow over nequaltes for node. For ths networ, the flow-over nequalty for both nodes aggregated wth flow over {,, 5} s y x + y x + y 5 x 5 6, (9) and the three-partton flow over nequalty wth threepartton flow over {,, 3, 5} s y x + y x + y 5 5x 5 + 3( x 3 ) + (y 6 7x 6 ) + mn {y 4,3x 4 } 3. Example (Type three-partton flow over nequaltes). Consder the networ depted n Fgure 4, and assume that the ar apaty s 0. Note that λ = = 6 and λ = 0 6 = 4, and sne λ < λ the type 6 0 three-partton flow over nequaltes redue to the flow over nequaltes for both nodes ollapsed. For ths networ the lfted flow-over nequalty for node wth flow over {} s y 4x + (y 3 4x 3 ) + + (y 4 4x 4 ) + mn {y 5,6x 5 } 0, and the three-partton flow over nequalty wth threepartton flow over {, } s y x 4x mn {y 3,4x 3 } + (y 4 6x 4 ) + mn {y 5,4x 5 } NETWORKS 06 DOI 0.00/net

5 and x j = for j S +. Under these ondtons, the lfted flow over nequalty for nodes and ollapsed, and wth flow over S + S+ s y(s + S+ ) ( λ)( x j ) (y j ( λ)x j ) + j N + \S+ mn { } y j, λx j d. (0) j N FIG. 5. Funton g. We now state and prove the man result of the artle. Theorem. The three-partton flow over nequaltes (8) are vald for T and T. To prove Theorem, we fx some of the varables at ther upper or lower bounds n order to apply the lfted flow over nequaltes of Seton. We then use superaddtve lftng to nlude the varables assumed to be fxed n the nequaltes. We frst state two propostons on superaddtve funtons that are used to prove Theorem. Proposton 4. (Gu et al. [7]) The funton g a, : R R, where 0 < a <, gven by { a f z ( + ) a, Z g a, (z) = z ( a) f a z, Z s superaddtve on R. Fgure 5 depts funton g a, (z). Proposton 5. Let g : R R be a superaddtve funton and let h : R R be defned as h(x, y) = g(x + y). Then, the funton h s superaddtve on R. Proof. For any (x, y ), (x, y ) R we have h(x + x, y + y ) = g((x + y ) + (x + y )) g(x + y ) + g(x + y ) = h(x, y ) + h(x, y ). We now proeed wth the proof of Theorem Valdty of Type Inequaltes. We prove the valdty of type three-partton flow over nequaltes when λ>λ. Let S be a mnmal three-partton flow over, and assume y j = for all j S (.e., y(s ) = (S ) = λ +λ λ), x j = 0 for j (N + \S+ ) N (N \S ) N, To obtan a vald nequalty for T, we lft nequalty (0) frst wth the varables assumed to be fxed n N + N, and then wth the varables n N N. The lftng funton assoated wth smultaneously lftng the nequalty wth the varables x j for j S + and pars (x j, y j ) for j (N + \ S+ ) N s gven by f (z, w ) = d + mn y(s+ S+ ) ( λ)( x j ) (y j ( λ)x j ) + mn { } y j, λx j s.t. j N + \S+ j N y(n + ) y(n ) d + λ + λ λ y(s + ) d λ λ + λ + w y(s + ) d + λ y(s + ) d + λ λ z 0 y j x j, x j {0, }, j N + N, where z s a nonnegatve multple of the apaty and stands for the apaty losed on ars n S +, w > 0 stands for the flow on ars n N and w < 0 stands for the flow on ars n N + \ S+. Note that the problem deomposes for the ars n N + N and the ars n S +. The objetve funton s nonnreasng n y(n + N ), and therefore, there exsts an optmal soluton where y(n + )y(n ) = d +λ +λ λ. The optmal hoe for parameters s to set y(n ) = x(n ) = 0, y(n+ \S+ ) = 0, and y(s + ) = d + λ λ + λ > (S + ) λ, sox j = for all j S +. Smlarly, n an optmal soluton we have y(s + ) = d + λ λ + mn {z, w λ }. Replang these values n the objetve, we get a losed form soluton for the lftng funton, { 0 fz + w λ f (z, w ) =w + z + w λ otherwse. The exat lftng funton f s not superaddtve n R. Note however that f and the superaddtve funton g λ, of Proposton 4 are losely related, as shown by Fgure 6. In partular, we show that ψ(z, w ) := w +g λ,(z +w ) s a superaddtve vald lftng funton for f. Proposton 6. bound of f. The funton ψ s a superaddtve lower NETWORKS 06 DOI 0.00/net 303

6 y(s + ) d + λ y(s + ) d + λ λ 0 y j x j, x j {0, }, j N + N, =,, FIG. 6. Funtons f (z + w ) + w and g λ,(z + w ). Proof. The funton ψ (z, w ) = w s lnear and therefore superaddtve. The funton ψ (z, w ) = g λ,(z + w ) s superaddtve by Proposton 5. We have that ψ = ψ + ψ s a sum of superaddtve funtons, and s therefore superaddtve tself. Moreover, sne ψ s a lower bound of f + w, ψ s a lower bound of f. Usng the lftng funton ψ of Proposton 6, we obtan the nequalty y(s + S+ ) + ( λ)( x j ) (y j ( λ)x j ) + j N + \S+ mn { } y j, λx j + ( λ )( x j ) j N j N + \S+ (y j ( λ )x j ) + mn { } y j, λ x j d. j N () Inequalty () assumes that y j = for j S and x j = 0 for j (N \ S ) N. To obtan a vald nequalty for T, we lft nequalty () wth the varable pars (x j, y j ) for j N N. The orrespondng lftng funton s f (w ) = d + mn y(s+ S+ ) ( λ)( x j ) j N + \S+ ( λ )( x j ) j N + \S+ (y j ( λ)x j ) + (y j ( λ )x j ) + mn { } y j, λx j mn { } y j, λ x j j N j N s.t. y(n + ) y(n ) d + λ + λ λ w () y(n + ) y(n ) d λ λ + λ + w (3) where w > 0 stands for the flow on ars n N plus the unused apaty on S (.e., y(n ) j S ( j y j )) and w < 0 stands for the flow on ars n N \ S. The problem deomposes agan and sne the objetve funton s nonnreasng n y(n + ) y(n ), =,, we an set these values to ther upper bounds, so that onstrants () and (3) are bndng. Note that for λ λ w λ,wehave(s + ) λ y(n + )y(n ) (S+ ). Therefore, n ths range, there exsts an optmal soluton wth y(s + ) = d +λ +λ λw and y(n N+ \ S+ ) = 0. For w λ we need to sequentally lose ars n S + or open ars n N, and for w λ λ we need to sequentally open ars n N + \ S+. Moreover, for 0 w λ,wehave(s + ) λ y(n + )y(n ) (S+ ). Therefore, n ths range, there exsts an optmal soluton wth y(s + ) = d λ λ +λ+w and y(n N+ \ S+ ) = 0. For w 0, we need to sequentally lose ars n S + or open ars n N, and for w λ we need to sequentally open ars n N + \ S+. The lftng funton, obtaned by subtratng from d the ontrbutons of ars n N + N, =,, s then (λ λ ) f w ( + ) λ + λ, Z f (w ) = w + (λ λ ) f λ + λ w, Z. Funton f s of the form g λλ, defned n Proposton 4, and s, therefore, superaddtve n R. Usng f, we obtan the type three-partton flow over nequalty y(s + S+ ) + ( λ)( x j ) (y j ( λ)x j ) + j N + \S+ mn { } y j, λx j + ( λ )( x j ) j N j N + \S+ (y j ( λ )x j ) + mn { } y j, λ x j j N { } (λ λ )( x j ) + max 0, yj + (λ λ )x j j S j N { } mn yj, (λ λ )x j d. (4) j N \S 3... Valdty of Type Inequaltes. The valdty of the type three-partton flow over nequaltes when λ >λs proved smlarly: We assume x j = for j S + S+ S and x j = 0 for j (N + \ S + ) N, =,. Under those assumptons, the lfted flow over nequalty for node yelds 304 NETWORKS 06 DOI 0.00/net

7 y(s + S ) ( λ )( x j ) j S { max yj ( λ )x j,0 } j N \S { } yj, λ x j d. j N mn We then lft frst varables n N + N and then N+ N to get a vald nequalty. The omplete proof s n Appendx. We now gve an alternatve haraterzaton of nequaltes (8). Remar 3. Note that we an rewrte nequaltes (8) as (y j ρ x j ) mn{y j, ( ρ )x j } =, =, j N max{y j ρ x j,0} (ρ ρ )x j =, j N + \S + j S { } mn yj, (ρ ρ )x j j N \S max(0, y j + (ρ ρ )x j ) j N d ρ S + (ρ ρ ) S. (5) =, Furthermore, from the ondtons defnng a mnmal three-partton flow over, we nfer that Condton : S + d Condtons and 4 : S + + S d = Condtons 3 and 4 : S + + S+ = d Condtons, 3, and 4 : S S + = S + S + S + S+ = d d Therefore, we get that nequaltes (8) are equvalent to (y j ρ x j ) mn{y j, ( ρ )x j } =, =, j N max{y j ρ x j,0} (ρ ρ )x j =, j N + \S + j S { } mn yj, (ρ ρ )x j j N \S max(0, y j + (ρ ρ )x j ) j N d d d + (ρ ρ ) ρ. (6) 3.3. Strength of the Three-Partton Flow Cover Inequaltes In ths seton, we study the strength of nequaltes (8). We prove that, under mld ondtons, they are faet defnng for T. We also show that, n some ases, they domnate flow over nequaltes Faet-Defnng Condtons Proposton 7. d > 0. T s full dmensonal when d > 0 and Proof. Let e be -th unt vetor and let 0 < ε mn {d, d, }. The set {0} n = {(x, y) = (e,0)} n = {(x, y) = (e, εe )} ontans n + affnely ndependent ponts belongng to T. Theorem. The three-partton flow over nequaltes (8) are faet defnng for T when d > 0, d > 0, S + = and S =. Proof. To prove that, under the ondtons of the theorem, nequaltes (8) are faet defnng for T, we provde, for eah type of three-partton flow over nequalty, n affnely ndependent ponts where the nequalty holds at equalty. Let = (,,...,), = (,,..., ) and let e be the -th unt vetor. For larty, we gve eah pont n the format ( x S+, y S +, x S +, y S +, x S, y S, x N\S, y N\S), where (y S, x S ) are the (y, x) values for the ars n the set S. Faet proof for type nequaltes Let C + N + \ S +, S N, =,, and let C N \ S, S N, and rewrte nequalty (4) as y(s + S+ ) ( λ)( x j ) (y j ( λ)x j ) j C + j S j C + λx j j N \S (y j ( λ )x j ) j S y j ( λ )( x j ) λ x j j N \S (λ λ )( x j ) + (y j + (λ λ )x j ) j S j S (λ λ )x j y j d. j C j N \(S C ) Note that λ λ<0 mples (S ) = λ + λ λ< λ < d + λ = (S + ). Therefore, f S = then we have S +. Let S +, =,, let S+ wth = and let S. Table shows the affnely ndependent ponts y j NETWORKS 06 DOI 0.00/net 305

8 TABLE. Affnely ndependent ponts for Type nequaltes S + S + S N \ S Condton y x y x y x y x S + λe λ e 0 0 e e λ e 0 0 S λ e λe e λ e e e N \ S λe λ e 0 e λ e λe e S e e ( λ λ )e 0 0 e (λ λ )e e e e 0 0 S e e ( λ + λ )e e e (λ λ )e e e e N \ S e e λ e 0 e e e λ e e S + (λ λ )e λ e 0 0 (λ λ )e e e 0 0 S (λ λ )e λ e e (λ λ )e e e N \ S (λ λ )e λ e 0 e (λ λ )e λ e e C + e e λ e ( λ)e e e e λ e ( λ + λ )e e N + \ (S+ C+ ) e e λ e 0 e e e λ e ( λ)e e C + (λ λ )e e e ( λ )e e λe e e λ e e e N + \ (S+ C+ ) (λ λ )e e e 0 e (λ λ )e e e ( λ )e e C λe e e ( λ )e e (λ λ )e λ e e e e e N \ (S C ) (λ λ )e e e 0 e e e (λ λ )e e for type nequaltes. Note that we provde two ponts for eah j N (and therefore there are n ponts). To he that the ponts are ndeed affnely ndependent, observe that the two ponts orrespondng to eah N \S ensure that x = 0 and y = x (for some R + ). Faet Proof for Type Inequaltes Let C + N + \ S +, S N, =,, and let C N \ S, S N. Rewrte nequaltes (8) as y(s + S+ ) (λ λ)( x j ) (y j (λ λ)x j ) j C + j S j C + ( (λ λ))x j j N \S (y j ( λ)x j ) j S y j ( λ)( x j ) λx j j N \S ( λ )( x j ) + (y j λ x j ) j S j S ( λ )x j y j d. j C j N \(S C ) Let S +, =,, and let S. Table shows the n affnely ndependent ponts for type nequaltes. To he y j that the ponts are ndeed affnely ndependent, observe that the pont orrespondng to = wth = s affnely ndependent from the prevously ntrodued ponts sne t s the frst pont where y(s + ) = y(s ) + d, and the two ponts orrespondng to eah N \ S ensure that x = 0 and y = x (for some R + ) Comparson wth the Flow Cover Inequaltes. Reall from Remar that, dependng on the values of λ and λ, ether the type or the type three-partton flow over nequaltes redue to lfted flow over nequaltes. We now gve ondtons under whh nequaltes (8) domnate the regular flow over nequaltes. Proposton 8. When λ > λ and S = N, the type three-partton nequaltes wth mnmal flow over (S +, S+, S ) domnate the orrespondng flow over nequaltes for both nodes aggregated wth flow over S + S +. Proof. Consder the type three-partton nequaltes when S = N y(s + S+ ) ( λ)( x j ) mn { } y j, λx j j N 306 NETWORKS 06 DOI 0.00/net

9 TABLE. Affnely ndependent ponts for Type nequaltes S + S + S N \ S Condton y x y x y x y x S λe λ e 0 0 ( λ + λ)e e e 0 0 S + λe (λ λ)e 0 0 e e (λ λ)e 0 0 = e e 0 0 e e e e 0 0 S + \ { } ( λ + λ)e e e 0 0 e e e e 0 0 S λe λ e e ( λ + λ)e e e N \ S λe λ e 0 e λe λ e S (λ λ)e λe e (λ λ)e e e N \ S λe (λ λ)e 0 (λ λ)e λ S e e ( λ + λ)e e e e e e N \ S ( λ + λ)e e e 0 e e e ( λ + λ)e e C e e e e ( λ + λ)e e e e e e e e N \ (S C ) λe λ e 0 e λe e e ( λ ) e C + e e λe (λ λ)e e e e e e (λ λ)e e N + \ (S+ C+ ) e e e e 0 e e e e e (λ λ)e e C + λe e e λ e e e e e (λ λ)e ( λ)e e N + \ (S+ C+ ) e e (λ λ)e 0 e e e (λ λ)e ( λ)e e ( λ )( x j ) mn { } y j, λ x j j N (λ λ )( x j ) j N { } 0, yj + (λ λ )x j d, (7) j N max and the flow over nequaltes y(s + S+ ) ( λ)( x j ) mn { } y j, λx j j N ( λ)( x j ) mn { } y j, λx j d. j N Observe that the oeffents multplyng the terms ( x j ) for j S + and x j for j N are stronger for the threepartton flow over nequalty. Moreover, nequalty (7) s further strengthened by the nonnegatve terms orrespondng to the ars n N N. Example (ontnued). Consder the networ depted n Fgure 3. Note that f we delete ar 4 from the networ, then the orrespondng three-partton flow over nequalty wth flow over {,, 5} y x + y x + y 5 5x 5 + 3( x 3 ) + (y 6 7x 6 ) + 3 domnates the flow over nequalty (9). Proposton 9. When λ < λ and S + = N +, the type three-partton nequaltes wth mnmal flow over (S +, S+, S ) domnate the orrespondng flow over nequaltes for node wth flow over S + S. Proof. Note that, after addng the flow onservaton onstrant (6) to the flow over nequalty for node, we get the equvalent nequalty y(n + S+ ) y(n ) ( λ )( x j ) mn { } y j, λ x j + ( λ )( x j ) j N j S { } mn yj, ( λ )x j + (y j λ x j ) + d, j N \S j N NETWORKS 06 DOI 0.00/net 307

10 and ompare t wth the three-partton flow over nequalty y(n + S+ ) y(n ) (y j ( (λ λ))x j ) + j N (λ λ)( x j ) ( λ)( x j ) j N + mn { } y j, λx j + ( λ )( x j ) j N j S { } mn yj, ( λ )x j + (y j λ x j ) + d. j N \S j N (8) Observe that the oeffents multplyng the terms ( x j ) for j S + and x j for j N are stronger for the threepartton flow over nequalty. Moreover, nequalty (8) s further strengthened by the nonnegatve terms orrespondng to the ars n N + N. Example (ontnued). Consder the networ depted n Fgure 4. Note that f we add the flow onservaton onstrant of node to the flow over nequalty, then we get the equvalent nequalty y 4x mn {y 3,4x 3 } mn {y 5,6x 5 }, whh s domnated by the three-partton flow over nequalty y x 4x mn {y 3,4x 3 } mn {y 5,4x 5 } 0. Remar 4. Note that the ondton S = N for the type nequaltes s naturally satsfed when there s a sngle ar between nodes and. The observaton suggests that the type nequaltes are partularly effetve (wth respet to the flow over nequaltes) when nodes and are sngle nodes (and the mplt node 0 orresponds to the aggregaton of all other nodes n the CFNF). Smlarly, type nequaltes may be partularly effetve when nodes 0 and are sngle nodes, and node orresponds to the aggregaton of all other nodes n the CFNF. Fnally, we lose the seton by notng that three-partton nequaltes and flow over nequaltes are not suffent to haraterze T or T. Example (ontnued). Consder the networ depted n Fgure 4. The nequalty 6x + 3y 6x + y 6x 3 + y 3 x 4 + 3y 4 x 5 y 5 0 s faet defnng for T and T, but s nether a three-partton flow over nequalty nor a flow over nequalty. 4. SEPARATION Gven a fratonal soluton, the separaton problem onssts of fndng a three-partton flow over nequalty that uts off that soluton f there exsts any. In Seton 4., we desrbe an algorthm that, gven a three-parttonng of the nodes, fnds a most volated three-partton flow over nequalty. In Seton 4., we provde dfferent heursts for hoosng the three-parttons. 4.. Choosng a Mnmal Cover In ths seton, we gve a polynomal separaton algorthm for fndng a most volated three-partton flow over nequalty gven a three-parttonng of the nodes. From nequalty (6), we observe that fndng a most volated nequalty for a gven fratonal soluton (y, x) onssts of hoosng sets C + N+, C+ N+ and S N suh that the left-hand-sde of (y j ρ x j ) mn{y j, ( ρ )x j } =, j C + =, j N { } (ρ ρ )x j mn yj, (ρ ρ )x j j S j N \S max(0, y j + (ρ ρ )x j ) j N d + (ρ ρ ) d s maxmzed, where C + d C + + C+ d C + + S d ρ, (9) d. (0) Let C + = { j N + : y j ρ x j 0 }, =,, and let S = { } j N : y j (ρ ρ )x j. If ondtons (0) are satsfed by ths hoe of C +, C+, S and the left-handsde of (9) s greater than or equal to the rght-hand-sde, then we have found a most volated nequalty. Otherwse, we need to add more elements to the sets C +, C+,orS so that ondtons (0) are satsfed. In order to do so, we eep three sorted lsts, for the ars n K = N + \ C +, =, and K = N \ S n whh the elements n the lst are sorted n desendng order of ɛ j = y j ρ x j for j K, =, and ɛ j = y j (ρ ρ )x j for j K.For {,, }, let K [0] =max j K ɛ j, and let frst(k ) be a funton that returns and removes the greatest element of K. The separaton algorthm s desrbed n Algorthm. If the left-hand-sde of (9) Note that C + orresponds to S + plus the ars n N + \ S + wth nonzero terms. 308 NETWORKS 06 DOI 0.00/net

11 s greater than the rght-hand-sde at termnaton of the algorthm, then we have found a volated nequalty. Otherwse, there exsts no volated three-partton flow over nequalty. Algorthm Separaton algorthm been hosen. In ths seton, we propose strateges for fndng three-parttons that may lead to volated nequaltes Sngle Nodes. When desrbng the mplementaton of flow over nequaltes, Van Roy and Wolsey [4] onsder n turn eah flow onservaton onstrant, whh s equvalent to onsderng two-parttons n whh one of the parttons s a sngle node. In a smlar sprt, we onsder all three-parttons n whh two of the parttons are sngle nodes Spannng Trees. Stallaert [] desrbes a heurst for fndng two-parttons to apply flow over nequaltes. We use an adaptaton to the ase of three-parttons, whh s desrbed next. Gven a soluton (x, y) to a CFNF defned on the graph G = (V, A), defne an atve ar as an ar j A that s nether vod nor saturated,.e., 0 < y j <, and let Ā be the set of atve ars n the urrent soluton. The algorthm proeeds as follows: Step. Construt a maxmum spannng forest wth ar weghts (x j y j ) for j Ā and otherwse. Step A (Two-partton). For eah ar n the forest, let T and T be two trees that result from the deleton of ar, wth node sets V T and V T. Then for eah =, there s a two-partton defned by V T and V \ V T. Ths s the orgnal method reported n []. Step B. For eah ar n the forest and =,, defne V T as above. Let v be the sngle node onneted to V T by. Then for eah =, there s a three-parttonng defned by V T, { v } and V \ (VT { v } ). Step C. For eah node, l n the forest wth at least two adjaent nodes, let { T l} be the olleton of trees that result from the deleton of node l. Then for eah par of elements and j n { T l} there s a three-parttonng defned by V T l, V T l j and V \ (V T V T j ). Note that sortng the lsts an be done n O(n log n) tme. If the lsts are sorted, then eah omputaton of K [0] and eah all of frst(k ) an be done n onstant tme. Therefore, the ommands nsde eah loop of Algorthm an be done n onstant tme. Sne an element s removed from the lsts at eah step, the loops fnsh n at most K K K n steps, and the omplexty of the algorthm after sortng the ars s O(n). Proposton 0. Gven a three-parttonng of the networ, there exsts an O(n log n) separaton algorthm for nequaltes (8). 4.. Choosng Three-Parttons For a general CFNF, the separaton algorthm desrbed n Seton 4. assumes a three-parttonng of the vertes has Extenson Methods. The dea of the extenson methods s to generate new parttons from a set of exstng parttons. Gven a three-parttonng P, defne the haraterst funton e P : V {0,, } as the funton that maps eah vertex to ts partton (labeled as 0, and ), and defne b P as the value of the volaton of a most volated threepartton flow over nequalty arsng from P. We generate new parttons n the followng ways: Mxture. Gven parttons P and P, generate a new partton P suh that for eah v V e P (v) f e P (v) = e P (v) or e P (v) = 0 e P (v) = e P (v) f e P (v) = 0 e Pm (v) otherwse, where m = fb P b P and m = otherwse. NETWORKS 06 DOI 0.00/net 309

12 Modfaton. Gven a partton P 0, generate a new partton P suh that e P (v) = e P0 (v) for v V \ {l}, and e P (l) = e P0 (l). The vertex l and the new value e P (l) are hosen randomly. Gven ζ Z +, the algorthm proeeds as follows: Step 0. Add all parttons n whh two of the parttons are sngle nodes to a pool of parttons. Step. Choose the ζ hghest parttons n terms of b P. Step. For eah par of the seleted parttons, generate a new partton usng the Mxture operaton. Add all new parttons to the pool. Step 3. For eah seleted partton, generate a new partton usng the Modfaton operaton. Add all new parttons to the pool. Step 4. If a termnaton rteron s met, then termnate the algorthm. Otherwse, return to Step. 5. COMPUTATIONAL EXPERIMENTS In ths seton, we report omputatonal experments wth solvng CFNF of varyng szes wth a ut-and-branh algorthm usng CPLEX v.6.0. All experments are onduted on one thread of a Dell omputer wth a.ghz Intel Core 7-670QM CPU and 8 GB man memory. We test the CPLEX branh-and-bound algorthm usng the followng onfguratons for generatng uts: ALL Adds three-partton flow over nequaltes for all three-parttons usng omplete enumeraton. FC Adds flow over nequaltes for sngle nodes and aggregatng two nodes. TP Adds three-partton flow over nequaltes n whh two of the parttons are sngle nodes. Note that onfguraton TP onsders the same parttons as FC (nstead of aggregatng the two nodes, eah node s ts own partton). FC* Adds the nequaltes from FC, plus flow over nequaltes derved from the spannng tree heurst from [] and from an adaptaton of the extenson heursts of Seton 4..3 to two parttons. TP* Adds three-partton flow over nequaltes usng the strateges proposed n Seton 4. for seletng the three-parttons. CP CPLEX n default settng. Wth the exepton of the onfguraton CP, CPLEX uts and heursts are turned off. For FC* and TP*, we use ζ = 50 for the partton ombnaton heurst. All nstanes are randomly generated as follows. Let α {40, 60, 80} be a densty parameter and let β {.5, } be a apaty parameter. In eah nstane, 40% of the nodes are demand nodes, 40% are supply nodes, and 0% are transhpment nodes. For eah demand node, the demand s randomly generated between and 0. The total supply, equal to the total demand, s dstrbuted equally among the supply nodes. The apaty of the ars s gven by = β d, where d denotes the average demand; note that nstanes wth hgh β result n weaer LP relaxatons, and thus n more dffult nstanes. Between eah par of nodes, there s an ar wth probablty α/00, wth fxed ost between and 000 and varable ost between and 00; for ths hoe of parameters the fxed and varable osts of an ar at full apaty are of the same order of magntude. 5.. ALL vs. TP and TP* Frst, to test the effetveness of the strateges for fndng three-parttons, we solve small nstanes wth up to 4 nodes. For eah nstane, we ompare the gap mprovement obtaned by usng only sngle node three-parttons (TP), usng the proposed heursts to fnd addtonal parttons (TP*), and onsderng all parttons by dong omplete enumeraton (ALL). Table 3 presents the results. Eah row represents the average over fve randomly generated nstanes of smlar haratersts. The table shows, from left to rght: the number of nodes n the nstane; the ntal gap; the algorthm onfguraton; the root gap mprovement, omputed as 00 (zroot znt)/(zub znt), where zroot s the LP lower bound after addng the uts, zub s the best nteger soluton found, and znt s the ntal LP soluton; the number of uts added by CPLEX; the end gap, as reported by CPLEX; the number of nodes proessed n the branh-and-bound tree; and the total tme used n seonds. Confguraton TP* stres a good balane between the qualty of gap mprovement and the soluton tmes. A gap mprovement lose to omplete enumeraton s aheved n only a fraton of the tme. 5.. FC vs. TP and FC* vs. TP* Next we test the mpat of the three-partton flow over nequaltes for larger nstanes wthout the nterferene of CPLEX uts. To evaluate the margnal mpat of addng threepartton uts on top of the flow over uts, we mplemented our verson of the lfted flow over nequaltes and tested the versons of the algorthm wthout separaton heursts (FC and TP), and usng the separaton heursts (FC* and TP*). Tables 4 and 5 present the results for 60-node nstanes wth dfferent apaty-to-demand ratos. Eah row represents the average over fve randomly generated nstanes of smlar haratersts. We set the tme lmt to 700 s and the memory lmt to 4 GB. The tables show, from left to rght: the ar densty; the ntal gap; the algorthm onfguraton; the root gap mprovement; the number of user uts and uts added by CPLEX; the end gap, as reported by CPLEX; the number of nodes proessed n the branh-and-bound tree; the total tme used; the results of the fve nstanes, where S denotes the number of nstanes solved to optmalty, T denotes the number of nstanes that tmed out and M denotes the number of nstanes that used all the avalable memory. Usng onfguratons TP and TP* results n an addtonal root gap mprovement of.7% and.% over onfguratons 30 NETWORKS 06 DOI 0.00/net

13 TABLE 3. Heursts ompared to omplete enumeraton Nodes. Intal gap. Confg. Gap mpr. Root tme Cuts End gap Nodes Tme 0 4.9% ALL 98.3% % 5 6 TP 85.6% 0 9 0% 37 0 TP* 98.% 63 0% 8 3.% ALL 97.5% % 8 64 TP 85.0% % 80 0 TP* 97.5% % % ALL 99.% 309 0% TP 85.4% % 7 0 TP* 99.% % % ALL 96.5% % TP 79.% % 83 0 TP* 96.% 8 0% % ALL 97.6% 4, % 06 4,600 TP 8.8% % 57 TP* 97.3% % 74 3 Average ALL 97.8%,84 0% 65,84 TP 83.4% % 77 0 TP* 97.6% % 56 0 TABLE 4. Instanes wth β =.5 and CPLEX uts off Cuts Result Dens. Intal gap. Confg. Gap mpr. User CPLEX End gap Nodes Tme S T M 40 4.% FC 67.6% % 739,986, FC+TP 80.5% % 35,35, FC* 86.7% % 305,80, 0 3 FC*+TP* 89.7% % 39,779, % FC 68.3% % 597,65, FC+TP 8.% % 38,04, FC* 87.7% % 63,66, FC*+TP* 90.0% % 40,076 3, % FC 69.7% % 95, FC+TP 83.4% % 83,4, FC* 88.4% % 85,477, FC*+TP* 90.0% % 0,075 3, Average FC 68.4% % 5,49,46 FC+TP 8.8% % 57,800,364 FC* 87.8% % 5,473,88 FC*+TP* 90.% % 83,977 3,07 FC and FC*, respetvely. We observe that opton TP also mproves over FC n terms of end gaps, resultng n redutons of.8% and 8.7% n the low and hgh apaty nstanes, respetvely. In the low apaty nstanes, opton TP* results n a small nrease of 0.% n end gaps wth respet to FC*, but n the hgh apaty nstanes TP* s learly superor, wth a derease of 7.0% n end gaps. Overall we see that three-partton flow over nequaltes are partularly effetve for the hgh apaty nstanes. Indeed, three-partton uts nrease the sze of the formulaton (mang the LPs harder to solve), but on the harder nstanes the stronger formulaton typally results n better overall performane. Note that sne the memory lmt s reahed n many of the nstanes, hgh run tmes ndate a better ablty to prune n the branh-and-bound tree (nstead of faster tmes to reah optmalty) CP vs. TP* Fnally, we test the beneft of addng three-partton flow over nequaltes to CPLEX wth default onfguraton. Note NETWORKS 06 DOI 0.00/net 3

14 TABLE 5. Instanes wth β = and CPLEX uts off Cuts Result Dens. Intal gap. Confg. Gap mpr. User CPLEX End gap Nodes Tme S T M % FC 73.% % 433, FC+TP 83.9% % 3,385, FC* 88.% % 457,307, FC*+TP* 90.0% % 465,086 3, % FC 73.5% % 53,709, FC+TP 83.8% % 47,659, FC* 87.8% % 3,06, FC*+TP* 89.7% % 3,304, % FC 69.8% % 8, FC+TP 83.0% % 47,863, FC* 88.3% % 0,030, FC*+TP* 90.6% % 34,303 4, Average FC 7.6% % 47,00,95 FC+TP 83.6% % 59,86,85 FC* 88.% 3 0.% 33,,08 FC*+TP* 90.% % 307,565 3,476 TABLE node nstanes wth CPLEX uts on Cuts Result Cap. Dens. Intal gap. Confg. Gap mpr. User CPLEX End gap Nodes Tme S T M % CP 87.4% % 00,354, CP+TP* 90.9% % 66, % CP 87.0% % 47, CP+TP* 9.0% % 34, % CP 86.7% % 7,40, CP+TP* 9.3% % 40,45, % CP 88.6% % 5,583, CP+TP* 90.9% % 85,6, % CP 87.% % 44,57 3, CP+TP* 90.% % 56,00 3, % CP 88.% %,98 4, CP+TP* 9.3% % 80,4, Average CP 87.5% % 00,586,396 CP+TP* 90.9% % 77,309,70 that CPLEX uses flow over nequaltes (among other uts) and onsders many levels of aggregaton. Tables 6 and 7 present the results for dfferent node szes. Three-partton flow over nequaltes help to lose an addtonal.9% of the root gap on top of default CPLEX, and result n a better overall performane. For the 60-node nstanes, onfguraton TP* solves 8/30 nstanes to optmalty (as opposed to 6/30 of CP), wth a reduton of 8% n soluton tmes and 3% of the branh-and-bound tree sze. For the 00-node nstanes, although none of the nstanes are solved to optmalty, onfguraton TP* results n a reduton of 8% n the end gaps. Note that for the 00-node nstanes, sne the memory lmt s reahed n many nstanes, hgh run tmes may ndate a better ablty to prune n the branh-and-bound tree. 6. CONCLUSIONS We derved new vald nequaltes for CFNF problems from three-parttons of a networ. The nequaltes share the same sprt as flow over nequaltes, but explot the nternal networ struture that aggregated flow over nequaltes gnore. We mplemented the nequaltes as uttng planes n a branh-and-bound approah usng CPLEX, and ompared the benefts of usng the three-partton flow over nequaltes under dfferent algorthm onfguratons. Aordng to our omputatonal experments, usng threepartton flow over nequaltes results n stronger formulatons, allowng addtonal gap mprovement at the root node. The mprovement often translates to lower end gaps at termnaton and faster soluton tmes. 3 NETWORKS 06 DOI 0.00/net

15 TABLE node nstanes wth CPLEX uts on Cuts Result Cap. Dens. Intal gap. Confg. Gap mpr. User CPLEX End gap Nodes Tme S T M % CP 85.5% 0,6.3% 6,84 7, CP+TP* 88.% % 9,88 6, % CP 86.3% 0,7.0% 68,705 7, CP+TP* 89.% %,76 6, % CP 85.7% 0,5.3% 5,4 7, CP+TP* 89.4% % 95,069 6, % CP 88.% 0,76.4% 8,59 6, CP+TP* 90.3% % 6,90 6, % CP 89.5% 0,3.% 7,460 7, CP+TP* 9.4% % 09,499 6, % CP 89.8% 0,97.0% 7,9 7, CP+TP* 9.5% %,486 6, Average CP 87.5% 0,.% 83,09 7,5 CP+TP* 89.9% % 5,76 6,476 The proof tehnque used n ths artle an n prnple be used to derve vald nequaltes for -parttons, or for the general three-partton polytope wth varyng apates: Fx varables n order to apply nown results for polytopes wth fewer parttons, and then lft the varables assumed to be fxed. Note, however, that omputng the neessary lftng funtons, and fndng sutable superaddtve lower bounds, may beome more dffult as the omplexty of the polytope nreases. APPENDIX: VALIDITY OF TYPE INEQUALITIES Let S be a over, and assume x j = for j S + S+ S and x j = 0 for j (N + \ S + ) N, =,. Under these assumptons, the lfted flow over nequalty for node yelds y(s + S ) ( λ )( x j ) j S { max yj ( λ )x j,0 } j N \S { } yj, λ x j d. j N mn By addng the flow onservaton for node, y(s + ) y(n ) + y(n ) d,weget y(s + S+ ) ( λ )( x j ) j S { } mn ( λ )x j, y j j N \S j N max { 0, yj λ x j } d. () The lftng funton assoated wth smultaneously lftng nequalty () wth varables x j for j S + and pars (y j, x j ) for j (N (N+ \ S+ )) s gven by f (w, z ) = mn d y(s + S+ ) ( λ )( x j ) j S { } { } mn ( λ )x j, y j max 0, yj λ x j j N \S j N s.t. y(s + ) y(n ) + y(n ) d () y(s + ) + y(n ) y(n ) d + w (3) y(s + ) d + λ (4) y(s + ) d + λ λ z (5) y(s ) λ + λ λ 0 y j x j, x j {0, } j N N, where z s a nonnegatve multple of the apaty and stands for the apaty losed on ars n S +, w 0 stands for the flow on ars n N, w < 0 stands for the flow on ars n N + \ S+. Note that ether () or (4) s bndng (otherwse we an nrease y(s + ) and obtan a better soluton), and ether (3) or (5) s bndng. We onsder then the followng four ases, dependng on whh equatons are bndng: (4) and (5): In ths ase, (S ) λ + λ = λ y(n )y(n ) λ λ+w +z (whh mples w +z λ). An optmal soluton exsts by settng y(s ) = λ,n whh ase f (w, z ) =λ + z. (4) and (3): In ths ase, an optmal soluton exsts where y(s + ) s as hgh as possble and y(n ) y(n ) s as low as possble. Therefore, we have y(s ) = λ and y(s + ) = d λ + w (whh mples w + z λ and s only feasble f d λ + w 0). In ths ase f (w, z ) =w. () and (5): In ths ase, y(n )y(n ) λ λ+w + z and y(s + ) d + λ λ + w + z. An optmal soluton exsts where y(s + ) s as hgh as possble and, therefore, we have y(s + ) = d +λ λ+w +z (whh mples w +z NETWORKS 06 DOI 0.00/net 33

16 Inequalty (6) stll assumes x j = for j S + and y j = 0 for j (N + \ S+ ) N. The lftng funton assoated wth smultaneously lftng (6) wth varables x j for j S + and pars (y j, x j ) for j (N + \ S+ ) N s gven by f (z, w ) = mn d y(s + S+ ) ( λ)( x j ) (y j ( λ)x j ) + mn { } y j, λx j FIG. 7. Funtons g λ, (z + w ) and f (z, w ) + w. λ) and y(n ) y(n ) = λ λ + w + z. Note that f z + w 0 then t may be optmal to sequentally open ars n N (N \ S ) or lose ars n S, but for z + w 0 we have f (w, z ) =w. () and (3): Sne y(s + S+ ) = d + w for all values y(n ) y(n ), an optmal soluton exsts where y(n ) y(n ) s as low as possble. Ths value s obtaned when y(s + ) s maxmal, and ths ase redues to ase () and (5). Combnng the dfferent ases, we get the lftng funton λ + z + w f λ w + z 0 f0 w + z λ λ f z + w f (z, w ) =w + + λ, Z + ( ) z w + ( )λ f + λ z + w ( ), Z +. The exat lftng funton f s not superaddtve n R, but we an use the superaddtve vald lftng funton ψ(z, w ) = w +g λ, (z +w ) where g λ, s the superaddtve funton of Proposton 4. Fgure 7 shows g λ, (z + w ) and f (z, w )+w. The proof that ψ s superaddtve s analogous to the proof of Proposton 6. Usng ψ, we get the vald nequalty y(s + S+ ) ( λ)( x j ) (y j ( λ)x j ) + j N + \S+ j N mn { y j, λx j } { } ( λ )( x j ) mn ( λ )x j, y j j S j N \S { } 0, yj λ x j d. (6) j N max j N + \S+ j N max j N { } ( λ )( x j ) + mn ( λ )x j, y j j S j N \S { } 0, yj λ x j s.t. y(s + ) y(n ) + y(n ) d + w y(n + ) y(n ) + y(n ) y(n ) d y(s + ) d + λ z y(s + ) d + λ λ y(s ) λ + λ λ 0 y j x j, x j {0, } j N N N + N, where z s a nonnegatve multple of the apaty and stands for the apaty losed on S +, w 0 stands for y(n ) and w < 0 stands for y(n + \ S+ ). Let γ (a) = mn ( λ )( x j ) j S s.t. { } mn ( λ )x j, y j j N \S { } 0, yj λ x j j N max y(n ) y(n ) = a be the ontrbuton of the ars n N N to the objetve of the lftng funton, gven that the flow on these ars s a. Note that for λ λ a λ + λ λ there exsts an optmal soluton where y(s ) = a, x(n N \ S ) = 0, and γ (a) = 0. If a >λ + λ λ then we need to sequentally open ars n N \ S, and for a <λ λ t s optmal to sequentally open ars n N or lose ars n S. We fnd that ( λ ) f λ λ + a λ λ + + λ, Z γ (a) = a λ + λ λ + ( λ ) f λ λ + + λ a λ λ + ( + ), Z. 34 NETWORKS 06 DOI 0.00/net

17 Moreover let γ (a) = mn y(s+ ) ( λ)( x j ) (y j ( λ)x j ) + s.t. j N + \S+ mn { } y j, λx j j N y(n + ) y(n ) = a be the ontrbuton of the ars n N + N to the objetve of the lftng funton, gven that the flow on these ars s a. Note that for d λ a d λ + λ there exsts an optmal soluton where y(s + ) = a, x(n N+ \S+ ) = 0, and γ (a) =a.ifa > d λ + λ then we need to sequentally open ars n N + \ S+, and for a < d λ t s optmal to sequentally open ars n N or lose ars n S+. Therefore γ (a) =(d λ ) (a d + λ ) λ f d λ + a d λ + + λ, Z + ( + )λ f d λ + + λ a d λ + ( + ), Z. Now n the lftng problem, for a fxed value of y(s + ) = y, an optmal soluton exsts where y(n ) y(n ) s as low as possble (y(n )y(n ) = yd w ) and y(n + )y(n ) s as hgh as possble (y(n + ) y(n ) = d + w y). In ths ase, f (z, w ) = d + mn {y + γ (y d w ) 0 y d +λ z + γ (d + w y)} (λ λ ) d w f λ λ + ( ) y d w λ = d + mn 0 y d +λ z y + + (λ λ ) d + λ f y d w λ + λ λ, where Z. The nner funton s nonnreasng n y and, therefore, the mnmum s attaned at y = d + λ z.we get (λ λ) f z + w λ λ + ( + ) f (z, w ) =w + z + w + (λ λ) f + λ λ z + w, whh s of the form w + g λ λ,(z + w ) and s superaddtve n R. Usng f, we get the three-partton flow over nequaltes y(s + S+ ) + (λ λ)( x j ) (y j (λ λ)x j ) + j N + \S+ mn { } y j, ( (λ λ))x j + ( λ)( x j ) j N j N + \S+ (y j ( λ)x j ) + mn { } y j, λx j j N { } ( λ )( x j ) + max 0, yj λ x j j S j N { } mn yj, ( λ )x j d. (7) j N \S REFERENCES [] Y.K. Agarwal, -partton-based faets of the networ desgn problem, Networs 47 (006), [] A. Atamtür, Flow pa faets of the sngle node fxed-harge flow polytope, Oper Res Lett 9 (00), [3] A. Atamtür and S. Küçüyavuz, Lot szng wth nventory bounds and fxed osts: Polyhedral study and omputaton, Oper Res 53 (005), [4] D. Bensto, S. Chopra, O. Günlü, and C.Y. Tsa, Mnmum ost apaty nstallaton for multommodty networ flows, Math Program 8 (998), [5] M.X. Goemans, Vald nequaltes and separaton for mxed 0- onstrants wth varable upper bounds, Oper Res Lett 8 (989), [6] M. Grötshel, L. Lovász, and A. Shrjver, The ellpsod method and ts onsequenes n ombnatoral optmzaton, Combnatora (98), [7] Z. Gu, G.L. Nemhauser, and M.W. Savelsbergh, Lfted flow over nequaltes for mxed 0- nteger programs, Math Program 85 (999), [8] O. Günlü, A branh-and-ut algorthm for apatated networ desgn problems, Math Program 86 (999), [9] T. Magnant, P. Mrhandan, and R. Vahan, The onvex hull of two ore apatated networ desgn problems, Math Program 60 (993), [0] M.W. Padberg, T.J. Van Roy, and L.A. Wolsey, Vald lnear nequaltes for fxed harge problems, Oper Res 33 (985), [] J. Stallaert, Vald nequaltes and separaton for apatated fxed harge flow problems, Dsr Appl Math 98 (000), [] T.J. Van Roy and L.A. Wolsey, Vald nequaltes and separaton for unapatated fxed harge networs, Oper Res Lett 4 (985), 05. [3] T.J. Van Roy and L.A. Wolsey, Vald nequaltes for mxed 0- programs, Dsr Appl Math 4 (986), [4] T.J. Van Roy and L.A. Wolsey, Solvng mxed nteger programmng problems usng automat reformulaton, Oper Res 35 (987), [5] L.A. Wolsey, Submodularty and vald nequaltes n apatated fxed harge networs, Oper Res Lett 8 (989), 9 4. NETWORKS 06 DOI 0.00/net 35