EXTENDED FORMULATIONS FOR PACKING AND PARTITIONING ORBITOPES

Transcription

1 EXTENDED FORMULATIONS FOR PACKING AND PARTITIONING ORBITOPES YURI FAENZA AND VOLKER KAIBEL Abrac. We give compac exended formulaion for he packing and pariioning orbiope (wih repec o he full ymmeric group) decribed and analyzed in [6]. Thee polyope are he convex hull of all 0/1-marice wih lexicographically ored column and a mo, rep. exacly, one 1-enry per row. They are imporan objec for ymmery reducion in cerain ineger program. Uing he exended formulaion, we alo derive a raher imple proof of he fac [6] ha baically hifed-column inequaliie uffice in order o decribe hoe orbiope linearly. 1. Inroducion Exploiaion of ymmerie i crucial for many very difficul ineger programming model. Over he la few year, ignifican progre ha been achieved wih repec o general echnique for dealing wih ymmerie wihin branchand-cu algorihm. Very nice and effecive procedure have been devied, like iomorphim pruning [10, 12, 11, 13] and orbial branching [7, 8]. There ha alo been progre in underanding linear inequaliie o be added o cerain ineger program in order o remove ymmery. Toward hi end, orbiope have been inroduced in [6]. The packing orbiope O p,q and he pariioning orbiope O = p,q are he convex hull of all 0/1-marice of ize p q whoe column are in lexicographically decreaing order having a mo or exacly, repecively, one 1-enry per row. In [6], complee decripion wih linear inequaliie have been derived for hee polyope (ee Thm. 16 and 17 in [6]). Knowledge on orbiope urn ou o be quie ueful in pracical ymmery reducion for cerain ineger programming model. For inance, in a well-known formulaion of he graph pariioning problem (for graph having p node o be pariioned ino q par) he ymmery on he 0/1- variable x ij indicaing wheher node i i pu ino par j of he pariioning he ymmery ariing from permuing he par can be removed by requiring x O = p,q. We refer o [5] and [6] for a more deailed dicuion of he pracical ue of orbiope. Dae: June 2, Thi work ha been uppored by he European Union, FP6, MRTN-CT (ADONET). 1

2 2 FAENZA AND KAIBEL The opic of hi paper are exended formulaion for hee orbiope, i.e., (imple) linear decripion of higher dimenional polyope which can be projeced o O p,q and O = p,q. In fac, uch exended formulaion play imporan role in polyhedral combinaoric and ineger programming in general, becaue raher han olving a linear opimizaion problem over a polyhedron in he original pace, one may olve i over a (hopefully impler decribed) polyhedron of which he fir one i a linear projecion. For inance, he lif-and-projec approach [1] and oher general reformulaion cheme (e.g., he one due o Lováz and Schrijver [9] a well a Adam and Sherali [15]) are baed on exended formulaion. Recen example include work on mixed ineger programming for dual of nework marice [3, 4, 2]. More claical i he general heory on exended formulaion obained from (cerain) dynamic programming algorihm [14]. The reul of hi paper are much in he piri of he laer work. In order o give an overview on he conribuion of hi paper le u fir recall a few fac on orbiope. A no 0/1-marix wih a mo one 1-enry per row and lexicographically (decreaing) ored column ha a one above i main diagonal, we may aume wihou lo of generaliy ha O = p,q O p,q R Ip,q wih I p,q = {(i, j) [p] [q] : i j} (where [n] = {1, 2,..., n}). In fac, O = p,q i he face of O p,q defined by requiring ha all row-um inequaliie x(row i ) 1 for i [p] are aified wih equaliy, where row i = {(i, j) I p,q : j [q]}. The main reul of [6] i a complee decripion of O p,q and O = p,q by mean of linear equaion and inequaliie. Thi yem of conrain (he SCI-yem) coni, nex o nonnegaiviy conrain and row-um inequaliie (or row-um equaion for O = p,q), of he exponenially large cla of hifed column inequaliie (SCI ) ha will be defined a he end of Sec. 2. In [6] i i alo proved ha, up o a few excepion, hee exponenially many SCI define face of he orbiope. The proof given in [6] of he fac ha he SCI-yem compleely decribe hee orbiope i raher lenghy and omewha echnical. Exending over page 18 o 27, i hardly leave (no only) he reader wih a good idea of he reaon for he SCI-yem being ufficien o decribe he orbiope. In conra o hi, he conribuion of he preen work are he following: We provide a quie imple exended formulaion for O p,q (along wih a raher hor proof eablihing hi) and, moreover, we how by ome imple and naural (no echnical) argumen ha he SCI-yem decribe he projecion of he feaible region of ha exended formulaion o he original pace, hu providing a new proof howing ha he SCI-yem decribe O p,q. Thi laer proof i much horer han he original one, and i eem o provide much beer inigh ino he reaon for he SCI-yem o decribe he orbiope. Clearly, a O = p,q i a face of O p,q, he reul for he laer polyope immediaely yield correponding reul for he fir one. However, beide leading o ha impler proof, we believe ha our exended formulaion for O p,q i inereing ielf. I provide a decripion of a quie naural polyope (he orbiope O p,q) by

3 EXTENDED FORMULATIONS FOR PACKING AND PARTITIONING ORBITOPES 3 a yem of conrain in a pace whoe dimenion i roughly wice he original dimenion I p,q wih only linearly (in I p,q ) many nonzero coefficien, while every linear decripion of he orbiope in he original pace require exponenially many inequaliie. Thi may alo urn ou o be compuaionally aracive. The baic idea of our exended formulaion i o aign o each verex of O p,q a direced pah in a cerain acyclic digraph. The addiional variable in our exended formulaion are ued o uiably expre hee pah. The digraph we work wih i e up in Sec. 2, where we alo fix ome noaion and define SCI. In Sec. 3 we hen decribe he exended formulaion for O p,q and O = p,q (Thm. 6 and Cor. 8). The main work i done in Sec. 3.1, where he exended formulaion for O p,q wih addiional variable encoding he pah menioned above i inroduced and proved o define an inegral polyhedron (Thm. 4). From hi i i eay o conclude ha he formulaion indeed define a polyope ha projec down o O p,q (Thm. 6). Boh he exenion of uch reul o he pariioning cae O = p,q (Cor. 8 in Sec. 3.2), and he ranformaion of he yem in order o reduce he number of variable and nonzero coefficien (Thm. 10 in Sec. 3.3) are obained wihou much work. On he way, we alo derive linear (in I p,q ) ime algorihm for opimizing linear objecive funcion over O p,q and O = p,q (Cor. 7 and 9). In Sec. 4 we finally prove ha he projecion of he feaible region defined by he exended formulaion conain he polyope defined by he SCI-yem (Thm. 12), hu providing he new proof of he fac (Thm. 11) ha he laer polyope equal O p,q. We conclude wih a few remark and acknowledgemen in Sec The Seup Le u aume p q 1 hroughou he paper. We define a direced acyclic graph D p,q = (V p,q, A p,q ) wih node e V p,q = I p,q ([p] 0 {0}) {} {} (where [n] 0 = [n] {0} and mean dijoin union). Uing he noaion q(i) = min{i, q}, he e of arc of D p,q i where A p,q = A p,q Ap,q {(, (0, 0))} {((p, j), ) : j [q] 0 }, A p,q = {((i, j), (i + 1, j)) : i [p 1] 0, j [q(i)] 0 } i he e of verical arc ha are denoed by (i, j) = ((i, j), (i + 1, j)), and A p,q = {((i, j), (i + 1, j + 1)) : i [p 1] 0, j [q(i + 1) 1] 0 } i he e of diagonal arc ha are denoed by (i, j) = ((i, j), (i + 1, j + 1)). The crucial propery of D p,q i ha every verex of O p,q induce an --pah in D p,q a indicaed in Fig. 1. Noe ha differen verice may induce he ame pah.

4 4 FAENZA AND KAIBEL Figure 1. The digraph D 8,6 and a verex of O 8,6 along wih i --pah. For a ube W V p,q we ue he following noaion: ou(w ) = {(w, u) A p,q : w W, u W } ou (W ) = {(w, u) A p,q : w W, u W } in(w ) = {(u, w) A p,q : w W, u W } in (W ) = {(u, w) A p,q : w W, u W } in (W ) = {(u, w) Ap,q : w W, u W } For a direced pah Γ in D p,q, we denoe by V(Γ) V p,q he e of node on he pah Γ, S(Γ) V(Γ) he e of node on Γ no enered by Γ via diagonal arc, and T(Γ) V(Γ) he e of node on Γ lef by Γ via diagonal arc (ee Fig. 2). Noe ha S(Γ) alway conain he ar node of Γ, and T(Γ) alway exclude he end node of Γ. Remark 1. For every direced pah Γ in D p,q wih end node (i, j) I i,j, we have in (V(Γ)) = in (S(Γ)) and ou (V(Γ) \ {(i, j)}) = ou (T(Γ)). A ube S I p,q i a hifed column if and only if S = S(Γ) for ome (l, l)-(i 1, j 1)-pah Γ in D p,q wih i [p]\{1}, j [q]\{1}, and l [q]. The aociaed hifed-column inequaliy i x(bar i,j ) x(s), where bar i,j = {(i, l) I p,q : l j}

5 EXTENDED FORMULATIONS FOR PACKING AND PARTITIONING ORBITOPES 5 Figure 2. A pah Γ along wih he e S(Γ) (lef) and T(Γ) (righ). and, a uual, we wrie z(n) = e N z e for ome vecor z R M and a ube N M (ee Fig. 3). 3. Exended formulaion 3.1. The packing cae. Denoe by F p,q R Ap,q he e of all --flow (wihou any capaciy rericion) in D p,q wih flow value one. Clearly, F p,q i an inegral polyope. Since D p,q i acyclic, he verice of F p,q are he incidence vecor of he direced --pah (viewed a ube of arc) in D p,q. For a flow y F p,q and a node (i, j) I p,q, we denoe by y(i, j) = y(in(i, j)) = y(ou(i, j)) he amoun of flow paing node (i, j). For a ube W row i of node in he ame row, y(w ) = w W y(w) i he oal amoun of flow enering W (or, equivalenly, leaving W ). Lemma 2. For a direced (k, l)-(i, j)-pah Γ in D p,q wih (i, j) I p,q he following aemen hold for all y F p,q (ee Fig. 4): (1) If k = l 1 hen (2) If l = 0 hen y(in (S(Γ))) y(ou (T(Γ))) = y(bar i,j ). 1 y(ou (T(Γ))) = y(bar i,j ).

6 6 FAENZA AND KAIBEL Figure 3. Coefficien vecor of wo SCI wih he ame bar bar 8,5. Proof. For boh cae, le W = {(a, b) V p,q \{, } : a a for ome (a, b) V(Γ) bar i,j } be he e of all node (differen from and ) in or above V(Γ) bar i,j. For cae (1), we ar by oberving (1) y(in(w )) = y(ou(w )) (a y F p,q i an --flow). Becaue of (0, 0) W (hu being no adjacen o W ) we have in(w ) = in (V(Γ)), which according o Remark 1 equal in (S(Γ)), yielding (2) y(in(w )) = y(in (S(Γ))). Similarly, we have ou(w ) = ou (V(Γ) \ {(i, j)}) ou(bar i,j ), where Remark 1 give ou (V(Γ) \ {(i, j)}) = ou (T(Γ)). Thu, we obain (3) y(ou(w )) = y(ou (T(Γ))) + y(ou(bar i,j )). Equaion (1), (2), and (3) imply he aemen on cae (1). For cae (2), we exploi he fac ha he --flow y F p,q of value one aifie (4) 1 = y(ou(w {})) y(in(w {})). We have in(w {}) = and ou(w {}) = ou(w ) (due o (0, 0) W in hi cae). From ou(w ) = ou (V(Γ) \ {(i, j)}) ou(bar i,j )

7 EXTENDED FORMULATIONS FOR PACKING AND PARTITIONING ORBITOPES 7 Figure 4. Illuraion of par (1) (lef) and par (2) (righ) of Lemma 2 wih (i, j) = (7, 4). and ou (V(Γ) \ {(i, j)}) = ou (T(Γ)) (ee Remark 1), we hu derive he aemen on cae (2) from (4). For (i, j) I p,q we denoe by col i,j = {(k, j) : j k i} he upper par of he j-h column from (j, j) down o (i, j), including boh node. For he direced pah Γ wih V(Γ) = col i,j, we have S(Γ) = col i,j and T(Γ) =. Thu, par (1) of Lemma 2 implie he following. Remark 3. For all y F p,q and (i, j) I p,q, we have y(in (col i,j )) = y(bar i,j ). The cenral objec of udy of hi paper i he polyope P p,q = {(x, y) R Ip,q R Ap,q : y F p,q and (x, y) aifie (5) and (6) below} wih (5) y (i 1,j 1) x ij for all (i, j) I p,q and (6) x(bar i,j ) y(in (col i,j )) for all (i, j) I p,q (ee Fig. 5). Since F p,q i he e of all --flow of value one, all (x, y) P p,q

8 8 FAENZA AND KAIBEL Figure 5. Coefficien vecor of inequaliie (5) (lef) and (6) (righ). aify 0 y 1, and hu 0 x 1 due o (5) and (6). Furhermore, inequaliie (6) imply he row-um inequaliie x(row i ) 1 for all (x, y) P p,q. Theorem 4. The polyope P p,q i inegral. For he proof of hi heorem, we need he following reul. Lemma 5. Le x 1,..., x n 0 and y 1,..., y n R wih n n x l y l for all 1 j n. l=j l=j For all number α 1,..., α n R and 0 β 1 β 2 β n wih he inequaliy hold. α j β j n α j x j for all 1 j n n β j y j Proof of Lemma 5. We prove he claim by inducion on n. The cae n = 1 i rivial, hu le n 1. Ignoring index 1 and decreaing he remaining α j and β j

9 EXTENDED FORMULATIONS FOR PACKING AND PARTITIONING ORBITOPES 9 by β 1, from he inducion hypohei we obain n n (α j β 1 )x j (β j β 1 )y j. j=2 Due o α 1 β 1 and x 1 0 we have (α 1 β 1 )x 1 0, hu we deduce n n (α j β 1 )x j (β j β 1 )y j. Hence we have n α j x j j=2 n ( n β j y j β 1 x j n y j ), wih a nonpoiive righ-hand ide due o β 1 0 and n x j n y j. Proof of Theorem 4. In order o how ha P p,q i inegral, we how ha for an arbirary objecive funcion vecor c R Ip,q R Ap,q he opimizaion problem (7) max{ c, (x, y) : (x, y) P p,q } ha an opimal oluion wih 0/1-componen. We define wo vecor c (1), c (2) R Ip,q R Ap,q wih he following excepion: c (1) (i 1,j 1) which are zero in all componen = c (i,j) for all (i, j) I p,q c (2) (i 1,j) = max{0, c (i,1),..., c (i,j) } for all (i, j) I p,q We are going o eablih he following wo claim: (1) For each (x, y) P p,q we have c, (x, y) c + c (1) + c (2), (0, y). (2) For each --flow y F p,q {0, 1} Ap,q here i ome x {0, 1} Ip,q wih (x, y) P p,q and c + c (1) + c (2), (0, y) = c, (x, y). Wih hee wo claim, he exience of an inegral opimal oluion o (7) can be eablihed a follow: Le c R Ap,q be he y-par of c + c (1) + c (2). A F p,q i a 0/1-polyope, here i a 0/1-flow y F p,q {0, 1} Ap,q wih c, y = max{ c, y : y F p,q }. Due o c + c (1) + c (2), (0, y) = c, y for all y F p,q, claim (1) implie ha he opimal value of (7) i a mo c, y. On he oher hand, claim (2) enure ha here i ome x {0, 1} Ip,q wih (x, y ) P p,q and c, (x, y ) = c + c (1) + c (2), (0, y ) = c, y. Thu, (x, y ) i an inegral opimal oluion o (7).

10 10 FAENZA AND KAIBEL In order o prove claim (2), le y F p,q {0, 1} Ap,q, i.e., y i he incidence vecor of an --pah in D p,q. For he conrucion of ome x {0, 1} Ip,q a required we ar by iniializing x = 0. For each (i, j) V p,q wih y (i,j) = 1 e x i+1,j+1 = 1. For every (i, j) V p,q wih j 1 and y (i,j) = 1 chooe l [j] wih c i+1,l = max{c (i+1),1,..., c (i+1),j }, and e x i+1,l = 1 if c i+1,l 0. For claim (1) le (x, y) P p,q. Define x R Ip,q x ij = x ij y (i 1,j 1) for all (i, j) I p,q. A (x, y) aifie (5), x 0 hold. Furhermore, we have via (8) c (1), (0, y) = c, (x x, 0). Therefore, i uffice o how (9) c (2), (0, y) c, (x, 0), becaue (8) and (9) yield (10) c + c (1) + c (2), (0, y) = c, (0, y) + c (1), (0, y) + c (2), (0, y) c, (0, y) + c, (x x, 0) + c, (x, 0) = c, (x, y). In order o eablih (9), we prove for every i [p] q(i) c ij x ij q(i 1) c (2) (i 1,j) y (i 1,j), which by ummaion over i [p] yield (9). To ee (10) for ome i [p], oberve ha, for every j [q(i)], we have x (bar i,j ) = x(bar i,j ) y(in (bar i,j )) = x(bar i,j ) y(bar i,j ) + y(in (bar i,j )). Due o Remark 3, and a (x, y) aifie (6), hi implie x (bar i,j ) y(in (bar i,j )). Defining y (i,q(i)) = 0 in cae of q(i 1) < q(i), we hu have q(i) q(i) x il l=j for every j [q(i)]. Seing c (2) (i 1,q(i)) q(i 1) < q(i), we furhermore have l=j y (i 1,l) o he bigge componen of c in cae of 0 c (2) (i 1,1) c (2) (i 1,q(i))

11 EXTENDED FORMULATIONS FOR PACKING AND PARTITIONING ORBITOPES 11 and c ij c (2) ((i 1),j) for all j [q(i)]. Thu we can ue Lemma 5 (wih n = q(i), x j = x ij 0, y j = y (i 1,j), α j = c ij, and β j = c (2) (i 1,j) ) o deduce q(i) q(i) c ij x ij c (2) (i 1,j) y (i 1,j), which yield (10). From Theorem 4 one obain ha P p,q i an exended formulaion for O p,q. Theorem 6. The orbiope O p,q R I p,q i he orhogonal projecion of he polyope P p,q R Ip,q R Ap,q o he pace R Ip,q. Proof. Le x O p,q {0, 1} Ip,q be an arbirary verex of O p,q. The incidence vecor y {0, 1} Ap,q of he unique --pah uing all arc (i 1, j 1) wih (i, j) I p,q and x ij = 1 aifie (x, y) P p,q. Thu, O p,q i conained in he projecion of P p,q. To ee ha vice vera he projecion of P p,q i conained in O p,q, by Theorem 4 i uffice o oberve ha every 0/1-poin (x, y) P p,q i conained in O p,q. Clearly, for uch a poin x ha a mo one one-enry per row (ince he rowum inequaliie are implied by he fac (x, y) P p,q ). Furhermore, if he j-h column of x wa lexicographically larger han he (j 1)- column of x wih i being minimal uch ha x ij = 1 hold, hen one would find ha y(col i,j 1 ) = 0 hold (becaue of (5)), conradicing (6) for (i, j 1). From he proof of Theorem 4, we derive a combinaorial algorihm for he linear opimizaion problem (11) max{ d, x : x O p,q} wih d I p,q. Indeed, wih c = (d, 0) R Ip,q R Ap,q every opimal oluion (x, y ) o (12) max{ c, (x, y) : (x, y) P p,q } yield an opimal oluion x o (11). From he proof of Theorem 4 we know ha we can compue an opimal oluion (x, y ) o (12) by fir compuing he incidence vecor y {0, 1} Ap,q of a longe --pah in he digraph D p,q wih repec o arc lengh given by c (1) + c (2) (which can be done in linear ime ince D p,q i acyclic) and hen eing x {0, 1} Ip,q a decribed in he proof of claim (2) (in he proof of Theorem 4). Corollary 7. Linear opimizaion over O p,q can be olved in ime O(pq).

12 12 FAENZA AND KAIBEL 3.2. The pariioning cae. The previou reul can be eaily exended o he pariioning cae. Since he row-um inequaliie x(row i ) 1 are valid for P p,q, P = p,q = {(x, y) P p,q : x(row i ) = 1 for all i [p]} i a face of P p,q. Clearly, due o Theorem 6 hi face map o he face (ee Sec. 1) {x O p,q : x(row i ) = 1 for all i [p]} = O = p,q of O p,q via he orhogonal projecion ono he x-pace R Ip,q. Corollary 8. P = p,q i an exended formulaion for O = p,q. Suppoe we wan o olve (13) max{ d, x : x O = p,q} for ome d R Ip,q. A all poin x O = p,q aify he row-um equaion x(row i ) = 1 for all i [p], we may add, for each i, an arbirary conan o he objecive funcion coefficien of he variable belonging o row i wihou changing he opimal oluion o (13) (hough, of coure, changing he objecive funcion value of he oluion). Therefore, we may aume ha d ha only poiive componen. Bu hen max{ d, x : x O = p,q} = max{ d, x : x O p,q}, and all opimal oluion o he opimizaion problem over O p,q are poin in O = p,q. Thu we derive he following from Corollary 7. Corollary 9. Linear opimizaion over O = p,q can be olved in ime O(pq) Reducing he number of variable and nonzero elemen. Le u manipulae he defining yem of P p,q in order o decreae he number of variable and nonzero coefficien. Thi may be advanageou for pracical purpoe. I furhermore emphaize he impliciy of he exended formulaion. For he ake of readabiliy, we define bar i,j = and col i,j = whenever j > q(i). Since every y F p,q aifie y (i,j) = y(bar i,j ) y(bar i+1,j+1 ), we deduce from Remark 3 ha for all (x, y) P p,q y (i,j) = y(in (col i,j )) y(in (col i+1,j+1 )) hold for all verical arc (i, j) wih (i, j) I p,q (and i < p) a well a y ((p,j),) = y(in (col p,j )) y(in (col p,j+1 )) for all arc ((p, j), ) wih j [q] 0, where we defined y(in (col p,q+1 )) = 0. Similarly o he derivaion of Remark 3, one furhermore deduce ha every (x, y) P p,q aifie y (i,0) = 1 y(in (col i+1,1 )) for all i [p 1] 0. Finally, every (x, y) P p,q clearly aifie y (,(0,0)) = 1. Therefore, we can eliminae from he yem decribing P p,q all arc variable excep for he one correponding o diagonal arc.

13 EXTENDED FORMULATIONS FOR PACKING AND PARTITIONING ORBITOPES 13 We finally apply he linear ranformaion defined by o R Ip,q R A z ij = x(bar i,j ) and w ij = y(in (col i,j )) for all (i, j) I p,q, p,q, whoe invere i given by x ij = z i,j z i,j+1 for all (i, j) I p,q. (defining z i,q(i)+1 = 0, for all i [p]) and y (i,j) = w i+1,j+1 w i,j+1 for all i [p 1] 0, j [q(i + 1) 1] 0. Few calculaion are needed o check ha he previou ranformaion (bijecively) map P p,q ono he polyope P comp p,q R Ip,q R Ip,q defined by he following very compac e of conrain: (14) (15) (16) (17) (18) (19) w i+1,j+1 w i,j+1 0 for i [p 1] 0, j [q(i + 1) 1] 0 w i,j w i+1,j+1 0 for (i, j) I p,q, i < p w p,1 1 w i,j w i 1,j z ij + z i,j+1 0 for (i, j) I p,q z i,j w i,j 0 for (i, j) I p,q w i,q(i) 0 for i [p] Here, (14) repreen he nonnegaiviy conrain on he diagonal arc. Nonnegaiviy on he verical arc (i, j) wih i [p 1] 0 i refleced by (15) for j [q(i)] and by (16) (ogeher wih he nonnegaiviy of w, which i implied by (19) and (15)) for j = 0. Finally, equaion (5) and (6) ranlae o (17) and (18), repecively. Ignoring he nonnegaiviy conrain, yem (14) (19) ha le han 2pq variable and 4pq conrain, for a oal number of nonzero coefficien ha i maller han 10pq. Noe ha w 1,1 1 i a valid inequaliy for P comp p,q. The face of P comp p,q defined by w 1,1 = 1 i he image of he face P = p,q of P p,q. Thu, adding w 1,1 = 1 o he yem (14) (19) one arrive a anoher exended formulaion of O = p,q. We ummarize he reul of hi ubecion. Theorem 10. The polyope P comp p,q R Ip,q R Ip,q defined by (14) (19) i an exended formulaion of O p,q. The face of P comp p,q defined by w 1,1 = 1 i an exended formulaion of O = p,q. 4. The projecion Le Q p,q R Ip,q be he polyope defined by he nonnegaiviy conrain x 0, he row-um inequaliie x(row i ) 1 for all i [p] and all hifed-column inequaliie. By checking he verice (0/1-vecor) of O p,q i i eay o ee ha O p,q Q p,q hold. Thu, in order o prove Theorem 11. O p,q = Q p,q

14 14 FAENZA AND KAIBEL (which i Prop. 13 in [6]) i uffice (due o Theorem 6) o how he following: Theorem 12. For each x Q p,q here i ome y F p,q wih (x, y) P p,q. Proof. For x Q p,q conider he nework D p,q wih capaciy x ij on he diagonal arc (i 1, j 1) for each (i, j) I p,q and infinie capaciie on all oher arc. In hi nework, we conruc a feaible flow y F p,q of value one wih he propery (20) y (i 1,j 1) > 0 y (i 1,j 1) = x ij for all (i, j) I p,q. Phraed verbally, he flow y ue a verical arc only if he diagonal arc emanaing from i ail i auraed. Such a flow can eaily be conruced in he following way: ar by ending one uni of flow from o along he verical pah in column zero. A each ep, if he flow y conruced o far violae (20) for ome (i, j) I p,q, chooe uch a pair (i, j) wih minimal j, breaking ie by chooing i minimally a well. Wih ϑ = min{y (i 1,j 1), x ij y (i 1,j 1) } (i.e., he minimum of he flow on he verical arc and he reidual capaciy on he diagonal arc aring a (i 1, j 1)) reroue ϑ uni of he flow currenly ravelling on he verical arc (i 1, j 1) along he pah aring wih he diagonal arc (i 1, j 1) and hen uing he verical arc in column j. Noe ha hi affec only arc leaving node (k, l) wih k i and l j. Afer hi rerouing, (20) hold for (i, j). The minimaliy requiremen in he choice of (i, j) enure ha he flow on he wo arc leaving (i, j) i no changed again aferward. Thu, (20) will alway be aified for (i, j) in he fuure. Therefore, he procedure evenually end wih a flow a required. A (x, y) aifie (5) by conrucion, i uffice o how (6) in order o prove (x, y) P p,q. To hi end, le (i, j) I p,q. Due o Remark 3, we only need o prove (21) y(bar i,j ) x(bar i,j ). We conruc a direced (k, l)-(i, j)-pah Γ in he reidual nework wih repec o he flow y (conaining only hoe arc of D p,q ha are no auraed by y) wih k = l or l = 0 in he following way: Saring from he rivial (lengh zero) w- (i, j)-pah wih w = (i, j), in each ep we exend he pah a i curren ar node w by he diagonal arc enering w if hi arc i par of he reidual nework, and by he verical arc enering w oherwie. A he reidual nework conain all verical arc, we clearly can proceed hi way unil he ar node of he curren pah i ome node (k, l) wih k = l or wih l = 0. Since Γ i a pah in he reidual nework and due o (20), we have (22) y(ou (T(Γ))) = 0.

15 EXTENDED FORMULATIONS FOR PACKING AND PARTITIONING ORBITOPES 15 If l = 0, par (2) of Lemma 2 ogeher wih (22) yield y(bar i,j ) = 1, from which (21) follow ince x aifie he row-um inequaliy x(row i ) 1 and he nonnegaiviy conrain. If l 0, hen k = l 1. Thu, according o par (1) of Lemma 2 and due o (22), we have (23) y(in (S(Γ))) = y(bar i,j ). Since we preferred diagonal arc from he reidual nework in our backward conrucion of Γ, we find ha all arc from in (S(Γ)) are auraed by he flow y. Therefore, for he hifed column S = S(Γ), we have (uing (23)) (24) x(s) = y(in (S(Γ)) = y(bar i,j ). Le a A p,q be he arc in Γ enering (i, j), and denoe by Γ he pah ariing from Γ by removing a. If a i diagonal, hen he (l, l)-(i 1, j 1)-pah Γ aifie S(Γ ) = S, and he hifed-column inequaliy x(bar i,j ) x(s) (aified by x) eablihe (21) via (24). If a i verical, by conrucion of Γ, we have y (i 1,j 1) = x ij. Furhermore, he (l, l)-(i 1, j)-pah Γ aifie S(Γ ) = S \{(i, j}). Thu uing he hifed-column inequaliy x(bar i,j+1 ) x(s \ {(i, j)}) one obain from (24) he inequaliy y(bar i,j ) = x(s) = x(s \ {(i, j)}) + x ij x(bar i,j+1 ) + x ij = x(bar i,j ). Thu, (21) i eablihed alo in hi cae, which finally prove Theorem Remark In our view, he exended formulaion for he orbiope O p,q and O = p,q preened in hi paper once more demonrae he power ha lie in he concep of exended formulaion. No only do he exended formulaion provide a very compac way of decribing he orbiope, bu alo do hey allow o derive raher imple proof of he fac ha nonnegaiviy conrain, row-um inequaliie/equaion, and SCI uffice in order o linearly decribe O p,q and O = p,q. To u i eem ha hee proof beer reveal he reaon why SCI are neceary and (baically) ufficien in hee decripion. The conrucion of he flow in he proof of Thm. 12 i quie naural. The re of he proof (i.e., he backward conrucion of he pah Γ) one may alo have done wihou knowing he SCI in advance. Thu, knowing he exended formulaion, one poibly could alo have deeced SCI on he way rying o do hi proof. An inereing pracical queion i wheher he very pare and compac exended formulaion for orbiope lead o performance gain in branch-and-cu algorihm compared o verion ha dynamically add SCI via he linear ime eparaion algorihm (decribed in [6]). Acknowledgemen. We would like o hank Laura Sanià for ueful dicuion and Marc Pfech for valuable commen on an earlier verion of hi paper.

16 16 FAENZA AND KAIBEL Reference 1. Egon Bala, Sebaián Ceria, and Gérard Cornuéjol, A lif-and-projec cuing plane algorihm for mixed 0-1 program, Mah. Programming 58 (1993), no. 3, Ser. A, Michele Confori, Marco Di Summa, Friedrich Eienbrand, and Laurence A. Woley, Nework formulaion of mixed ineger program, CORE Dicuion Paper 2006/ Michele Confori, Marco Di Summa, and Laurence A. Woley, The mixing e wih flow, SIAM J. Dicree Mah. 21 (2007), no. 2, Michele Confori, Ber Gerard, and Giacomo Zambelli, Mixed-ineger verex cover on biparie graph, Proceeding of IPCO XII (Maeo Fichei and David Williamon, ed.), LNCS, vol. 4513, Springer-Verlag, 2007, pp Volker Kaibel, Mahia Peinhard, and Marc E. Pfech, Orbiopal fixing, Proceeding of IPCO XII (Maeo Fichei and David Williamon, ed.), LNCS, vol. 4513, Springer- Verlag, 2007, pp Volker Kaibel and Marc E. Pfech, Packing and pariioning orbiope, Mah. Programming, Ser. A 114 (2008), no. 1, Jeff Linderoh, Jame Orowki, Fabrizio Roi, and Sefano Smriglio, Orbial branching, Proceeding of IPCO XII (Maeo Fichei and David Williamon, ed.), LNCS, vol. 4513, Springer-Verlag, 2007, pp , Conrain orbial branching, Proceeding of IPCO XIII (Andrea Lodi and Giovanni Rinaldi, ed.), LNCS, Springer-Verlag, 2008, o appear. 9. L. Lováz and A. Schrijver, Cone of marice and e-funcion and 0-1 opimizaion, SIAM J. Opim. 1 (1991), no. 2, Franćoi Margo, Pruning by iomorphim in branch-and-cu, Mah. Program. 94 (2002), no. 1, , Exploiing orbi in ymmeric ILP, Mah. Program. 98 (2003), no. 1 3, , Small covering deign by branch-and-cu, Mah. Program. 94 (2003), no. 2 3, , Symmeric ILP: Coloring and mall ineger, Dicree Op. 4 (2007), no. 1, R. Kipp Marin, Ronald L. Rardin, and Brian A. Campbell, Polyhedral characerizaion of dicree dynamic programming, Oper. Re. 38 (1990), no. 1, Hanif D. Sherali and Warren P. Adam, A hierarchy of relaxaion and convex hull characerizaion for mixed-ineger zero-one programming problem, Dicree Appl. Mah. 52 (1994), no. 1, (Faenza) Diparimeno di Ingegneria dell Imprea, Univerià di Roma Tor Vergaa, Rome, Ialy addre, Faenza: faenza@dip.uniroma2.i (Kaibel) Oo-von-Guericke Univeriä Magdeburg, Fakulä für Mahemaik, Univeriäplaz 2, Magdeburg, Germany addre, Kaibel: kaibel@ovgu.de