A poynomay sovabe case of the poong probem Natasha Boand Georga Insttute of Technoogy, Atanta, U.S.A. Thomas Kanowsk Faban Rgternk The Unversty of Newcaste, Austraa arxv:1508.03181v4 [math.oc] 5 Apr 2016 September 24, 2018 Abstract Answerng a queston of Haugand, we show that the poong probem wth one poo and a bounded number of nputs can be soved n poynoma tme by sovng a poynoma number of near programs of poynoma sze. We aso gve an overvew of known compexty resuts and remanng open probems to further characterze the border between (strongy N-hard and poynomay sovabe cases of the poong probem. Keywords oong probem Computatona compexty 1 Introducton, motvaton and probem defnton The poong probem s a nonconvex nonnear programmng probem wth appcatons n the refnng and petrochemca ndustres [9, 16], mnng [5, 7], agrcuture, food manufacturng, and pup and paper producton [18]. Informay, the probem can be stated as foows: gven a set of raw matera suppers (nputs and quates of the matera, fnd a cost-mnmzng way of bendng these raw materas n ntermedate poos and outputs so as to satsfy requrements on the fna output quates. The bendng n poos and outputs ntroduces bnear constrants and makes the probem hard. Whe the poong probem has been known to be hard n practce ever snce ts proposa by Havery n 1978 [15], t was ony formay proven to be strongy N-hard by Afak and Haugand n 2013 [1]. Ther proof of strong N-hardness, however, consdered a very genera case of the probem, wth arbtrary parameters and an arbtrary network structure. Once the parameters and the network structure are more specfc, e.g., by boundng the number of vertces, ther n- and out-degrees, or the number of quates, the compexty of the probem needs to be re-examned. Ths way, severa poynomay sovabe cases of the poong probem were proven [2, 12, 13]. However, the border between (strongy N-hard and poynomay sovabe cases of the poong probem s st ony partay characterzed. Ths s many due to the combnatora exposon of parameter choces for the probem. In ths paper, we sove an open probem that has been ponted out n [12, 13]: the poong probem wth one poo and a bounded number of nputs s n fact poynomay sovabe. We consder a drected graph G = (V,A where V s the set of vertces and A s the set of arcs. V s parttoned nto three subsets I,L,J V: I s the set of nputs, L s the set of poos and J s the set of outputs. Fows are bended n poos and outputs. The poong probem terature addresses a varety of probem nstances wth A (I L (L L (L J (I J. Instances wth A (L L = 1
Tabe 1: Notaton for the poong probem Sets V I L J K A A I A J A out v A n v Set of vertces Set of nputs Set of poos Set of outputs Set of quates Set of arcs Set of nput-to-poo arcs: A I := A (I L Set of poo-to-output arcs: A J := A (L J Set of outgong arcs of v I L Set of ncomng arcs of v L J arameters c a Cost of fow on arc a A λ k Quaty vaue of nput I for quaty k K λ ak λ ak λ k, a A out, I, k K µ jk Upper bound on quaty vaue of output j J for quaty k K C v Upper bound on tota fow through vertex v V Upper bound on fow on arc a A u a Varabes x a Fow on arc a A I y a Fow on arc a A J p k Quaty vaue of poo L for quaty k K p ak p ak p k, a A out, L, k K are referred to as standard poong probems (Ss, and nstances wth A (L L are referred to as generazed poong probems (Gs. Both Ss and Gs can be modeed as bnear programs, whch are speca cases of nonnear programs. Instances wth L = are referred to as bendng probems and can be modeed as near programs. In ths paper (as n [2, 12, 13], we study the compexty of Ss where A (I L (L J,.e., a arcs are ether nput-to-poo or poo-to-output arcs. For notatona smpcty, we denote the set of the former by A I := A (I L and the set of the atter by A J := A (L J. We do not consder nput-to-output arcs snce for every such arc (,j, we can add an auxary poo and repace (,j by an nput-to-poo arc (, and a poo-to-output arc (,j. Throughout ths paper, we use the term poong probem to refer to a S wthout nput-to-output arcs. We consder a set of quates K whose quaty vaues are tracked across the network. We assume near bendng,.e., the quaty vaue of a poo or output for a quaty s the convex combnaton of the ncomng quaty vaues weghted by the ncomng fows as a fracton of the tota ncomng fow. For nputs and poos v I L, we denote the set of outgong arcs of v by A out v, and for poos and outputs v L J, we denote the set of ncomng arcs of v by A n v. Let x a be the fow on nput-to-poo arc a A I, and et y a be the fow on poo-to-output arc a A J. The cost of fow on arc a A (whch may be negatve s gven by c a. The tota fow through vertex v V (resp. the fow on arc a A s bounded above by C v (resp. u a. For every nput I and quaty k K, the quaty vaue of the ncomng raw matera s gven by λ k. Let p k denote the quaty vaue of the bended raw materas n poo L for quaty k K. For every output j J and quaty k K, the upper bound on the quaty vaue of the outgong bend s gven by µ jk. In addton to λ k and p k, t s sometmes more convenent to have arc-based rather than node based quaty parameters and varabes. Snce the quaty of fow on arc (v,w s equa to the bended quaty of the tota fow through vertex v, we have λ k λ ak for a nputs I, ther outgong arcs a A out and quates k K. Anaogousy, we have p k p ak for a poos L, ther outgong arcs a A out and quates k K. Tabe 1 summarses the notaton for the poong probem. We now present the cassca formuaton of the poong probem, commony referred to as the - formuaton [15]. There are numerous aternatve formuatons of the poong probem, ncudng the Q- [4], Q- [17] and HYB-formuatons [3], and most recenty mut-commodty fow formuatons [1, 2, 6]. A formuatons are equvaent n the sense that there s a one-to-one correspondence between a feasbe 2
souton of one formuaton and another, and they a have the same optma objectve vaue. However, the aternatve formuatons often show a better computatona performance than the -formuaton, as studed e.g. n [6]. A recent paper by Gupte et a. [11] gves an exceent overvew of topcs that have been studed n the context of the poong probem. Wthn the scope of ths paper, however, we chose to prove compexty resuts usng the cassca -formuaton. In the -formuaton, a fow (x, y satsfes the foowng constrants: a A n a A out a A n a A n j x a = a A out y a, L, (1 x a C, I, (2 x a C, L, (3 y a C j, j J, (4 x a,y a u a, a A I, A J, resp. (5 Constrant (1 s fow conservaton whch ensures that at every poo, the tota ncomng fow equas the tota outgong fow. (2 (4 are vertex capacty constrants and (5 s an arc capacty constrant. For notatona smpcty, we denote the set of fows by F := {(x,y R AI 0 R AJ 0 : (1 (5 are satsfed}. The -formuaton can now be stated as foows: mn x,y,p a A I c a x a + a A J c a y a s.t. (x,y F, λ ak x a = p k a A n a A n j a A out p ak y a µ jk a A n j y a, L, k K, (6 y a, j J, k K. (7 Equaty (6 s the poo bendng constrant whch ensures that the p varabes track the quaty vaues across the network. Inequaty (7 s the output bendng constrant. We take the requrements that λ ak λ k for a a A out, I and k K, and that p ak p k for a a A out, L and k K, to be mpct n the mode. 2 Known compexty resuts Tabe 2 provdes an overvew of known compexty resuts, and Fgure 1 shows most of these compexty resuts n a tree structure. A of these resuts were formay proven n [2, 10, 12, 13]. When boundng the number of vertces, the cases of one nput or output are poynomay sovabe. Furthermore, the cases of one poo and a bounded number of outputs or quates are poynomay sovabe. If we ony have one poo (and no other restrctons, then the probem remans strongy N-hard. The same hods f we have ony one quaty. The probem remans strongy N-hard f we have one quaty and two nputs or two outputs. Ony f we have one quaty, two nputs and two outputs, then the probem becomes N-hard. The probem aso remans strongy N-hard f the out-degrees of nputs and poos are bounded above by two, or f the n-degrees of poos and outputs are bounded above by two. Fnay, t was shown n 3
[10] that there exsts a poynoma tme agorthm whch guarantees an n-approxmaton (where n s the number of output nodes. The authors of ths paper aso showed that f there exsts a poynoma tme approxmaton agorthm wth guarantee better than n 1 ε for any ε > 0, then N-compete probems have randomzed poynoma tme agorthms. oong probem bounded #vertces K = 1 bounded n-/out-degrees I = 1 I [1, max ] L = 1 J [1,j max ] K [1,k max ] J = 1 I = 2 J = 2 N J = 2 I = 2 N A out v 2, v I L A n v 2, v L J ths paper Fgure 1: Overvew of known compexty resuts n a tree structure. For smpcty, we omt #11 and #14 from Tabe 2. 4
Tabe 2: Overvew of known compexty resuts bounded #vertces bounded n-/out-degrees # I L J K I L j J Compexty Reducton Reference(s 1 1 trva 2 1 MIS 3 1 trva [2], Coroary 1; [12], roposton 1; [13], Theorems 1 2 4 1 X3C see #8, #9 and #11 5 [1, max] 1 ths paper 6 1 [1,j max] [12], roposton 2 7 1 [1,k max] [2], roposton 2; [12], roposton 3 5 8 2 1 X3C [13], Theorem 4 9 2 1 X3C [13], Theorem 5 [12], roposton 5; 10 2 2 1 N B2 [13], Theorem 3 11 mn{ I, J } = 2 1 max{ A n, A out } 6 X3C [13], Coroary 1 12 A out 2 A out 2 MAX 2-SAT 13 A n 2 14 mn{ A n, A out } = 1 A n j 2 MIN 2-SAT [12], roposton 7; [13], Theorem 6 [12], roposton 6; [13], Theorem 7 [10], Coroary 1; [12], roposton 4; [13], roposton 3 = poynoma, N = N-hard, = strongy N-hard, B2 = bn packng wth 2 bns, MAX 2-SAT = maxmum 2-satsfabty, MIN 2-SAT = mnmum 2-satsfabty, MIS = maxma ndependent set, X3C = exact cover by 3-sets
3 The poong probem wth one poo and a bounded number of nputs In ths secton, we consder the poong probem wth m nputs (et I = {v 1,...,v m }, one poo (et L = {}, n outputs (et J = {w 1,...,w n }, q quates (et K = {1,...,q}, the set of nput-to-poo arcs A I = {a 1,...,a m } = {(v 1,,...,(v m,}, and the set of poo-to-output arcs A J = {a m+1,...,a m+n } = {(,w 1,...,(,w n }. We wrte x for the fow on nput-to-poo arc a ( = 1,...,m, y j for the fow on poo-to-output arc a m+j (j = 1,...,n, c for the cost of fow on arc a ( = 1,...,m+n, λ k for the k-th quaty vaue at the ta node of nput-to-poo arc a ( = 1,...,m, and µ jk for the bound on the k-th quaty vaue at the head node of arc a m+j (j = 1,...,n. For a postve nteger N, we use [N] to denote the set {1,2,...,N}. If for some j [n], there exsts a k [q] such that mn{λ k : [m]} > µ jk, then y j = 0 n every feasbe souton. Hence, wthout oss of generaty, we assume Note that y j > 0 mpes j [n] k [q] mn{λ k : [m]} µ jk. (8 k [q] m m λ k x µ jk x. (9 It has been observed, for nstance n [13], that for a fxed set J [n] of outputs, an optma souton that satsfes the quaty constrants for a j J and has y j = 0 for a j [n] \ J, can be found by sovng the foowng near program whch we denote by L(J : mn x,y m c x + c j y j j J s.t. (x,y F, m x = y j, j J (λ k λ mk x (µ jk λ mk (x 1 + +x m, j J, k [q]. Let va(j denote the optma vaue of probem L(J. An optma souton for the poong probem can be obtaned by sovng L(J for every J [n], and choosng one wth mnmum va(j. Beow we argue that f the number m of nputs s fxed, then t s suffcent to consder a poynoma number of subsets J, where the poynoma s of degree n both n and q. 6
Introducng varabes z = x / m =1 x for [], condton (9 can be rewrtten as k [q] (λ k λ mk z µ jk λ mk. (10 The vector z s an eement of the smpex = {z [0,1] : z 1 + +z 1}. For z, we defne the reachabe output set J(z as J(z = {j [n] : (10 s satsfed}. (11 Lemma 1. The objectve vaue for any fow correspondng to z s at east va(j(z. roof. For a fxed z, we can fnd the optma fow by sovng the near program mn x,y m c x + j J(z s.t. (x,y F, m x = j J(z c j y j y j, x = z (x 1 + +x m, [m]. Every feasbe souton for ths probem s aso feasbe for L(J(z and the cam foows. The nequates (10 defne a partton of R (and therefore of nto regons of constant J(z. To be more precse, et H be the hyperpane arrangement H = {H jk : j [n], k [q]}, where { } H jk = z R : (λ k λ mk z = µ jk λ mk. The system H nduces a partton of R. Let Hjk 0 and H1 jk be defned by } Hjk {z 0 = R : (λ k λ mk z µ jk λ mk, } Hjk {z 1 = R : (λ k λ mk z > µ jk λ mk. If, for every vector ε = (ε jk j [n],k [q] {0,1} nq, we defne the set (ε = n q j=1 k=1 H ε jk jk, then the space R s the dsjont unon of the sets (ε, and for every z the set J(z s determned by the vector ε wth z (ε. Lemma 2. For ε {0,1} nq, et J(ε = {j [n] : k [q] ε jk = 0}. Then, for a ε {0,1} nq and for a z (ε, we have J(z = J(ε. 7
roof. Let ε {0,1} nq and z (ε. Then j J(z (11 k [q] z H 0 jk z (ε k [q] ε jk = 0 j J(ε. It s we known that the number of nonempty sets (ε s bounded by a poynoma of degree m n nq (see for exampe [8]. However, drect appcaton of [8] yeds the upper bound ( nq =0, whch s weaker than the bound n the foowng emma. We derve a stronger bound than [8] snce the nq hyperpanes are parttoned nto q subsets of each n parae hyperpanes. ( q Lemma 3. There are at most n vectors ε {0, 1} nq such that (ε. =0 roof. We denote the cam of the emma, parameterzed by the nput cardnaty m and the quaty cardnaty q, by C(m,q, and we prove ths cam by nducton on m and q. Base case and nductve step are as foows: 1. Base case: q,m {1,2,...} : C(1,q, C(2,q and C(m,1 2. Inductve step: q {2,3,...}, m {3,4,...} : C(,q 1 C(m,q 1 = C(m,q For m = 1, note that R 0 = {0} contans ony a snge pont, and snce the sets (ε are dsjont there can be at most 1 = ( q 0 n 0 nonempty sets (ε. In fact, usng assumpton (8, we have (ε ε = 0. For m = 2, the nq nequates partton R 1 nto at most 1+nq = ( q 0 n 0 + ( q 1 n 1 ntervas. For q = 1 and m 3, the n parae hyperpanes H 11,...,H n1 partton R nto at most 1+n = ( 1 0 n 0 + ( 1 1 n 1 parts. Now et q 2, m 3, and assume that C(,q 1 and C(m,q 1 are true. From C(m,q 1 t foows that the system {H jk : j [n], k [q 1]} cuts R nto at most =0 parts. For every j [n], the hyperpane H jq s somorphc to R m 2, and for every j [n], k [q 1], the ntersecton H j k H jq s ether empty or an (m 3-dmensona affne subspace of H jk. Snce the map H j k H j k H jq preserves paraesm, C(,q 1 mpes that the hyperpane H jq s cut by n the system {H j k H jq : j [n], k [q 1]} nto at most m 2 =0 parts. If we start wth the partton of R gven by the system {H jk : j [n], k [q 1]} and add the hyperpanes H 1q, H 2q,..., H nq one by one, then every hyperpane adds at most m 2( q 1 =0 n parts to the partton, and the number of parts nto whch R s cut by H s at most =0 n +n m 2 =0 n = =0 = 0 n 0 + n + n 1 ( + n ( q 1 n = 1 =0 ( q n. Remark 1. Note that the proof of Lemma 3 aso provdes a recursve method to determne the vectors ε wth (ε n poynoma tme. 8
Remark 2. The upper bound gven n Lemma 3 s best possbe,.e., for a m, q and n, there exst nstances n whch the number of vectors ε wth (ε equas ( q =0 n. In fact, ths bound s obtaned by amost a systems H. To make ths statement more precse, we say that a system H of nq hyperpanes H jk n R, consstng of q sets of n parae hyperpanes, s n genera poston f the ntersecton of every set of m of these hyperpanes s empty and t [] (j 1,...,j t [n] t (k 1,...,k t [q] t wth k 1 < k 2 < < k t H j1k 1 H j2k 2 H jtk t s an ( t-dmensona affne subspace of R. The bound n Lemma 3 s obtaned whenever the system H s n genera poston, and ths can be seen by checkng that n ths case a estmates n the nducton proof are tght. For m = 1, we have that (0 = {0}. For m = 2, the system H s a st of nq ponts, and H s n genera poston f these ponts are dstnct, n whch case t parttons R nto 1 + nq parts as requred. For q = 1, the n parae hyperpanes H 11,...,H n1 n genera poston partton R nto exacty 1+n parts. For the nductve step, note that the system of ntersectons {H j k H jq : j [n], k [q 1]} forms a system of hyperpanes n genera poston n H jq, and therefore the nequates n the nductve step are satsfed wth equaty. Theorem 1. For every postve nteger m, the poong probem wth one poo and m nputs can be soved n poynoma tme. More precsey, t can be reduced to sovng at most =0 ( q n near programs wth m+n varabes and m+n(q+1+2 constrants, where q s the number of quates and n s the number of outputs. roof. We cam that the poong probem can be soved by choosng a mnmum cost souton obtaned from sovng the probem L(J(ε for every ε wth (ε, and by Lemma 3 the number of these near programs s bounded as camed. Ceary, B = mn{va(j(ε : (ε } s an upper bound because a souton for L(J(ε s aways feasbe for the poong probem. By Lemma 2, for every z there exsts some ε wth J(z = J(ε, and usng Lemma 1 t foows that B s aso a ower bound. We note that ths resut was obtaned, ndependenty, by Haugand and Hendrx [14]. 4 Remanng open probems To further characterze the compexty of the poong probem, the foowng open probems coud be addressed n the future [12, 13]: 1. For a the cases that can be soved n poynoma tme by reducton to poynomay many near programs of poynoma sze, does there exst a strongy poynoma agorthm,.e., an agorthm that s poynoma n the number of vertces and quates? 2. Is the poong probem wth one quaty and n-degrees at most two poynomay sovabe? 3. Is the poong probem wth one quaty and out-degrees at most two poynomay sovabe? 4. Do poynoma agorthms exst for the poong probem wth two poos and some bounds on the number of nputs, outputs, and quates? 9
Acknowedgements Ths research was supported by the ARC Lnkage Grant no. L110200524, Hunter Vaey Coa Chan Coordnator (hvccc.com.au and Trpe ont Technoogy (tpt.com. We woud ke to thank the two anonymous referees for ther hepfu comments whch mproved the quaty of the paper. References [1] M. Afak and D. Haugand. A mut-commodty fow formuaton for the generazed poong probem. Journa of Goba Optmzaton, 56(3:917 937, 2013. [2] M. Afak and D. Haugand. Strong formuatons for the poong probem. Journa of Goba Optmzaton, 56(3:897 916, 2013. [3] C. Audet, J. Brmberg,. Hansen, S. Le Dgabe, and N. Madenovć. oong robem: Aternate Formuatons and Souton Methods. Management Scence, 50(6:761 776, 2004. [4] A. Ben-Ta, G. Eger, and V. Gershovtz. Goba mnmzaton by reducng the duaty gap. Mathematca rogrammng, 63(1 3:193 212, 1994. [5] N. Boand, T. Kanowsk, and F. Rgternk. Dscrete fow poong probems n coa suppy chans. In T. Weber, M. J. Mchee, and R. S. Anderssen, edtors, MODSIM2015, 21 st Internatona Congress on Modeng and Smuaton, pages 1710 1716, God Coast, Austraa, December 2015. Modeng and Smuaton Socety of Austraa and New Zeaand. [6] N. Boand, T. Kanowsk, and F. Rgternk. New mut-commodty fow formuatons for the poong probem. Journa of Goba Optmzaton, 2016. Advance onne pubcaton, 42 pages. DOI: 10.1007/s10898-016-0404-x. [7] N. Boand, T. Kanowsk, F. Rgternk, and M. Savesbergh. A speca case of the generazed poong probem arsng n the mnng ndustry. Optmzaton Onne e-prnts, Juy 2015. optmzaton-onne:5025. [8] R. C. Buck. artton of Space. The Amercan Mathematca Monthy, 50(9:541 544, 1943. [9] C. W. dewtt, L.S. Lasdon, A. D. Waren, D. A. Brenner, ands.a. Mehem. OMEGA:An Improved Gasone Bendng System for Texaco. Interfaces, 19(1:85 101, 1989. [10] S. S. Dey and A. Gupte. Anayss of MIL Technques for the oong robem. Operatons Research, 63(2:412 427, 2015. [11] A. Gupte, S. Ahmed, S. S. Dey, and M. S. Cheon. Reaxatons and dscretzatons for the poong probem. Journa of Goba Optmzaton, to appear. reprnt: optmzaton-onne:4883. [12] D. Haugand. The hardness of the poong probem. In L. G. Casado, I. García, and E. M. T. Hendrx, edtors, roceedngs of the XII goba optmzaton workshop, mathematca and apped goba optmzaton, MAGO 2014, pages 29 32, Máaga, Span, September 2014. [13] D. Haugand. The computatona compexty of the poong probem. Journa of Goba Optmzaton, 64(2:199 215, 2015. [14] D. Haugand and E. M. T. Hendrx. oong robems wth oynoma-tme Agorthms. Journa of Optmzaton Theory and Appcatons, 2016. Advance onne pubcaton, 25 pages. DOI: 10.1007/s10957-016-0890-5. 10
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