The Minimum Universal Cost Flow in an Infeasible Flow Network

Transcription

1 Journal of Scences, Islamc Republc of Iran 17(2): (2006) Unversty of Tehran, ISSN The Mnmum Unversal Cost Flow n an Infeasble Flow Network H Saleh Fathabad * M Bagheran Department of Mathematcs, Statstcs, Computer Scence, Faculty of Scences, Unversty of Tehran, Tehran, Islamc Republc of Iran Abstract In ths paper the concept of the Mnmum Unversal Cost Flow (MUCF) for an nfeasble flow network s ntroduced A new mathematcal model n whch the objectve functon ncludes the total costs of changng arc capactes sendng flow s bult analyzed A polynomal tme algorthm s presented to fnd the MUCF Keywords: Mnmum cost network flow problem; Infeasble flow network 1 Introducton Let G = (N,A, u,c) be a network wth node set N, arc set A, capacty vector u cost vector c Each components u c of u c represent the arc capacty flow cost of (, j) A, respectvely A mnmum cost flow problem (MCF) on G s defned as: mn c x (1-1a) (, j) A (1-1b) s t : x k = b N 0 x u (, j) A (1-1c) A flow x that satsfes Equatons (1-1b) (1-1c) s called a feasble flow the network s feasble f such a flow exsts Feasblty of a flow network depends on node numbers, b s arc capactes u s Flow network nfeasblty was frst consdered by Hoffman [1] He proved that a flow network s nfeasble f only f there exsts a cut ( S, S ) such that b > u (1-2) S (, j ) ( S, S ) In such a case, S s called solaton by Greenberg [2] a wtness by Aggarwal, Hao Orln [3] Aggarwal et al [3] showed that the problem of fndng a mnmum wtness n an nfeasble flow network s Nphard But rather effcent heurstc procedures have been ntroduced for practcal nstances, Greenberg [2,4,5] Havng a flow network been dagnosed as nfeasble, the next task s to convert t to a feasble one by the least cost For ths purpose McCormck [6] ntroduced the followng model: c (1-3a) mn (, j ) A (1-3b) s t : x k = b N where 0 x u + (, j) A (1-3c) c s the cost of changng the capacty of arc AMS Subject Classfcaton: 90C35, 90B10 * E-mal: saleh@khayamutacr 175

2 Vol 17 No 2 Sprng 2006 Saleh Fathabad Bagheran J Sc I R Iran (, j) A by one unts When the above problem s solved the optmal amount of capacty change a feasble flow n the resultng network would be obtaned Such a feasble flow was called a Least Infeasble Flow (LIF) [6] As t s seen, the objectve functon of McCormck's model conssts of the modfcaton cost only The optmal flow cost has to be computed by solvng the mnmum cost flow problem usng the resulted arc capactes In ths paper we frst construct a comprehensve model that ncludes both modfcaton flow costs Then a polynomal tme algorthm for computng the optmal capactes flow s presented Ths algorthm has two applcatons, e modfyng the arc capactes obtanng the optmal flow n the resultng feasble flow network at the same tme, mnmzng the sum of modfcaton flow costs nstead of the modfcaton cost only In Sectons 2 3 we construct the model obtan the optmalty condtons In Secton 4 a polynomal tme algorthm that fnds the optmal arc capactes flow s ntroduced s verfed 2 Problem Formulaton Analyss Suppose that due to the current arc capactes the flow network G = (N,A) the mnmum cost flow problem defned by (1-1) are nfeasble Thus there exsts a wtness S such that (1-2) s true In order to change the network to a feasble one, arc capactes have to be modfed so that there does not exsts any wtness satsfyng (1-2) In the procedure of computng LIF, arc capactes are changed so as the total changng cost s mnmzed The computed LIF s merely a feasble flow n the resulted network hence ts sendng cost may be non-optmal, when both costs are taken nto account The followng example proves ths clam s 88 If we let x 12 = 1, 12 = 3, x 13 = 1 13 = 0, then c x + ( c + c ) becomes 87 (, j ) A The above example shows that a LIF s not the best soluton of an nfeasble flow network In order to change the nfeasble network G to a feasble one, accordng to (2-1) the current arc capactes have to be ncreased, as much as all wtnesses are vanshed Snce the amount of ncrements must ncur the least possble cost, any feasble soluton of the resultng flow network has a flow amount equal to the new arc capactes In other words f x s a flow satsfyng the current capacty constrants, the amount of change on x, n order to satsfy the conservaton constrants, s equal to the ncreasng amount of u Now let c respectvely denote the number of capacty unts to be added to u the cost of each unt, the mnmum unversal cost flow model s defned as: (2-1a) mn c x + ( c + c ) (, j ) A (, j ) A (2-1b) s: t ( x + ) ( x + ) = b N k k 0 x u (, j) A (2-1c) 0 (, j) A (2-1d) It s obvous that the problem (2-1) s not a mnmum cost flow problem can't be solved by the related algorthms Snce c 0, problem (2-1) may be solved by a mnmum convex cost flow algorthm But we ntroduce a dfferent algorthm, to solve the problem, wthout duplcaton of arcs Frst we obtan the optmalty condtons Example 21 Consder the flow network shown n Fgure (21), where c denotes the flow cost, c denotes the capacty changng cost u denotes the current capacty of arc (, j) A Let be the amount of the ncrement of the capacty of arc (, j) A The LIF that mnmzes c s (, j ) A x 12 = 1, x 23 = 4, x 13 = 1, 13 = 1, 12 = 2, 23 = 0 x 13 = 1, the mnmum cost of the convertng the flow network to a feasble one s 48 The total costs of convertng flow cost, c x + ( c + c ) (, j ) A b 5 (15,18,1) c c u (2,15,1) 1 3 Fgure (1,14,4) 1 176

3 J Sc I R Iran Saleh Fathabad Bagheran Vol 17 No 2 Sprng Optmalty Condtons Let λ denote the dual varables correspondng to (2-1b) (2-1c), respectvely The dual problem of (2-1) s defned as: λ (2-2a) max b u N (, j ) A s t : λ c (, j) A (2-2b) j c + c (, j) A (2-2c) j unrestrcted (2-2d) λ 0 (2-2e) Theorem 21 Suppose c = c + j c = c + c denote the reduced costs of arc (, j) A wth c c + c costs respectvely ( x, ) are respectvely optmal f only f the followng condtons hold: c 0 0 = 1) c < 0 x = u, c = 0 0 (2-3a) 2) c > 0 x = 0, = 0 (2-3b) 3) c = 0 0 x u, = 0 (2-3c) Proof Consderng the problems (2-1) (2-2) the complementary slackens condtons we have: x ( c ) 0 x ( c + + λ = + λ ) = 0 j ( c c ) 0 c + + = = 0 j λ ( u ) = 0 Snce (2-2a) s to be maxmzed the coeffcent of λ n (2-2a) s u, n an optmal soluton, λ should be at the least possble value Thus f x 0 λ c = If c 0 then λ 0 hence we then have x u = Notng c = 0, (2-3a) s obtaned Now f c > 0, Snce λ 0, then c + λ > 0 the condton x ( c + λ ) = 0 mples x = 0, c > 0 mples = 0 If c = 0, then x λ = 0, c > 0 Hence 0 x u = 0 3 Computng the Optmal Soluton In the model (2-1) the flow on an arc may exceed ts current capacty The extra flow s denoted by If an arc has a postve resdual capacty t s no need to ncrease ts capacty In other words whenever arc (,j) s saturated, then embarks to be postve Ths fact can be easly extracted from the complementary slackens condtons (2-3) Sendng unts of extra flow on arc (,j), causes a cost equal to ( c + c ), snce the capacty ncrement cost s c sendng flow cost s c Therefore whenever the flow on an arc reaches ts capacty, the arc cost becomes c + c Now let B = b x = x+, for a pseudo flow x b > 0 0 It s very easy to see that, wthn any feasble flow network t s no need for arc capactes to be greater than B The model (2-1) can now be reformulated as: (3-1a) mn c x + c (, j ) A (, j ) A (3-1b) s t : x k = b N 0 x B (, j) A (3-1c) 0 (, j) A (3-1d) 4 The Mnmum Unversal Cost Flow (MUCF) Algorthm As many mnmum cost flow algorthms, MUCF algorthm uses resdual network for adjustng the arc capactes The resdual network that s used n our algorthm dffers n arc capactes costs from the resdual networks defned so far Frst we descrbe ths network whch we call t exped resdual network As t wll be seen later the algorthm at each teraton generates a pseudo flow x n whch = 0 whle x < u Now defne: { } { } A1 = (, j):0 < x < u = (, j):0< x < u { } { } A2 = (, j): x u = (, j): x = u The exped resdual network, G ( x ) s constructed as follows: 177

4 Vol 17 No 2 Sprng 2006 Saleh Fathabad Bagheran J Sc I R Iran G ( x ) Contans all nodes of G For each arc (, j) A two arcs (,j) (j,) belong to G ( x ) The former one has resdual capacty r = u cost c r The resdual capacty cost of the latter one are = x c = c, respectvely For each arc (, j) A2 there are two arcs (,j) (j,) n G( x ) The resdual capacty cost of (,j) are B c + c, respectvely For arc (j,), f x = u then r = x c = c, otherwse r = x u c = ( c + c ) Now let x be a pseudo flow 0, for A let: e b x x () = + k Defne E = { : e( ) > 0} as the set of excess nodes D = { : e( ) < 0} as the set of defct nodes The algorthm begns wth a pseudo flow x = 0, extra capacty vector = 0 node potentals = 0 At each step t selects a node s E a node t D fnds a shortest path P from s to t n G( x ), usng c c as the length of arcs wth x u > 0, respectvely Then δ = mn{ es ( ), et ( ), mn{ r : (, j) P}} unts of flow s augmented along P n G ( x ) The equvalent operatons n G are as follows: Suppose (, j) P n G ( x ) Two cases are possble: Case I: (, j) G, f (, j) A1, set x = x + δ, otherwse set = + δ Case II: ( j, ) G, f 0 < x < u, set x = x δ, otherwse f = 0, set x = x δ, f > 0, set = δ After updatng x the algorthm updates to -d, where d, denotes the shortest path dstances from s to all other nodes The above steps are repeated untl E /or D are empty network, G ( x ), contans no negatve cost drected cycle Proof In G ( x ) defne A 3 = {( j, ) : (, j) A 1 } A = {( j, ) : (, j) A } For a gven x, suppose that 4 2 G ( x ) contans a negatve cost drected cycle as w The cost of w equals to Cw ( ) = c + c + c (, j ) A1 (, j) A2 + c + ( c + c ) (, j ) A3 (, j) A4 whch s negatve Let δ = mn { r : (, j ) w }, augmentng δ unts of flow along w reduces the objectve functon of problem(3-1) by δ Cw ( ) unts Thus x could not be optmal Conversely suppose that for a gven feasble flow x of (3-1), G ( x ) contans * no negatve cost drected cycle Let x denotes the optmal soluton of problem (3-1) Then accordng to * [7], x s a feasble soluton of problem (3-1) can be decomposed n to at most m drected cycles n G ( x ) The sum of the cost of flows on these cycles accordng to (3-1a) s ( ) ( ) c x + c (, j ) A (, j ) A Snce the cost of all cycles n G ( x ) are nonnegatve we have or ( ) + ( ) 0 c x x c (, j ) A (, j ) A c x + c c x + c (, j ) A (, j ) A (, j ) A (, j) A Accordng to the optmalty of * x we get c x + c c x + c (, j ) A (, j ) A (, j ) A (, j) A 41 Algorthm Verfcaton Lemma 41 A feasble flow x s an optmal soluton of problem (3-1) f only f the exped resdual Hence (3-1) x s also an optmal soluton of problem 178

5 J Sc I R Iran Saleh Fathabad Bagheran Vol 17 No 2 Sprng 2006 Remark 41 A feasble flow for problem (3-1) s optmal f only f, some set of node potentals,, satsfy the followng condtons: c 0, (, j) A A 1 3 (4-1) c 0, (, j) A A n G ( x ) 2 4 Proof Gven a feasble flow x for problem (3-1), suppose that there exsts some node potentals, satsfyng condton (4-1) Therefore for every drected cycle w, c + c 0 (, j ) ( A1 A3) (, j ) ( A2 A4) Thus we have (, j ) ( A1 A3) c + j + c + c + 0 (, j ) ( A2 A4) Snce w s a drected cycle we get: (, j) ( A1 A3) (, j) ( A2 A4) j c + c + c 0 e the cost of cycle w s nonnegatve Thus G ( x ) contans no negatve cost drected cycle Conversely, f contans no negatve cost drected cycle, then the shortest path dstances from node 1 to all other nodes, d, are well defned satsfy the condtons d( j) d( ) + c, (, j) A A 1 3 d( j) d( ) + c + c, (, j) A A If we let = d, then 2 4 c ( d( ) + ( d( j)) = c 0, (, j) A A 1 3 c + c ( d( ) + ( d( j)) = c 0, (, j) A A 2 4 The followng two lemmas are counterparts of Lemmas n [7] are smlarly proved Lemma 42 Suppose that x satsfes (4-1) wth respect to some node potentals Let vector d denotes the shortest path dstances from node s to all other nodes n G ( x ) wth c c as the lengths of arcs (, j) A1 A3 (, j) A2 A4, respectvely Then the followng propertes are vald a) The pseudo flow x also satsfes the optmalty condtons (4-1) wth respect to = d b) c 0, (, j) ( A1 A3) P c 0, (, j) ( A2 A4) P, where P denotes the shortest path from s to all other nodes Lemma 43 Suppose that a pseudo flow x satsfes condton (4-1) we obtan x from x by sendng flow along a shortest path from node s to some other node k n G ( x ), then x also satsfes condton (4-1) Theorem 41 Algorthm MUCF solves problem (2-1) n polynomal tme Proof The algorthm begns wth a pseudo flow ( x, ) satsfyng (2-3) or equvalently a pseudo flow x satsfyng (4-1) wth respect to some node potentals At each step, the algorthm attempts to reduce the nfeasblty of the soluton meanwhle attempts to preserve the optmalty condtons Lemmas (4-2) (4-3) show that the algorthm at each step preserves the optmalty condtons reduces the nfeasblty of the soluton by sendng flow from an excess node to a defct node The algorthm termnates, when an optmal feasble flow s found Let U = max{ u : (, j) A} + B, C = max{ c : (, j) A} S(m,n,C) denotes the tme requred to solve a shortest path problem wth m arcs, n nodes nonnegatve cost whose values are no more than C In each teraton, the algorthm fnds a shortest path from an excess node to a defct node whch takes o(s(m,n,nc)) tme, the number of teratons s bounded by o(nu) Thus the runnng tme of the algorthm s o(nu S(m,n,nC)) By scalng the capacty smlar to [7], the polynomal verson of the algorthm, o(mlog U S(m,n,nC)), s obtaned 179

6 Vol 17 No 2 Sprng 2006 Saleh Fathabad Bagheran J Sc I R Iran 5 Concluson Ths paper showed that the total costs of convertng an nfeasble flow network to a feasble one solvng the obtaned problem may not be mnmzed by a LIF A flow that mnmzes the sum of these costs, MUCF, called Mnmum Unversal Cost Flow, was ntroduced computed by a proposed algorthm of polynomal tme order References 1 Hoffman AJ Some recent applcatons of the theory of lnear nequaltes to extremal combnatoral analyss In: Bellman R Hall M, Jr (Eds), Proceedngs of Symposa n Appled Mathematcs Vol x, combnatoral Analyss (Amercan Mathematcal Socety, Provdence, RI): p (1960) 2 Greenberg HJ Dagnosng nfeasblty n mn-cost network flow problem part I: Dual nfeasblty IMA Journal of Mathematcs n Management, 1: (1987) 3 Aggarwal C, Hao J, Orln JB Dagnosng nfeasbltes n network flow problems MIT Sloan School Workng Paper wp #3696, Cambrdge, MA (1994) 4 Greenberg HJ Dagnosng nfeasblty n mn-cost network flow problems part II: Prmal nfeasblty IMA Journal of Mathematcs Appled n Busness & Industry, 2: (1989) 5 Greenberg HJ Consstency, redundancy mpled equaltes n lnear systems Techncal Report, Mathematcs Department, Unversty of Colorado at Denver, Co (1993) 6 McCormck ST How to compute least nfeasble flows Mathematcal Programmng 78: (1997) 7 Ahuja RK, Magnaat TL, Orln JB Network Flows: Theory, Algorthms Applcaton Prentce Hall, New York, NY (1993) 180