Faster and Simpler Algorithms for Multicommodity Flow and other Fractional Packing Problems

Transcription

1 Faser and Simpler Algorihms for Muliommodiy Flow and oher Fraional Paking Problems aveen Garg Compuer Siene and Engineering Indian Insiue of Tehnology, ew Delhi, India Johen Könemann GSIA, Carnegie Mellon Universiy Pisburgh, PA Absra This paper onsiders he problem of designing fas, approximae, ombinaorial algorihms for muliommodiy flows and oher fraional paking problems We provide a differen approah o hese problems whih yields faser and muh simpler algorihms Our approah also allows us o subsiue shores pah ompuaions for minos flow ompuaions in ompuing maximum onurren flow and minos muliommodiy flow; his yields muh faser algorihms when he number of ommodiies is large 1 Inroduion Consider he problem of ompuing a maximum flow in a graph wih uni edge apaiies While here are many differen algorihms known for his problem we disuss one whih views he problem purely as one of paking pahs so ha onsrains imposed by edgeapaiies are no violaed The algorihm assoiaes a lengh wih eah edge and a any sep i roues a uni flow along he shores pah I hen muliplies he lengh of every edge on his pah by for a fixed Thus he longer an edge is he more is he flow hrough i Sine we always hoose he shores pah o roue flow along, we essenially ry o balane he flow on all edges in he graph One an argue ha, if, afer suffiienly many seps, is he maximum flow hrough an edge, hen he flow ompued is almos imes he maximum flow Therefore saling he flow by gives a feasible flow whih is almos maximum oe ha he lengh of an edge a any sep is exponenial in he oal flow going hrough he edge Suh a lengh funion was firs proposed by Shahrokhi and Maula [13] who Suppored by he EU ESPRIT LTR Proje 2244 (ALCOMIT) Work done while he auhor was a he MaxPlankInsiu für Informaik, Im Sadwald, Saarbrüken, Germany Work done while he auhor was a he Universiä des Saarlandes, Im Sadwald, Saarbrüken, Germany used i o ompue he hroughpu of a given muliommodiy flow insane While his problem (and all oher problems onsidered in his paper) an be formulaed as a linear program and solved o opimaliy using fas marix mulipliaion [16], in [13], people were mainly ineresed in providing fas, possibly approximae, ombinaorial algorihms Their proedure, whih applied only o he ase of uniform edge apaiies, ompued a approximaion o he maximum hroughpu in ime polynomial in The key idea of heir proedure, whih was adoped in a lo of subsequen work, was o ompue an iniial flow by disregarding edge apaiies and hen o reroue his, ieraively, along shor pahs so as o redue he maximum ongesion on any edge The running ime of [13] was improved signifianly by Klein eal [9] I was hen exended and refined o he ase of arbirary edge apaiies by Leighon eal [1], Goldberg [4] and Radzik [12] o obain beer running imes; see Table 1 for he urren bes bound Plokin, Shmoys and Tardos [11] and Grigoriadis and Khahiyan [7] observed ha a similar ehnique ould be applied o solve any fraional paking or overing problem Their approah, for paking problems, sars wih an infeasible soluion The amoun by whih a paking onsrain is violaed is apured by a variable whih is exponenial in he exen of his violaion A any sep he paking is modified by a fixed amoun in a direion deermined by hese variables Hene, he running ime of he proedure depends upon he maximum exen o whih any onsrain ould be violaed; his is referred o as he widh of he problem [11] The running ime of heir algorihm for paking problems being only pseudopolynomial, [11] sugges differen ways of reduing he widh of he problem In a signifian deparure from his line of researh and moivaed by ideas from randomized rounding, Young [17] proposed an oblivious rounding approah o paking problems Young s approah has he essenial ingredien of previous approahes a lengh funion whih measures, and is exponenial in, he exen o whih eah onsrain is vio 1

2 b laed by a given soluion However, [17] builds he soluion from srah and a eah sep adds o he paking a variable whih violaes only suh paking onsrains ha are no already oo violaed In pariular, for muliommodiy flow, i implies a proedure whih does no involve rerouing flow (he flow is only saled a he end) and whih for he ase of maximum flow redues o he algorihm disussed a he beginning of his seion Our Conribuions In his paper we provide a unified framework for a hos of muliommodiy flow and paking problems whih yields signifianly simpler and faser algorihms han previously known Our approah is similar o Young s approah for paking problems However, we develop a new and simple ombinaorial analysis whih has he added flexibiliy ha i allows us o make he greaes possible advane a eah sep Thus for he seing of maximum flows wih inegral edge apaiies, Young s proedure roues a uni flow a eah sep while our proedure would roue enough flow so as o saurae he minimum apaiy edge on he shores pah This simple modifiaion is surprisingly powerful and delivers beer running imes and simpler proofs In pariular, i les us argue ha he onribuion of a onsrain o he running ime of he proedure anno exeed a erain bound whih is independen of he widh This yields a new and sraighforward sronglypolynomial ombinaorial approximaion algorihm for he fraional paking problem (Seion 3) The earlier algorihm for his problem, due o Grigoriadis and Khahiyan [5] redued he problem o wo resoure sharing problems Our approah yields a new, very naural, algorihm for maximum onurren flow (Seion 5) whih exends in a sraighforward manner o minos muliommodiy flows (Seion 6) Boh hese algorihms use a minos flow ompuaion as a subrouine as do all earlier algorihms Conradiing popular belief ha using minos flow as a subrouine is beer, we provide algorihms for hese wo problems whih use shores pah ompuaions as a subrouine and are faser han previous algorihms by a leas a faor where are he number of ommodiies, edges and veries respeively Table 1 summarizes our resuls and "$# are he imes o ompue singlesoure shores pahs and singleommodiy minos flow in a graph wih posiive edge lenghs and oss while % " is he ime aken for eah all o an orale as in [11] All our algorihms are deerminisi and ompue a approximaion o he opimum soluion For breviy we define & ')( *, # / where ;: < * and & < '=( *> # $? ? 7@? 8 where A: < B * oe ha in he running ime of our algorihms for onurren C flow problems we an replae & < 2143 by 213 < 2143 using a rik from earlier papers; we remark on his in Seion 5 2 Maximum muliommodiy flow Given a graph D * /E F wih edge apaiies GIH FKJ LM5 and pairs of erminals O 6, wih one ommodiy assoiaed wih eah pair, we wan o find a muliommodiy flow suh ha he sum of he flows of all ommodiies is maximized The dual of he maximum muliommodiy flow problem is an assignmen of lenghs PQH FKJ he edges suh ha S P '=( *UTWV # G $X LR5 o is minimized This is subje o he onsrain ha he shores pah beween any pair ) 6 under he lengh funion P, whih we denoe by dis P, is a leas one Le Y P '=( * # 2 dis P be he minimum lengh pah beween any pair of erminals Then he dual problem is equivalen o finding a lengh is minimized Le funion PIH FZJ L5 suh ha []\_^a '=( *M 2 # \a^a ^ S P ed Y P The algorihm proeeds in ieraions Le Pf be he lengh funion a he beginning of he g$hi mlolnl ieraion and jk be he oal flow roued in ieraions g : Le o be a pah of lengh Y P beween a pair of erminals and le G be he apaiy of he minimum apaiy edge on o In he g hi ieraion we roue G unis of flow along o Thus j * j G The funion P differs from P only in he lenghs of he edges along o ; hese are modified as * P2 $X G d where is a onsan o be hosen laer Iniially every edge X has lengh p, ie, P@q$X * p for some onsan p o be hosen laer For breviy we denoe Y P2 S P2 by Y S respeively The proedure sops afer ieraions where is he smalles number suh sr ha Y 21 Analysis For every ieraion g r S $g * V P $X * V G $X Vuv G * : S : jo Y : whih implies ha S $g * S $w x : j x : Y 2z (1) 2

3 Problem Previous Bes Our running ime Improvemen Max muliomm O( B < 2143 ) [14] & flow Fraional Paking [5] & C % " B Spreading meris f [3] & C Maximum $ < "$# 13 & < "$# In onsans onurren flow [12, 1] 2143 & < For r d C Max < & < "$# 2143 onurren flow "$# ) [6] 2143 & < For r d Table 1 A summary of our resuls Consider he lengh funion P : Pq oe ha S P : Pq * S O: S $w and YfP : Pq mr Y O: p where is he maximum number of edges on pah along whih flow is roued Hene S P2 : P q YfP : P q Subsiuing his bound on S $g : ge S $g s: Y $g : p in equaion ( 1) we Y p xny $j x : j x : Y 2z oe ha for any g, Y $g is maximum when for all z w z : g, Y 2z is he larges possible Hene Y$g Y : jo 6d Y : X \ Sine p his implies Y p mx By our sopping ondiion and hene Y j p mx fp Claim 21 There is a feasible flow of value (2) Proof: Consider an edge X For every G $X unis of flow roued hrough X he lengh of X inreases by a faor of a leas The las ime is lengh was inreased, X was on a pah of lengh srily less han 1 Sine every inrease in edgelengh is by a faor of a mos, P $X Sine P q $X * p i follows ha he oal flow hrough X is a mos G $X Saling he flow, j, by hen gives a feasible flow of laimed value Thus he raio of he values of he opimum dual and he By subsiuing he d bound on j from (2) we obain * 657 fp fp The raio \ /57 A: equals for \ p * Hene wih his hoie of p we have A: A: s: < d primal soluions,, is A: < Sine his quaniy should be no more han our approximaion raio we hoose appropriaely 22 Running ime In he g hi ieraion we inrease he lengh of he minimum apaiy edge along o by a faor of Sine for any edge X, P * p and P $X " he number of ieraions in whih X is he minimum apaiy edge on he pah hosen in ha ieraion is a mos 2143 /57 8 Using he fa ha here are edges we ge he following heorem : Theorem 21 There is an algorihm ha ompues a < approximaion o he maximum muliommodiy flow in ime where is he maximum number of edges on a pah beween any souresink pair and f is he ime required o ompue he shores pah in a graph wih nonnegaive edgeweighs 3 Paking LP A paking LP is a linear program of he kind $#&%(' G)*, /* 1 * r w32 where and G are 54 3

4 ^ \ \ 4 and 4 maries all of whose enries are posiive We also assume ha for all g z, he $g z hfi enry of, z, is a mos The dual of his LP is M 2/' ), ) r G r w32 We view he rows of as edges and he olumns as pahs is he apaiy of edge g and every uni of flow roued along he z hi olumn onsumes z unis of apaiy of edge g while providing a benefi of G z unis ) Then he dual program is equiva The dual variable ables as lengh 2z '=( * # T $g z $g ed G z lengh is minimum; define Y '=( * # x lengh z Also define S '=( * # len o finding a variable assignmen suh ha S 6d Y is minimized orresponds o he lengh of edge g Define he lengh of a olumnz wih respe o he dual vari Finding a shores pah now orresponds o finding a olumn whose One again our proedure will be ieraive Le be he dual variables and j he value of he primal soluion a he beginning of he hi ieraion Le be he minimum lengh olumn of ie, Y * lengh his orresponds o he pah along whih we roue flow in his ieraion The minimum apaiy edge is he row for whih 6d $g is minimum; le his be row Thus in his ieraion we will inrease he primal variable * by an ed amoun so ha j * j G 6d The dual variables are modified as ed * where is a onsan o be hosen laer $g ed The iniial values of he dual variables are given by q $g * p d $g, for some onsan p o be hosen laer For breviy we denoe Y S by S $ respeively Thus * p The proedure sops a he firs ieraion suh ha S r 31 Analysis The analysis here proeeds almos exaly as in he ase of maximum muliommodiy flow For every ieraion r * * $g * : j Y $ : whih, as before, implies ha * y $j ^ : j ^ YfP : 4 Le ')( *M 2 # S 6d Y Then S P : 6d Y P : and so p whih implies ha By our sopping ondiion and hene ^ ^ p4x S j p4x : j ^ p S fp : Claim 31 There is a feasible soluion o he paking LP of value Proof: The primal soluion * we onsrued has value j& However, i may no be feasible sine some paking onsrain T x $g z * z 6d may be violaed When we pik olumn and inrease * 6d by we inrease he lefhandside (LHS) of he g hi onsrain by \ \ (* say) Simulaneously we inrease he dual variable by a mulipliaive faor of By our definiion of i follows ha and hene inreasing he LHS of he g hfi onsrain by 1 auses an inrease in $g by a mulipliaive faor of a leas oe ha Od $g and so 6d Sine q$g * p d $g i follows ha he final value of he LHS of he g hi onsrain is no more han Sine his is rue for every g, saling he primal soluion by gives a feasible soluion of value as in he laim The res of he analysis is exaly he same as in Seion 21 wih replaing Thus p * 32 Running ime In he hfi ieraion we inrease he dual variable of he minimum apaiy row by a faor of Sine for any row g, q * p d $g and $g 6d and here are rows in all, he oal number of ieraions is a mos Theorem 31 There is an algorihm ha ompues a : 7< approximaion o he Paking LP in ime 2143 /57 8 % " where is he number of rows and % " is he ime required o ompue he minimum lengh olumn

5 ) T 4 Spreading meris,, :ed Given a graph D * E F wih edge oss GH F J L 5, a spreading meri is an assignmen of lenghs o he edges, PH F J LM5, so as o minimize T V subje o he onsrain ha for any se E and verex, T u dis P r j where dis fp is he disane from o under he lengh funion P and j is a funion only of he size of For he linear arrange *,, : men problem j he problem of ompuing a separaor j,, :, E, [3] [2] while for is defined as Sine he lengh funion P is posiive, he shores pahs from o he oher veries in forms a ree he shores pah ree rooed a Thus he above onsrains an be equivalenly saed as: for any ree, for any subse of veries in and for any verex u dis P Ar j where dis P denoes he disane from o in ree under he lengh funion P Le V be he number of veries of in he subree below edge X when is rooed a Then he above onsrain an be rewrien again o obain he LP T V G $X minimize subje o T V u ) V f r j The primal program, whih is a paking LP, has a nonnegaive variable * for every ree, subse and verex and is as follows maximize ) *f j subje o X F T Vu * V f ) oe ha he paking LP has exponenially many variables However, he approximaion o he opimum fraional soluion, in he previous Seion, only needed an orale ha reurned he mos violaed onsrain of he dual LP In his seing, his orale is a subrouine, whih, given a lengh funion P finds a riple f for whih TWVu ) V 6d j T u dis fp 6d j, is minimum, or equivalenly Our subrouine will ry ou all hoies for verex and for eah of hese i will deermine he bes hoie of For a given and every subse, he expression a minimum os se of edges whose removal disonnes he graph ino onneed omponens eah of whih has a mos veries T u dis P is minimized when is he ree of shores pahs from and under he lengh funion P Therefore, for a given, our hoie of will be he shores pah ree rooed a Sine j depends only on,,, given ha,, *, he raio T u dis P 6d j is minimized when is he se of neares veries o Amongs he differen hoies for, and hene for, we hoose he se for whih he above raio is minimum The subrouine hus requires singlesoure shores pah ompuaions The running ime of he proedure is obained by noing ha he subrouine is invoked one in eah of he ieraions Theorem 41 There is an algorihm ha ompues a : < approximaion o Spreading meris in ime 2143 /57 8n where is he ime required o ompue singlesoure shores pahs in a graph wih nonnegaive edgeweighs 5 Maximum onurren flow One again we are given a graph wih edge apaiies F J L5 G H and ommodiies wih n 6 being he soure, sink for ommodiy g ow eah ommodiy has a demand assoiaed wih i and we wan o find he larges " suh ha here is a muliommodiy flow whih roues " $g unis of ommodiy g Le min osx P be he minimum os of shipping 2z unis of flow from x o x where P $X is he os of shipping one uni of flow along edge X and he oal flow hrough X is a mos Furher le Y P '=( * # T xny min osx fp The dual problem now is an assignmen of lenghs o he edges, P H F J L5, suh ha S fp 6d Y P is minimized r Le be his minimum For now we assume ha and shall remove his assumpion laer The algorihm now proeeds in phases; eah phase is omposed of ieraions Consider he z hi ieraion of he g hi phase and le P x be he lengh funion before his ieraion In his ieraion we roue 2z unis of ommodiy z along he pahs given by min osx P x Le j be he flow hrough edge X The lengh funion is modified as P2 x $X * 6d Then S * V P G * S V P j * S g P 5 * P X P * p d G * P q fp x $X P x x $X $# P2x min osx fp2 x The lenghs a he sar of he hi phase are he same as ha a he end of he hi phase, ie, q Iniially, for any edge, 5

6 * V \ V \ * 51 The Analysis We shall be ineresed in he values of he funions S Y only for he lengh funions P g r w For by S Y respe breviy we denoe S fp Y P2 ively Wih hese new noaions we have for g r S $g * S P * S fp q xny min osx P x Sine he edgelenghs are monoonially inreasing min osx fp x min osx P and hene S S fp q Sine [ b \ \ xny r we have min osx fp S S $g : A: d Sine * p we have for g r S $g A: d p A: d p A: d p X A: p X * S $g : Y $g : r where he las inequaliy uses our assumpion ha The proedure sops a he firs phase for whih S Ar Therefore, S p A: X whih implies : A: : In he firs phases, for every ommodiy z, we have : roued 2z unis However, his flow may violae apaiy onsrains Claim 51 " Proof: Consider an edge X For every G $X unis of flow roued hrough X, we inrease is lengh by a leas a faor Iniially, is lengh is p d G $X : and afer phases, sine S : (3), he lengh of X saisfies nd Therefore he oal amoun of flow hrough X : in he firs phases is srily less han * Od p imes is apaiy Saling he flow by nd p implies he laim Thus he raio of he values of he dual and primal soluions,, is srily d less han nd p Subsiuing : he bound on from (3) we ge Od A: p A: nd p For p * d ]: he raio equals and hene A: < A: A: < : < 7: B d ]: B is a ow i remains o hoose suiably so ha mos our desired approximaion raio 52 Running ime By weakdualiy we have : and hene he number of phases in he above proedure,, is srily less han 213 /57 nd p whih implies ha * The running ime of our ompuaion depends on whih an be redued/inreased by muliplying he demands/apaiies appropriaely Le be he maximum possible flow of ommodiy g and le ')( *M 2 # d Then denoes he maximum fraion of he demands ha an be roued independenly and hene d U1 We sale he apaiies/demands so ha d * hus saisfy r ing our assumpion ha oe however ha ould now be as large as If our proedure does no sop wihin (* r, say) phases hen we know ha We double he demands of all ommodiies and oninue he proedure oe ha is now half is value in he previous phase and is a leas 1 We run he proedure for an addiional phases and if i does no hal we again double demands Sine we halve he value of afer every phases, he oal number of phases is a mos 2143 : Theorem 51 There is an algorihm ha ompues a B approximaion o he maximum onurren flow in "$# where "$# is he ime re ime quired o ompue a minimum os flow in a graph wih nonnegaive edgeoss p 6

7 The number of phases an be redued furher using an idea from [11] We firs ompue a 2approximaion o using he proedure oulined above C This requires phases and reurns, d ow reae a new insane by muliplying demands by ; his insane has Therefore we need a mos an addiional phases o obain C $ a approximaion Thus he number of phases is whih muliplied by gives he number of single ommodiy minos flow ompuaions required 6 Minimum os muliommodiy flow Given an insane of he muliommodiy flow problem, as in he previous seion, edge oss H F L 5, where represens he os inurred in shipping 1 uni of flow along edge X, and a bound, we onsider he problem of maximizing " subje o he addiional onsrain ha he os of he flow is no more han The dual of his linear program is an assignmen of lenghs o he edges, P H F J L5, and a salar whih we view as a lengh assoiaed wih a pseudoedge of apaiy suh ha is a leas suh is minimum; le denoe his mini S P '=( * # T V G $X is minimized subje o he onsrain ha YfP '=( * # T x min osx P 1 This is equivalen o finding a lengh funion P ha S fp 6d Y fp mum value As in he ase r of maximum onurren flow we begin by assuming ha One again he algorihm proeeds in phases eah of whih is omposed of ieraions In he z hi ieraion of he g hi phase we begin wih lengh funions fpx x roue 2z define P 5 q$x * P $X and P * * p d G $X Similarly 5 q * and q * d p and unis of ommodiyz As before, for all edges X, The flow in eah ieraion is roued in a sequene of seps; in eah sep we only roue so muh flow ha is os does no exeed he bound Le fp x x be he lengh funions a he sar of he hfi sep (see Fig 1); he lenghs a he sar of he firs sep are given by P q x * P2 x and q x * x Furher, le x be he flow of ommodiy z ha remains o be roued in his ieraion We ompue j x ')( * # min osx P x x whih roues 2z unis of flow of ommodiy z Sine we need o roue only x unis of flow we muliply he flow funion j x by x d 2z If x is he os of flow j x hen he os of he saled flow is x x d 2z If his quaniy exeeds hen we muliply he original flow funion j x by d x We reuse noaion and denoe he final saled flow and is os by j x x respeively ow j x roues a mos x unis of flow a os x The lengh funions are modified in a similar manner as before Thus P x * P x j 6d and x * d x x Furher, only x * x : j x, more unis of ommodiy z remain o be roued in his ieraion The ieraion ends a he sep for whih x * w The proedure sops a he firs sep a whih S exeeds 1; le his happen in he hfi phase 61 Analysis oe ha now S P x x * S P x x $# min osx fp S P x x $# min osx fp x x x j x d z x j x d where he las inequaliy holds beause he edgelenghs are monoonially inreasing over seps The oal flow roued in he seps equals he demand of ommodiy z, ie, T y j x * 2z Summing over all seps we ge S P x x S P x q q # x min osx P x x The lengh funions a he sar of he z hfi ieraion are given by P x * P x and x * x Moving from seps o ieraions we have S P x x S P2x x S P x x $# $# min osx fp min osx fp2 x x where he las inequaliy uses he fa ha he edgelenghs are monoonially inreasing over ieraions Summing over all ieraions in he g hfi phase we have S P S P q q min xny osx P * S P Y P As before we abbreviae S P Y P o S Y respeively o obain S $g S $g : Y The remainder of he analysis is exaly as in Seion 51 The only modifiaion is in he laim abou he hroughpu of he flow roued ow we need o argue ha he os of he flow afer we sale i by nd p is a mos, or : equivalenly, ha he os of he flow roued in he firs ieraions is a mos 213 /57 nd p This follows from he nd fa ha (sine S : ), ha q * d p and ha in our proedure every ime we roue flow whose oal os is we inrease by a leas a faor 2z 7

8 phase ier ier sep sep sep ier Figure 1 The noaion used in Seions 6 and 7 The lengh funions above he enral axis are he lenghs before he box on he righ and he ones below are he lenghs afer he box on he lef 62 Running ime oe ha exep for he las sep in eah ieraion, in all oher seps we inrease he lengh funion by a faor This implies ha he oal number of seps exeeds he number of ieraions by a mos 13 /57 nd p ow define as he maximum possible flow of ommodiy g of os no more han Again '=( *M 2 # d denoes he maximum fraion of he demands ha an be roued if he apaiy onsrains and he bound on he os of he flow applied independenly o eah ommodiy Thus d K and we muliply demands suiably so ha for he new insane As before we double he demands, hereby halving, afer every phases Thus he number of ieraions is 2143 and so our proedure for minimum os muliommodiy flow needs a mos singleommodiy minos flow ompuaions Theorem 61 There is an algorihm ha ompues a B approximaion o he maximum osbounded onur 8 : ren flow in ime /57 7 "$# where "$# is he ime required o ompue a minimum os flow in a graph wih nonnegaive edgeoss 7 Avoiding minos flow ompuaions We now use ideas from our algorihm for minos muliommodiy flow o give algorihms for he maximum onurren flow and minos muliommodiy flow problems whih use shores pah ompuaions insead of minos flow ompuaions and are faser han he algorihms in Seion 5 and 6 by a leas a faorm Maximum onurren flow revisied Define Y P '=( * # T x z disx P where disx fp denoes he shores pah disane beween x and x under he lengh funion P The dual o he maximum onurren flow problem an also be viewed as an assignmen of lenghs o edges, P H F J L5, suh ha S fp 6d Y P is minimized Le be his minimum The sruure of his new algorihm is similar o ha in he previous seion Thus he algorihm runs in phases eah of whih is omposed of ieraions In he z hi ieraion of he g hi phase we roue z unis of ommodiy z in a sequene of seps Le P x be he lengh funion before he hi sep and le o x be he shores pah beween x and x, ie, o x has lengh disx fp x In his sep we roue x *M 2 ' G x where G is he j x 2 unis of flow along o apaiy of he minimum apaiy edge on his pah We now se x o where x * w x : j x ; he ieraion ends afer seps Thus a eah sep we perform a shores pah ompuaion insead of a minos flow ompuaion as in Seion 6 The lengh funions are modified in exaly he same manner as before and he analysis is almos exaly he same Thus afer rouing all flow of ommodiyz we have $# S fp x 2z disx fp x x S fp q and afer rouing all ommodiies in he g hi phase we have S fp S fp q xny 2z disx fp Using he same abbreviaions as before we again obain S $g S $g : Y Beyond his poin we follow he analysis of Seion 51 o argue ha we have a approximaion for he same hoie of and p 8

9 For he running ime we again noe ha in eah sep, exep he las one in an ieraion, we inrease he lengh of a leas one edge by a faor Sine eah edge has an iniial lengh of p and a final lengh less han, he number of seps exeeds he number of ieraions by a mos Thus he oal number of seps is a mos and eah of hese involves one shores pah ompuaion : Theorem 71 There is an algorihm ha ompues a B approximaion o he maximum onurren flow in ime where is he ime required o ompue he shores pah in a graph wih nonnegaive edgeweighs 72 Minos muliommodiy flow revisied We now define Y P '=( * # T x 2z disx P The dual o he minos muliommodiy flow problem is an assignmen of lenghs o edges, PmH FUJ L 5, and a salar suh ha S P 6d Y fp is minimized Le be his minimum The algorihm differs from he one developed in Seion 6 in ha a any sep we roue flow along only one pah, whih, if his is he hi sep of he z hi phase of he g hi ieraion, is he shores pah beween x and x under he lengh fun ion P pah has apaiy G hen he flow funion a his sep, j x, orresponds o rouing G unis of flow along his pah If G x and he os of his flow is less han we roue his flow ompleely Else we sale i so ha he flow roued in his sep has os no more han and he oal flow roued in his ieraion does no exeed 2z x x If he minimum apaiy edge on his The analysis of he algorihm proeeds as in Seion 61 wih he only modifiaion ha min osx l is replaed wih 2z disx l For he running ime we need only observe ha in eah sep, exep he las sep in an ieraion, we inrease, eiher he lengh of some edge or he value of by a faor The lenghs of he edges and an eah be inreased by a faor a mos 213 /57 /57 imes Hene he number of seps exeeds he number of ieraions by a mos : Theorem 72 There is an algorihm ha ompues a B approximaion o he maximum onurren flow in ime where is he ime required o ompue he shores pah in a graph wih nonnegaive edgeweighs 73 Inegraliy A muliommodiy flow has inegraliy if he flow of every ommodiy on every edge is a nonnegaive ineger muliple of In his seion we show how small modifiaions o he algorihms disussed in previous seions lead o flows ha have small inegraliy Our algorihm for maximum muliommodiy flow roues flow along a pah o in he g hi ieraion If G is he apaiy of a minimum apaiy edge on o hen we require ha he flow roued in his ieraion be no more han G However, noe ha if we roue G unis along o and inrease he lengh of an edge X on o d by a faor hen 7: 7< he algorihm sill delivers a approximaion o he maximum muliommodiy flow, albei wih a worse running ime To obain a feasible flow we sale he flow onsrued in his manner by Od p Thus if we were rouing unis in a erain ieraion hen only appear in he feasible soluion unis Theorem 73 Le X be he minimum apaiy edge in D and G $X Then one an in polynomial ime ompue a flow j : 7< whih is a approximaion o he maximum muliommodiy flow and j has inegraliy 2143 /57 ;: hen here is an inegral flow whih is a Corollary 74 If all edges in D have apaiy a leas < approximaion o he maximum muliommodiy flow For maximum onurren flow we use he algorihm from Seion 7 Reall ha in he hi sep of he z hi ieraion in he g hi phase we roue j x *M 2 ' G x 2 unis of flow along pah o x where G is he apaiy of he minimum apaiy edge on his pah and x is he residual demand of he z hi ommodiy As in he ase of maximum muliommodiy flow we roue j x unis of flow in his sep and inrease he lengh of an edge X on o by a faor d To ensure ha exaly unis of flow an be roued in eah sep of he z hfi ieraion we require ha z be an inegral muliple of To obain a feasible flow we sale he flow onsrued by 13 /57 nd p Hene in he final soluion he flow appears in unis of =\ 7 Theorem 75 Le X be he minimum apaiy edge in D and If all demands are inegral muliples of hen one an, in polynomial ime, ompue a flow j whih is a : B approximaion o he maximum onurren flow and j has inegraliy )\ Corollary 76 If all edges in D have apaiy a leas 2143 /57 7 and all demands are inegral muliples of 2143 /57 7 hen here is an inegral flow whih is a A: B approximaion o he maximum onurren flow The above heorem and is orollary also hold for he seing of minos muliommodiy flows 9

10 Aknowledgemens The firs auhor would like o Philip Klein, Cliff Sein and eal Young for useful disussions Referenes [1] B Awerbuh and FT Leighon Improved approximaion algorihms for he muliommodiy flow problem and loal ompeiive rouing in dynami neworks In Proeedings, ACM Symposium on Theory of Compuing, pages , 1994 [2] G Even, J aor, S Rao, and B Shieber Divideandonquer approximaion algorihms via spreading meris In Proeedings, IEEE Symposium on Foundaions of Compuer Siene, pages 62 71, 1995 [3] G Even, J aor, S Rao, and B Shieber Fas approximae graph pariioning algorihms In Proeedings, ACMSIAM Symposium on Disree Algorihms, pages , 1997 [4] AV Goldberg A naural randomizaion sraegy for muliommodiy flow and relaed algorihms Inform Proess Le, 42: , 1992 [5] M Grigoriadis and LG Khahiyan An exponenial funion reduion mehod for blok angular onvex programs Tehnial Repor LCSRTR211, Deparmen of Compuer Siene, Rugers Universiy, ew Brunswik, U, 1993 [6] M Grigoriadis and LG Khahiyan Approximae C 7< minimumos muliommodiy flows in 7 ime Mah Programming, 75: , 1996 [11] S Plokin, D Shmoys, and É Tardos Fas approximaion algorihms for fraional paking and overing problems Mah Oper Res, 2:257 31, 1995 [12] T Radzik Fas deerminisi approximaion for he muliommodiy flow problem In Proeedings, ACM SIAM Symposium on Disree Algorihms, pages , 1995 [13] F Shahrokhi and D Maula The maximum onurren flow problem J ACM, 37(2): , 199 [14] David Shmoys Approximaion algorihms for Phard problems, haper Cu problems and heir appliaion o divide and onquer, pages PWS Publishing Company, 1997 [15] C Sein Approximaion algorihms for muliommodiy flow and sheduling problems PhD hesis, MIT, 1992 [16] PM Vaidya Speeding up linear programming using fas marix mulipliaion In Proeedings, IEEE Symposium on Foundaions of Compuer Siene, pages , 1989 [17] Young Randomized rounding wihou solving he linear program In Proeedings, ACMSIAM Symposium on Disree Algorihms, 17178, 1995 [7] MD Grigoriadis and LG Khahiyan Fas approximaion shemes for onvex programs wih many bloks and oupling onsrains SIAM J Opimizaion, 4(1):86 17, 1994 [8] D Karger and S Plokin Adding muliple os onsrains o ombinaorial opimizaion problems, wih appliaions o muliommodiy flows In Proeedings, ACM Symposium on Theory of Compuing, pages 18 25, 1995 [9] P Klein, S Plokin, C Sein, and E Tardos Faser approximaion algorihms for he uni apaiy onurren flow problem wih appliaions o rouing and finding sparse us SIAM J Compu, 23(3): , 1994 [1] T Leighon, F Makedon, S Plokin, C Sein, S Tragoudas, and E Tardos Fas approximaion algorihms for muliommodiy flow problems J Compu Sysem Si, 5: ,