Electrical Flows, Laplacian Systems, and Faster Approximation of Maximum Flow in Undirected Graphs

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1 Elctrical Flows, Laplacian Systms, and Fastr Approximation of Maximum Flow in Undirctd Graphs Paul Christiano MIT Jonathan A. Klnr MIT Alksandr Mądry MIT Shang-Hua Tng Univrsity of Southrn California Octobr 19, 010 Danil Spilman Yal Univrsity Abstract W introduc a nw approach to computing an approximatly maximum s-t flow in a capacitatd, undirctd graph. This flow is computd by solving a squnc of lctrical flow problms. Each lctrical flow is givn by th solution of a systm of linar quations in a Laplacian matrix, and thus may b approximatly computd in narly-linar tim. Using this approach, w dvlop th fastst known algorithm for computing approximatly maximum s-t flows. For a graph having n vrtics and m dgs, our algorithm computs a (1 )- approximatly maximum s-t flow in tim 1 O mn 1/3 11/3. A dual vrsion of our approach computs a (1 + )-approximatly minimum s-t cut in tim O m + n 4/3 8/3,whichisth fastst known algorithm for this problm as wll. Prviously, th bst dpndnc on m and n was achivd by th algorithm of Goldbrg and Rao (J. ACM 1998), which can b usd to comput approximatly maximum s-t flows in tim O m n 1, and approximatly minimum s-t cuts in tim O m + n 3/ 3. Rsarch partially supportd by NSF grant CCF Rsarch partially supportd by NSF grant CCF and by ONR grant N This matrial is basd upon work supportd by th National Scinc Foundation undr Grant Nos , and Any opinions, findings, and conclusions or rcommndations xprssd in this matrial ar thos of th authors and do not ncssarily rflct th viws of th National Scinc Foundation. This rsarch is in part supportd by NSF grants , , and a USC Vitrbi School of Enginring startup grant which is in turn supportd by a Powll Foundation Award. 1 W rcall that O (f(m)) dnots O(f(m)log c f(m)) for som constant c.

2 1 Introduction Th maximum s-t flow problm and its dual, th minimum s-t cut problm, ar two of th most fundamntal and xtnsivly studid problms in Oprations Rsarch and Optimization [16, 1]. Thy hav many applications (s []) and ar oftn usd as subroutins in othr algorithms (s [3, 17]). Many advancs hav bn mad in th dvlopmnt of algorithms for this problm (s Goldbrg and Rao [11] for an ovrviw). Howvr, for th basic problm of computing or (1 )- approximating th maximum flow in undirctd, unit-capacity graphs with m = O(n) dgs, th asymptotically fastst known algorithm is th on dvlopd in 1975 by Evn and Tarjan [9], which taks tim O(n 3/ ). Dspit 35 yars of xtnsiv work on th problm, this bound has not bn improvd. In this papr, w introduc a nw approach to computing approximatly maximum s-t flows and minimum s-t cuts in undirctd, capacitatd graphs. Using it, w prsnt th first algorithms that brak th O(n 3/ ) complxity barrir dscribd abov. In addition to bing th fastst known algorithms for this problm, thy ar simpl to dscrib and introduc tchniqus that may b applicabl to othr problms. In thm, w rduc th problm of computing maximum flows subjct to capacity constraints to th problm of computing lctrical flows in rsistor ntworks. An approximat solution to ach lctrical flow problm can b found in tim O (m) using rcntly dvlopd algorithms for solving systms of linar quations in Laplacian matrics [13, 18]. Thr is a simpl physical intuition that undrlis our approach, which w dscrib hr in th cas of a graph with unit dg capacitis. W bgin by thinking of ach dg of th input graph as a rsistor with rsistanc on, and w comput th lctrical flow that rsults whn w snd currnt from th sourc s to th sink t. Ths currnts oby th flow consrvation constraints, but thy may not rspct th capacitis of th dgs. To rmdy this, w incras th rsistanc of ach dg in proportion to th amount of currnt flowing through it thrby pnalizing dgs that violat thir capacitis and comput th lctrical flow with ths nw rsistancs. Aftr rpating this opration O m 1/3 poly(1/) tims, w will b abl to obtain a (1 )- approximatly maximum s-t flow by taking a crtain avrag of th lctrical flows that w hav computd, and w will b abl to xtract a (1 + )-approximatly minimum s-t cut from th vrtx potntials. This will giv us algorithms for both problms that run in tim O m 4/3 poly(1/). By combining this with th graph smoothing and sampling tchniqus of Kargr [1], w can gt a (1 )-approximatly maximum s-t flow in tim O mn 1/3 11/3. Furthrmor, by applying th cut algorithm to a sparsifir [4] of th input graph, w can comput a (1 + )-approximatly minimum s-t cut in tim O m + n 4/3 8/3. W rmark that th rsults in this papr immdiatly improv th running tim of algorithms that us th computation of an approximatly maximum s-t flow on an undirctd, capacitatd graph as a subroutin. For xampl, combining our work with that of Shrman [17] allows us to achiv th bst currntly known approximation ratio of O( log n) for th sparsst cut problm in tim O m + n 4/3. W ar hopful that our approach can b xtndd to dirctd graphs and can also vntually lad to an algorithm that approximatly solvs th maximum flow problm in narly-linar tim. For clarity, w will analyz ths two cass sparatly, and thy will us slightly diffrnt ruls for updating th rsistancs. 1

3 1.1 Prvious Work on Maximum Flows and Minimum Cuts Th bst prviously known algorithms for th problms studid hr ar du to Goldbrg and Rao. In a brakthrough papr, Goldbrg and Rao [11] dvlopd an algorithm for computing xact maximum s-t flows in dirctd or undirctd capacitatd graphs in tim O(m min(n /3,m 1/ ) log(n /m) log U), assuming that th dg capacitis ar intgrs btwn 1 and U. Whn w ar intrstd in finding (1 )-approximatly maximum s-t flow, th dpndnc on log U can b rmovd and, by mploying th smoothing tchniqu of Kargr [1], on can obtain a running tim of O m n 1. By applying thir algorithm to a sparsifir, as constructd by Bnczúr and Kargr [4], Goldbrg and Rao show how to comput a (1 + )-approximatly minimum s-t cut in an undirctd graph in tim O m + n 3/ 3. Thir work was th culmination of a long lin of paprs on th problm; w rfr th radr to thir papr for an xtnsiv survy of arlir dvlopmnts in algorithms for computing maximum s-t flows. In mor rcnt work, Daitch and Spilman [7] showd that fast solvrs for Laplacian linar systms [18, 13] could b usd to mak intrior-point algorithms for th maximum flow and minimum-cost flow problms run in tim O m 3/ log U, and Mądry [14] showd that on can approximat a wid rang of cut problms, including th minimum s-t cut problm, within a polylogarithmic factor in almost linar tim. 1. Outlin W bgin th tchnical part of this papr in Sction with a rviw of maximum flows and lctrical flows, along with svral thorms about thm that w will nd in th squl. In Sction 3 w giv a simplifid vrsion of our approximat maximum-flow algorithm that has running tim O m 3/ 5/. In Sction 4, w will show how to improv th running tim of our algorithm to O m 4/3 3 ; w will thn dscrib how to combin this with xisting graph smoothing and sparsification tchniqus to comput approximatly maximum s-t flows in tim O mn 1/3 11/3 and to approximat th valu of such flows in tim O m + n 4/3 8/3. In Sction 5, w prsnt a variant of our algorithm that computs approximatly minimum s-t cuts in tim O m + n 4/3 8/3. Maximum Flows, Elctrical Flows, and Laplacian Systms.1 Graph Thory Dfinitions Throughout th rst of th papr, lt G =(V,E) b an undirctd graph with n vrtics and m dgs. W distinguish two vrtics, a sourc vrtx s and a sink vrtx t. W assign ach dg a nonzro intgral capacity u Z +, and w lt U := max u / min u b th ratio of th largst to th smallst capacitis. W arbitrarily orint ach dg in E; this divids th dgs incidnt to a vrtx v V into th st E + (v) of dgs orintd towards v and th st E (v) of dgs orintd away from v. Ths orintations ar mrly for notational convninc. W us thm to intrprt th maning of a positiv flow on an dg. If an dg has positiv flow and is in E + (v), thn th flow is towards v.

4 Convrsly, if it has ngativ flow thn th flow is away from v. On should kp in mind that our graphs ar undirctd and that th flow on an dg can go in ithr dirction, rgardlss of this dg s orintation. W now dfin our primary objcts of study, s-t cuts and s-t flows. Dfinition.1 (Cuts). An s-t cut is a partition (S, V \S) of th vrtics into two disjoint sts such that s S and t V \S. Thcapacity u(s) of th cut is dfind to b th sum u(s) := E(S) u, whr E(S) E is th st of dgs with on ndpoint in S and on ndpoint in V \ S. Dfinition. (Flows). An s-t flow is a function f : E IR that obys th flow-consrvation constraints f() f() =0 for all v V \{s, t}. E (v) E + (v) Th valu f of th flow is dfind to b th nt flow out of th sourc vrtx, f := E (s) f() E + (s) f(). It follows asily from th flow consrvation constraints that th nt flow out of s is qual to th nt flow into t, so f may b intrprtd as th amount of flow that is snt from s to t.. Maximum Flows and Minimum Cuts W say that an s-t flow f is fasibl if f() u for ach dg, i.., if th amount of flow routd through any dg dos not xcd its capacity. Th maximum s-t flow problm is that of finding a fasibl s-t flow in G of maximum valu. W dnot a maximum flow in G (with th givn capacitis) by f, and w dnot its valu by F := f. W say that f is a (1 )-approximatly maximum flow if it is a fasibl s-t flow of valu at last (1 )F. To simplify th xposition, w will tak to b a constant indpndnt of m throughout th papr, and m will b assumd to b largr than som fixd constant. Howvr, our analysis will go through unchangd as long as > Ω(m 1/3 ). In particular, our analysis will apply to all for which our givn bounds ar fastr than th O(m 3/ ) tim rquird by xisting xact algorithms. On can asily rduc th problm of finding a (1 )-approximation to th maximum flow in an arbitrary undirctd graph to that of finding a (1 /)-approximation in a graph in which th ratio of th largst to smallst capacitis is polynomially boundd. To do this, on should first comput a crud approximation of th maximum flow in th original graph. For xampl, on can comput th s-t path of maximum bottlnck in tim O(m + n log n) [16, Sction 8.6], whr w rcall that th bottlnck of a path is th minimum capacity of an dg on that path. If this maximum bottlnck of an s-t path is B, thn th maximum flow lis btwn B and mb. This mans that thr is a maximum flow in which ach dg flows at most mb, so all capacitis can b dcrasd to b at most mb. On th othr hand, if on rmovs all th dgs with capacitis lss B/m, th maximum flow can dcras by at most B/. So, w can assum that th minimum capacity is at last B/m and th maximum is at most Bm, for a ratio of m /. Thus, by a simpl scaling, w can assum that all dg capacitis ar intgrs btwn 1 and m /. Th minimum s-t cut problm is that of finding th s-t cut of minimum capacity. Th Max Flow-Min Cut Thorm ([10, 8]) stats that th capacity of th minimum s-t cut is qual to F, th valu of th maximum s-t flow. 3

5 In particular, th Max Flow-Min Cut Thorm implis that on can us th capacity of any s-t cut as an uppr bound on th valu of any fasibl s-t flow, and that th task of finding th valu of th maximum flow is quivalnt to th task of finding th capacity of a minimal s-t cut. On should not, howvr, that th abov quivalnc applis only to th valus of th flow and th capacity and that although on can asily obtain a minimum s-t cut of a graph givn its maximum flow, thr is no known procdur that obtains a maximum flow from minimum s-t cut mor fficintly than by just computing th maximum flow from a scratch..3 Elctrical Flows and th Narly Linar Tim Laplacian Solvr In this sction, w rviw som basic facts about lctrical flows in ntworks of rsistors and prsnt a thorm that allows us to quickly approximat ths flows. For an in-dpth tratmnt of th background matrial, w rfr th radr to [6]. W bgin by assigning a rsistanc r > 0 to ach dg E, and w collct ths rsistancs into a vctor r IR m. For a givn s-t flow f, wdfinitsnrgy (with rspct to r) as E r (f) := r f (). Th lctrical flow of valu F (with rspct to r, froms to t) is th flow that minimizs E r (f) among all s-t flows f of valu F. This flow is asily shown to b uniqu, and w not that it nd not rspct th capacity constraints. From a physical point of viw, th lctrical flow of valu on corrsponds to th currnt that is inducd in G if w viw it as an lctrical circuit in which ach dg has rsistanc of r, and w snd on unit of currnt from s to t, say by attaching s to a currnt sourc and t to ground..3.1 Elctrical Flows and Linar Systms Whil finding th maximum s-t flow corrsponds to solving a linar program, w can comput th lctrical flow by solving a systm of linar quations. To do so, w introduc th dg-vrtx incidnc matrix B, which is an n m matrix with rows indxd by vrtics and columns indxd by dgs, such that 1 if E (v), B v, = 1 if E + (v), 0 othrwis. If w trat our flow f as a vctor f IR m, whr w us th orintations of th dgs to dtrmin th signs of th coordinats, th v th ntry of th vctor B T f will b th diffrnc btwn th flow out of and th flow into vrtx v. As such, th constraints that on unit of flow is snt from s to t and that flow is consrvd at all othr vrtics can b writtn as B T f = χ s,t, whr χ s,t is th vctor with a 1 in th coordinat corrsponding to s, a 1 in th coordinat corrsponding to t, and all othr coordinats qual to 0. W dfin th (wightd) Laplacian L of G (with rspct to th rsistancs r) tobthn n matrix L := BCB T, 4

6 whr C is th m m diagonal matrix with C, = c =1/r. On can asily chck that its ntris ar givn by E + (u) E (u) c if u = v, L u,v = c if =(u, v) is an dg of G, and 0 othrwis. Lt R = C 1 b th diagonal matrix with R, = r. Th nrgy of a flow f is givn by E r (f) := r f () = f T Rf = R 1/ f. Th lctrical flow of valu 1 thus corrsponds to th vctor f that minimizs R 1/ f subjct to Bf = χ s,t. If f is an lctrical flow, it is wll known that it is a potntial flow, which mans that thr is a vctor φ IR V such that That is, f (u, v) = φ v φ u r u,v. f = CB T φ = R 1 B T φ. Applying Bf = χ s,t,whavbf = BCB T φ = χ s,t, and hnc th vrtx potntials ar givn by φ = L χ s,t, whr L dnots th Moor-Pnros psudo-invrs of L. Thus, th lctrical flow f is givn by th xprssion f = CB T L χ s,t. This lts us rwrit th nrgy of th lctrical flow of valu 1 as E r (f )=f T Rf = χ T s,t L T BC T R CB T L χ s,t = χ s,t L LL χ s,t = χ T s,t L χ s,t = φ T Lφ..3. Effctiv s-t Rsistanc and Effctiv s-t Conductanc Our analysis will mak rpatd us of two basic quantitis from th thory of lctrical ntworks, th ffctiv s-t rsistanc and ffctiv s-t conductanc. Lt f b th lctrical s-t flow of valu 1, and lt φ b its vctor of vrtx potntials. Th ffctiv s-t rsistanc of G with rspct to th rsistancs r is givn by R ff (r) =φ(s) φ(t). Throughout this papr, w will only look at th ffctiv rsistanc btwn th vrtics s and t in th graph G, so w supprss ths lttrs in our notation and simply writ R ff (r). Using our linar algbraic dscription of th lctrical flow and Equation (1), w hav R ff (r) =φ(s) φ(t) =χ s,t T φ = χ s,t T L χ s,t = E r (f ). This givs us an altrnativ dscription of th ffctiv s-t rsistanc as th nrgy of th lctrical flow of valu 1. (1) 5

7 It will somtims b convnint to us th rlatd notion of th ffctiv s-t conductanc of G with rspct to th rsistancs r, whichwdfinby C ff (r) =1/R ff (r). W not that this is qual to th valu of th lctrical flow in which φ(s) φ(t) = Approximatly Computing Elctrical Flows From th algorithmic point of viw, th crucial proprty of th Laplacian L is that it is symmtric and diagonally dominant, i.., for any u, v =u L u,v L u,v. This allows us to us th rsult of Koutis, Millr, and Png [13], which builds on th work of Spilman and Tng [18], to approximatly solv our linar systm in narly-linar tim. By rounding th approximat solution to a flow, w can prov th following thorm (s Appndix A for a proof). Thorm.3 (Fast Approximation of Elctrical Flows). For any δ>0, any F > 0, and any vctor r of rsistancs in which th ratio of th largst to th smallst rsistanc is at most R, w can comput, in tim O (m log R/δ), a vctor of vrtx potntials φ and an s-t flow f of valu F such that a. E r ( f) (1 + δ)e r (f), whr f is th lctrical s-t flow of valu F,and b. for vry dg, whr f is th tru lctrical flow. r f r f δ mr E r (f), c. φ s φ t δ 1 FR ff (r). 1nmR W will rfr to a flow mting th abov conditions as a δ-approximat lctrical flow..4 How th Rsistanc of an Edg Influncs th Effctiv Rsistanc In this sction, w will study how changing th rsistanc of an dg affcts th ffctiv rsistanc of th graph. This will b a ky componnt of th analysis of our O m 4/3 poly(1/) algorithms. W will mak us of th following standard fact about ffctiv conductanc; w rfr th radr to [6, Chaptr IX., Corollary 5] for a proof. Fact.4. For any G =(V,E) and any vctor of rsistancs r, C ff (r) = min φ φs=1, φ t =0 (u,v) E (φ u φ v ) r (u,v). Furthrmor, th quality is attaind for φ bing vctor of vrtx potntials corrsponding to th lctrical s-t flow of G (with rspct to r) of valu 1/R ff (r). Corollary.5 (Rayligh Monotonicity). If r r for all E, thn R ff (r ) R ff (r). 6

8 Proof. For any φ, (u,v) E (φ u φ v ) r (u,v) (u,v) E (φ u φ v ) r (u,v), so th minima of ths xprssions ovr possibl valus of φ oby th sam rlation, and thus C ff (r ) C ff (r). Invrting both sids of this inquality yilds th dsird rsult. Our analysis of th O m 4/3 algorithm will rquir th following lmma, which givs a lowr bound on th ffct that incrasing th rsistanc of an dg can hav on th ffctiv rsistanc. Lmma.6. Lt f b an lctrical s-t flow on a graph G with rsistancs r. Suppos that som dg h =(i, j) accounts for a β fraction of th total nrgy of f, i.., f(h) r h = βe r (f). For som γ>0, dfin nw rsistancs r such that r h = γr h,andr = r for all = h. Thn In particular: R ff (r ) If w cut th dg h by stting γ =, thn γ β + γ(1 β) R ff(r). R ff (r ) R ff(r) 1 β. If w slightly incras th ffctiv rsistanc of th dg h by stting γ =(1+) with 1, thn R ff (r 1+ ) β +(1+)(1 β) R ff(r) 1+ β R ff (r). Proof. Th assumptions of th thorm ar unchangd if w multiply f by a constant, so w may assum without loss of gnrality that f is th lctrical s-t flow of valu 1/R ff (r). If φ f is th vctor of vrtx potntials corrsponding to f, this givs φ f s φ f t =1. Sinc adding a constant to th potntials dosn t chang th flow, w may assum that φ s =1and φ t =0. By Fact.4, C ff (r) = (u,v) E φ f u φ f v = r (u,v) φ f i φf j + r h (u,v) E\{h} φ f u φ f v. r (u,v) Th assumption that h contributs a β fraction of th total nrgy implis that, in th abov xprssion, φ f i φf j = βc ff (r), and thus (u,v) E\{h} r h φ f u φ f v r (u,v) 7 =(1 β)c ff (r).

9 W will obtain our bound on C ff (r ) by plugging th original vctor of potntials φ f into th xprssion in Fact.4: C ff (r )= = min φ φs=1, φ t =0 φ f i φf j r h (u,v) E + φu φ v r (u,v) (u,v) E\{h} φ f u φ f v r (u,v) (u,v) E φ f u φ f v = r (u,v) φ f i φf j + γr h = β β + γ(1 β) γ C ff(r)+(1 β)c ff (r) = C ff (r). γ (u,v) E\{h} Sinc R ff (r) =1/C ff (r) and R ff (r )=1/C ff (r ), th dsird rsult follows. 3 A Simpl O m 3/ 5/ -Tim Flow Algorithm φ f u φ f v Bfor dscribing our O m 4/3 3 algorithm, w will dscrib a simplr algorithm that finds a (1 )-approximatly maximum flow in tim O m 3/ 5/. Our final algorithm will b obtaind by carfully modifying th on dscribd hr. Th algorithm will mploy th multiplicativ wights updat mthod [3, 15]. In our stting, on can undrstand th multiplicativ wights mthod as a way of taking an algorithm that solvs a flow problm vry crudly and, by calling it rpatdly, convrts it into an algorithm that givs a good approximation for th maximum flow in G. Th crud algorithm is calld as a black-box, so it can b thought of as an oracl that answrs a crtain typ of qury. In this sction, w provid a slf-containd dscription of th multiplicativ wights mthod whn it is spcializd to our stting. In Sction 3.1, w will dscrib th rquirmnts on th oracl, giv an algorithm that itrativly uss it to obtain a (1 )-approximatly maximum flow, and stat how th numbr of itrations rquird by th algorithm dpnds on th proprtis of th oracl. In Sction 3., w will dscrib how to implmnt th oracl using lctrical flows. Finally, in Sction 3.3 w will provid a simpl proof of th convrgnc bound st forth in Sction Multiplicativ Wights Mthod: From Elctrical Flows to Maximum Flows For an s-t flow f, wdfinthcongstion of an dg to b th ratio cong f () := f() u btwn th flow on an dg and its capacity. In particular, an s-t flow is fasibl if and only if cong f () 1 for all E. Th multiplicativ wights mthod will us a subroutin that w will rfr to as an (, )-oracl. This oracl will tak as input a numbr F and a vctor w of dg wights. For any F F, w know that thr xists a way to rout F units of flow in G so that all of th dg capacitis ar rspctd. Our oracl will provid a wakr guarant: Whn F F, it will satisfy all of th capacity constraints up to a multiplicativ factor of, 3 and it will satisfy th avrag of ths r (u,v) 3 Up to polynomial factors in 1/, th valu of will b Θ( m) in this sction, and Θ(m 1/3 ) latr in th papr. 8

10 constraints, wightd by th w i, up to a (much bttr) multiplicativ factor of (1 + ). Whn F > F, th oracl will ithr output an s-t flow satisfing th conditions abov, or it will rturn fail. Formally, w will us th following dfinition: Dfinition 3.1 ((, ) oracl). For >0 and >0, an(, ) oracl is an algorithm that, givn a ral numbr F>0 and a vctor w of dg wights with w 1 for all, rturns an s-t flow f such that: 1. If F F, thn it outputs an s-t flow f satisfying: (i) f = F ; (ii) w cong f () (1 + ) w 1, whr w 1 := w ; (iii) max cong f ().. If F > F, thn it ithr outputs a flow f satisfying conditions (i), (ii), (iii) or outputs fail. Our algorithm will b givn a flow valu F as an input. If F F, it will rturn a flow of valu at last (1 O())F. If F>F, it will ithr rturn a flow of valu at last (1 O())F (which may occur if F is only slightly gratr than F ) or it will rturn fail. This allows us to find a (1 O())-approximation of F using binary sarch. As outlind in Sction., w can obtain a crud bound B in tim O(m + n log n) such that B F mb, so th binary sarch will only call our algorithm a logarithmic numbr of tims. In Figur 1, w prsnt our simpl algorithm, which applis th multiplicativ wights updat routin to approximat th maximum flow by calling an (, )-flow oracl. Th algorithm initializs all of th wights to 1 and calls th oracl with ths wights. Th call rturns a flow that satisfis conditions (i), (ii), and (iii) dfind abov. It thn multiplis th wight of ach dg by (1 + cong f i()). Not that if th congstion of an dg is poor, say clos to, thn its wight will incras by a factor of (1 + ). On th othr hand, if th flow on an dg is no mor than its capacity, thn th nw wight of th dg is ssntially unchangd. This will put a largr fraction of th wight on th violatd constraints, so th oracl will b forcd to rturn a solution that coms closr to satisfying thm (possibly at th xpns of othr dgs). In th nd, w rturn th avrag of all of th flows as our answr. Th ky point in analyzing this algorithm is that th total wight on G dos not grow too quickly, du to th avrag congstion constraint on th flows rturnd by th oracl O. Howvr, if an dg consistntly suffrs larg congstion in a squnc of flows rturnd by O, thnits wight incrass rapidly rlativ to th total wight, which will significantly pnaliz any furthr congstion th dg in th subsqunt flows. If this wr to occur too many tims, its wight would xcd th total wight, which obviously cannot occur. In Sction 3.3, w will prov th following thorm by showing that our algorithm convrgs in ln m/ itrations. Thorm 3. (Approximating Maximum Flows by Multiplicativ Wights). For any 0 <<1/ and >0, givn an (, )-flow oracl with running tim T (m, 1/, U), on can obtain an algorithm that computs a (1 O())-approximat maximum flow in a capacitatd, undirctd graph in tim O T (m, 1/, U). 9

11 Input : A graph G =(V,E) with capacitis {u }, a targt flow valu F,andan(, )-oracl O Output: Eithraflowf, or fail indicating that F>F ; ln m Initializ w 0 1 for all dgs, andn for i := 1,...,N do Qury O with dg wights givn by w i 1 and targt flow valu F if O rturns fail thn rturn fail ls Lt f i b th rturnd flow nd nd w i w i 1 rturn f (1 ) (1+)N ( i f i ) (1 + cong f i()) for ach E Figur 1: Multiplicativ-wights-updat routin Not that th numbr of itrations of th algorithm abov grows linarly with th valu of, which w call th width of th oracl. Intuitivly, this should b ncssary bcaus th final flow across an dg is qual to th avrag of th flows snt ovr it by all f i.ifwsnd u units of flow across th dg in som stp, thn w will nd at last Ω() itrations to drop th avrag to Constructing an Oracl of Width 3 m/ Using Elctrical Flows Givn Thorm 3., our problm is thus rducd to dsigning an fficint oracl that has a small width. In this subsction, w will giv simpl O m log 1 tim implmntation of an (, 3 m/)- flow oracl for any 0 <<1/. By Thorm 3., this will immdiatly yild an O m 3/ 5/ tim algorithm for finding an approximatly maximum flow. To build such an oracl, w st r := 1 u w + w 1 () 3m for ach dg, and w us th procdur from Thorm.3 to approximat th lctrical flow that snds F units of flow from s to t in a ntwork whos rsistancs ar givn by th r. Th psudocod for this oracl is shown in Figur. W now show that th rsulting flow f has th proprtis rquird by Dfinition 3.1. Sinc f = F by construction, w only nd to dmonstrat th bounds on th avrag congstion (wightd by th w ) and th maximum congstion. W will us th basic fact that lctrical flows minimiz th nrgy of th flow. Our analysis will thn compar th nrgy of f with that of an optimal max flow. Intuitivly, th w trm in Equation () guarants th bound on th avrag congstion, whil th w 1 /(3m) trm guarants th bound on th maximum congstion. 4 Strictly spaking, it is possibl that w could do bttr than this by snding a larg amount of flow across th dg in th opposit dirction. Howvr, nothing in our algorithm aims to obtain this kind of cancllation, so w shouldn t xpct to b abl to systmatically xploit it. 10

12 Input : A graph G =(V,E) with capacitis {u }, a targt flow valu F, and dg wights {w } Output: Eithraflowf,or fail indicating that F>F r 1 u w + w 1 3m for ach E Find an (/3)-approximat lctrical flow f using Thorm.3 on G with rsistancs r and targt flow valu F if E r ( f) > (1 + ) w 1 thn rturn fail ls rturn f Figur : A simpl implmntation of an, 3 m/ oracl Suppos f is a maximum flow. By its fasibility, cong f () 1 for all, so E r (f )= w + w 1 f () 3m u = w + w 1 congf () 3m w + w 1 3m = 1+ w 1. 3 Sinc th lctrical flow minimizs th nrgy, E r (f ) is an uppr bound on th nrgy of th lctrical flow of valu F whnvr F F. In this cas, Thorm.3 implis that our (/3)- approximat lctrical flow satisfis E r ( f) 1+ E r (f ) 1+ w 1 (1 + ) w 1. (3) 3 3 This shows that our oracl will nvr output fail whn F F. It thus suffics to show that th nrgy bound E r ( f) > (1 + ) w 1, which holds whnvr th algorithm dos not rturn fail, implis th rquird bounds on th avrag and worst-cas congstion. To s this, w not that th nrgy bound implis that and, for all E, w 1 3m By th Cauchy-Schwarz inquality, w cong f () (1 + ) w 1, (4) cong f () (1 + ) w 1. (5) w cong f () w 1 w cong f (), (6) 11

13 so Equation (4) givs us that w cong f () 1+ w 1 < (1 + ) w 1, (7) which is th rquird bound on th avrag congstion. Furthrmor, Equation (5) and th fact that <1/ implis that 3m(1 + ) cong bf () 3 m/ for all, which stablishs th rquird bound on th maximum congstion. So our algorithm implmnts an (, 3 m/)-oracl, as dsird. To bound th running tim of this oracl, rcall that w can assum all dg capacitis li btwn 1 and U = m / and comput R = max, r r U max r + w 1 w 1 U m + = O (m/) O(1). (8) This stablishs an uppr bound on th ratio of th largst rsistanc to th smallst rsistanc. Thus, by Thorm.3, th running tim of this implmntation is O (m log R/) = O (m log 1/). Combining this with Thorm 3., w hav shown Thorm 3.3. For any 0 <<1/, th maximum flow problm can b (1 )-approximatd in O m 3/ 5/ tim. 3.3 Th Convrgnc of Multiplicativ Wights In this sction, w prov Thorm 3. by analyzing th multiplicativ wights updat algorithm shown in Figur 1. Our analysis will b basd on th potntial function µ i := w i 1. Clarly, µ 0 = m and this potntial only incrass during th cours of th algorithm. It follows from condition (1.ii) of Dfinition 3.1 that if w run th (, )-oracl O with F = F,thn By condition (1.iii) of Dfinition 3.1, wcong i f i+1() (1 + ) w i 1, for all i 1. (9) cong f i(), for all i 1 and any dg. (10) W start by uppr bounding th total growth of µ i thoughout th algorithm. Lmma 3.4. For any i 0, (1 + ) µ i+1 µ i xp. In particular, w N 1 = µ N m xp (1+) N = n O(1/). 1

14 Proof. For any i 0, whav µ i+1 = w i+1 = w i 1+ cong f i+1() = w i + w i cong f i+1() µ i + (1 + ) w i 1, whr th last inquality follows from (9). Thus, w can conclud that (1 + ) µ i+1 µ i + w i (1 + ) (1 + ) 1 = µ i (1 + ) µ i xp, as dsird. Th lmma follows. On of th consquncs of th abov lmma is that whnvr w mak a call to th oracl, th total wight w i 1 is at most n O(1/). Thus, th running tim of our algorithm is O T (m, 1/, U, m O(1/) ) as claimd. Nxt, w bound th final wight w N of a particular dg with th congstion cong f () that this dg suffrs in our final flow f. Lmma 3.5. For any dg and i 0, w i xp (1 ) i cong f j(). j 1 In particular, w N xp (1+)N (1 ) cong f. () Proof. For any i 0, whav w i = i 1+ cong f j() j 1 whr w usd (10) and that for any 1/ >>0 and x [0, 1]: Now, th lmma follows sinc w i i j 1 i j 1 xp((1 )x) (1 x). (1 ) xp cong f j() (1 ) xp cong f j(), =xp (1 ) i cong f j(), j 1 and for i = N w N xp (1 ) N cong f j() (1 + )N =xp (1 ) cong f (). j 1 13

15 Finally, by Lmmas 3.4 and 3.5, w conclud that for any dg, (1 + )N (1 + )N m xp µ N = w N 1 w N xp (1 ) cong f (). This implis that cong f () 1 + (1 ) ln m (1 + )N =1 + (1 ) (1 + ) 1 for vry dg. Thus, w s that f is a fasibl s-t flow and, sinc ach f i has throughput F, th throughput f of f is (1 ) (1+) F (1 O())F for 1/ >>0, as dsird. 4 An O mn 1/3 11/3 Algorithm for Approximat Maximum Flow In this sction, w modify our algorithm to run in tim O m 4/3 3. W thn combin this with th smoothing and sampling tchniqus of Kargr [1] to obtain an O mn 1/3 11/3 -tim algorithm. For fixd, th algorithm in th prvious sction rquird us to comput O m 1/ lctrical flows, ach of which took tim O (m), which ld to a running tim of O m 3/.Torducthisto O m 4/3, w ll show how to find an approximat flow whil computing only O m 1/3 lctrical flows. Our analysis of th oracl from Sction 3. was fairly simplistic, and on might hop to improv th running tim of th algorithm by proving a tightr bound on th width. Unfortunatly, th graph in Figur 3 shows that our analysis was ssntially tight. Th graph consists of k paralll paths of lngth k conncting s to t, along with a singl dg that dirctly conncts s to t. Th max flow in this graph is k +1. In th first call mad to th oracl by th multiplicativ wights routin, all of th dgs will hav th sam rsistanc. In this cas, th lctrical flow of valu k +1 will snd (k + 1)/k units of flow along ach of th k paths and (k + 1)/ units of flow across. Sinc th graph has m =Θ(k ), th width of th oracl in this cas is Θ(m 1/ ). 4.1 Th Improvd Algorithm Th abov xampl shows that it is possibl for th lctrical flow rturnd by th oracl to xcd th dg capacitis by Θ(m 1/ ). Howvr, w not that if on rmovs th dg from th graph in Figur 3, th lctrical flow on th rsulting graph is much bttr bhavd, but th valu of th maximum flow is only vry slightly rducd. This dmonstrats a phnomnon that will b cntral to our improvd algorithm: whil instancs in which th lctrical flow snds a hug amount of flow ovr som dgs xist, thy ar somwhat fragil, and thy ar oftn drastically improvd by rmoving th bad dgs. This motivats us to modify our algorithm as follows. W ll st to b som valu smallr than th actual worst-cas bound of O m 1/. (It will nd up bing O m 1/3.) Th oracl will bgin by computing an lctrical flow as bfor. Howvr, whn this lctrical flow xcds th capacity of som dg by a factor gratr than, w llrmov from th graph and try again, kping all of th othr wights th sam. W ll rpat this procss until w obtain a flow in which all dgs flow at most a factor of tims thir capacity (or som failur condition is rachd), and w ll us this flow in our multiplicativ wights routin. Whn th oracl rmovs an dg, it is addd to a 14

16 s k paths of lngth k 1 dg t Figur 3: A graph on which th lctrical flow snds approximatly m units of flow across an dg whn snding th maximum flow F from s to t. st H of forbiddn dgs. Ths dgs will b prmanntly rmovd from th graph, i.., thy will not b includd in th graphs supplid to futur invocations of th oracl. In Figurs 4 and 5, w prsnt th modifid vrsions of th oracl and ovrall algorithm, whr w hav highlightd th parts that hav changd from th simplr vrsion shown in Figurs 1 and. 4. Analysis of th Nw Algorithm Bfor procding to a formal analysis of th nw algorithm, it will b hlpful to xamin what is alrady guarantd by th analysis from Sction 3, and what w ll nd to show to dmonstrat th algorithm s corrctnss and bound its running tim. W first not that, by construction, th congstion of any dg in th flow f rturnd by th modifid oracl from Figur 4 will b boundd by. Furthrmor, it nforcs th bound E r ( f) (1 + ) w 1 ; by Equations (4), (6), and (7) in Sction 3., this guarants that f will mt th wightd avrag congstion bound rquird for a (, )-oracl. So, as long as th modifid oracl always succssfully rturns a flow, it will function as an (, )-oracl, and our analysis from Sction 3 will show that th multiplicativ updat schm mployd by our algorithm will yild an approximat maximum flow aftr O () itrations. Our problm is thus rducd to undrstanding th bhavior of th modifid oracl. To prov corrctnss, w will nd to show that whnvr th modifid oracl is calld with F F,itwill rturn som flow f (as opposd to rturning fail ). To bound th running tim, w will nd to provid an uppr bound on th total numbr of lctrical flows computd by th modifid oracl throughout th xcution of th algorithm. To this nd, w will show th following uppr bound on th cardinality H and th capacity u(h) of th st of forbiddn dgs, whos proof w postpon until th nxt sction: 15

17 Input : A graph G =(V,E) with capacitis {u }, a targt flow valu F, dg wights {w }, and a st H of forbiddn dgs Output: Eithraflowf and a st H of forbiddn dgs, or fail indicating that F>F 8m1/3 ln 1/3 m r 1 u w + w 1 3m for ach E \H Find an approximat lctrical flow f using Thorm.3 on G H := (V,E \ H) with rsistancs r, targt flow valu F,andparamtrδ = /3. if E r ( f) > (1 + ) w 1 or s and t ar disconnctd in G H if thr xists with cong f () >thn add to H and start ovr rturn f thn rturn fail Figur 4: Th modifid oracl O usd by our improvd algorithm Input : A graph G =(V,E) with capacitis {u }, and a targt flow valu F ; Output: Eithraflowf, or fail indicating that F>F ; Initializ w 0 1 for all dgs, H, 8m1/3 ln 1/3 m,and N ln m for i := 1,...,N do Qury O with dg wights givn by w i 1, targt flow valu F, and forbiddn dg st H if O rturns fail thn rturn fail ls Lt f i b th rturnd answr Rplac H with th rturnd (augmntd) st of forbiddn dgs w i w i 1 (1 + cong f i()) for ach E nd nd rturn f (1 ) (1+)N ( i f i ) Figur 5: An improvd (1 O())-approximation algorithm for th maximum flow problm 16

18 Lmma 4.1. Throughout th xcution of th algorithm, and H u(h) 30m ln m 30mF ln m 3. If w plug in th valu =(8m 1/3 ln 1/3 m)/ usd by th algorithm, Lmma 4.1 givs th bounds H 15 3 (m ln m)1/3 and u(h) 15 56F < F/1. Givn th abov lmma, it is now straightforward to show th following thorm, which stablishs th corrctnss and bounds th running tim of our algorithm. Thorm 4.. For any 0 <<1/, iff F th algorithm in Figur 5 will rturn a fasibl s-t flow f of valu f =(1 O())F in tim O m 4/3 3. Proof. To bound th running tim, w not that, whnvr w invok th algorithm from Thorm.3, w ithr advanc th numbr of itrations or w incras th cardinality of H, sothnumbr of linar systms w solv is at most N + H N (m ln m)1/3. Equation (8) implis that th valu of R from Thorm.3 is O((m/) O(1) ), so solving ach linar systm taks tim at most O (m log 1/). This givs an ovrall running tim of O N (m ln m)1/3 m = O m 4/3 3, as claimd. It thus rmains to prov corrctnss. For this, w nd to show that if F F, thn th oracl dos not rturn fail, which would occur if w disconnct s from t or if E r ( f) > (1 + ) w 1.By Lmma 4.1 and th commnt following it, w know that throughout th whol algorithm G H has maximum flow valu of at last F F/1 (1 /1) F and thus, in particular, w will nvr disconnct s from t. Furthrmor, this implis that thr xists a fasibl flow in our graph of valu (1 /1) F, vn aftr w hav rmovd th dgs in H. Thr is thus a flow of valu F in which vry dg has congstion at most 1/ (1 /1). W can thrfor us th argumnt from Sction 3. (Equation (3) and th lins dirctly prcding it) to show that w always hav as rquird. E r ( f) (1 + /1) (1 + /3) w 1 (1 + ) w 1, Th abov thorm allows us to apply th binary sarch stratgy that w usd in Sction 3.1. This yilds our main thorm: Thorm 4.3. For any 0 <<1/, th maximum flow problm can b (1 )-approximatd in O m 4/3 3 tim. 17

19 4.3 Th Proof of Lmma 4.1 All that rmains is to prov th bounds givn by Lmma 4.1 on th cardinality and capacity of H. To do so, w will us th ffctiv rsistanc of th circuits on which w comput lctrical flows as a potntial function. Th ky insight is that w only cut an dg whn its flow accounts for a nontrivial fraction of th nrgy of th lctrical flow, and that cutting such an dg will caus a substantial chang in th ffctiv rsistanc. Combining this with a bound on how much th ffctiv rsistanc can chang during th xcution of th algorithm will guarant that w won t cut too many dgs. Lt r j b th rsistancs usd in th j th lctrical flow computd during th xcution of th algorithm 5. If an dg is not in E or if has bn addd to H by stp j, str j =. Wdfin th potntial function Φ(j) =R ff (r j )=E r j(f r j), whr f r j is th (xact) lctrical flow of valu 1 arising from r j. Lmma 4.1 will follow asily from: Lmma 4.4. Suppos that F F mf. Thn: 1. Φ(j) nvr dcrass during th xcution of th algorithm.. Φ(1) m 4 F. 3. If w add an dg to H btwn stps j 1 and j, thn (1 5m )Φ(j) > Φ(j 1). Proof. Proof of (1) Th only way that th rsistanc r j of an dg can chang is if th wight w is incrasd by th multiplicativ wights routin, or if is addd to H so that r j is st to. As such, th rsistancs ar nondcrasing during th xcution of th algorithm. By Rayligh Monotonicity (Corollary.5), this implis that th ffctiv rsistanc is nondcrasing as wll. Proof of () In th first linar systm, H = and r 1 = 1+/3 for all E. Lt (S, V \S) b th minimum s-t cut u of G. By th Max Flow-Min Cut Thorm ([10, 8]), w know that th capacity u(s) = E(S) u of this cut is qual to F. In particular, r 1 = 1+/3 u 1+/3 F > 1 F for all E(S). As f r 1 is an lctrical s-t flow of valu 1, it snds 1 unit of nt flow across (S, V \ S); so, som dg E(S) must hav f r 1( ) 1/m. This givs Φ(1) = E r 1(f r 1)= E Sinc F mf by assumption, th dsird inquality follows. f r 1() r 1 f r 1( ) r 1 > 1 m F. (11) 5 Not that r j is not just th st of rsistancs arising from w j, sinc a singl call to th oracl may comput multipl lctrical flows as dgs ar addd to H. 18

20 Proof of (3) Suppos w add th dg h to H btwn stps j 1 and j. W will show that h accounts for a substantial fraction of th total nrgy of th lctrical flow with rspct to th rsistancs r j 1, and our rsult will thn follow from Lmma.6. Lt w b th wights usd at stp j 1, and lt f b th flow w computd in stp j 1. Bcaus w addd h to H, w know that cong f (h) >. Sinc our algorithm did not rturn fail aftr computing this f, w must hav that E r j 1( f) (1 + ) w 1. (1) Using th dfinition of r j 1 h, th fact that cong f (h) >, and Equation (1), w obtain: f(h) r j 1 h = f(h) w + w 1 3m u f(h) w 1 3mu = f(h) w 1 3m u = 3(1 + )m cong f (h) ((1 + ) w 1 ) > 3(1 + )m E r j 1( f). Th abov inqualitis stablish that dg h accounts for mor than a 3(1+)m fraction of th total nrgy E r i( f) of th flow f. Th flow f is th approximat lctrical flow computd by our algorithm, but our argumnt will rquir that an inquality lik this holds in th xact lctrical flow f r j 1. This is guarantd by part b of Thorm.3, which, along with th facts that E r ( f) E r (f r j 1), 1, and <1/, givs us that f r j 1(h) r j 1 h > f(h) r j 1 h /3 mr E r (f r j 1) > Th rsult now follows from Lmma.6. W ar now rady to prov Lmma (1 + )m /3 mr E r (f r j 1) > 5m E r (f r j 1). Proof of Lmma 4.1. Lt k b th cardinality of th st H at th nd of th algorithm. Lt f b th flow that was producd by our algorithm just bfor k-th dg was addd to H, ltj b th tim whn this flow was output, and lt w b th corrsponding wights. As th nrgy rquird by an s-t flow scals with th squar of th valu of th flow, Φ(j) E r j(f) F E r j( f) F. (13) 19

21 By th construction of our algorithm, it must hav bn th cas that E r j( f) (1 + ) w 1. This inquality togthr with quation (13) and part of Lmma 4.4 implis that Φ(j) =E r j(f r j) E r j( f) F (1 + ) w 1 m 4 Φ(1). Now, sinc up to tim j w had k 1 additions of dgs to H, parts 1 and 3 of Lmma 4.4, and Lmma 3.4 imply that (k 1) 1 Φ(j) 5m Φ(1) (1 + ) w 1m 4 (1 + )m 4 m xp (1+) N m 5 xp(3 1 ln m), whr th last inquality usd th fact that <1/. Rarranging th trms in th abov inquality givs us that k ln + 5 ln m +3 1 ln m +1< 6 1 ln m < ln 1 5m ln 1 5m 30m ln m, whr w usd th inqualitis <1/ and log(1 c) < c for all c (0, 1). This stablishs our bound on cardinality of th st H. To bound th valu of u(h), lt us not that w add an dg to H only whn w snd at last u units of flow across it. But sinc w nvr flow mor than F units across any singl dg, w hav that u F/. Thrfor, w may conclud that u(h) H F 30mF ln m 3, as dsird. 4.4 Improving th Running Tim to O mn 1/3 11/3 W can now combin our algorithm with xisting mthods to furthr improv its running tim. In [1] (s also [5]), Kargr prsntd a tchniqu, which h calld graph smoothing, that allows on to us random sampling to spd up an xact or (1 )-approximat flow algorithm. Mor prcisly, his tchniqus yild th following thorm, which is implicit in [1] and statd in a mor similar form in [5]: Thorm 4.5 ([1, 5]). Lt T (m, n, ) b th tim ndd to find a (1 )-approximatly maximum flow in an undirctd, capacitatd graph with m dgs and n vrtics. Thn on can obtain a (1 )-approximatly maximal flow in such a graph in tim O m/n T ( O n,n,ω()). By applying th abov thorm to our O m 4/3 3 algorithm, w obtain our dsird running tim bound: Thorm 4.6. For any 0 <<1/, th maximum flow problm can b (1 )-approximatd in O mn 1/3 11/3 tim. 0

22 4.5 Approximating th Valu of th Maximum s-t Flow in Tim O m + n 4/3 8/3 Givn any wightd undirctd graph G =(V,E,w) with n vrtics and m dgs, Bnczúr and Kargr [4] showd that on can construct a graph G =(V,E,w ) (calld a sparsifir of G) on th sam vrtx st in tim O (m) such that E = O(n log n/ ) and th capacity of any cut in G is btwn 1 and (1+) tims its capacity in G. Applying our algorithm from Sction 4 to a sparsifir of G givs us an algorithm for (1 )-approximating th valu of th maximum s-t flow on G in tim O m + n 4/3 3. W not that this only allows us to approximat th valu of th maximum s-t flow on G. It givs us a flow on G, not on on G. W do not know how to us an approximatly maximum s-t flow on G to obtain on on G in lss tim than would b rquird to comput a maximum flow in G from scratch using th algorithm from Sction 4. For this rason, thr is a gap btwn th tim w rquir to find a maximum flow and th tim w rquir to comput its valu. W not, howvr, that this gap will not xist for th minimum s-t cut problm, sinc an approximatly minimum s-t cut on G will also b an approximatly minimum s-t cut G. W will prsnt an algorithm for finding such a cut in th nxt sction. By th Max Flow-Min Cut Thorm, this will provid us with an altrnat algorithm for approximating th valu of th maximum s-t flow. It will hav a slightly bttr dpndnc on, which will allow us to approximat th valu of th maximum s-t flow in tim O m + n 4/3 8/3. 5 A Dual Algorithm for Finding an Approximatly Minimum s-t Cut in Tim O m + n 4/3 8/3 In this sction, w ll dscrib a dual prspctiv that yilds to an vn simplr algorithm for computing an approximatly minimum s-t cut. Rathr than using lctrical flows to obtain a flow, it will us th lctrical potntials to obtain a cut. Th algorithm will schw th oracl abstraction and multiplicativ wights machinry. Instad, it will just rpatdly comput an lctrical flow, incras th rsistancs of dgs according to th amount flowing ovr thm, and rpat. It will thn us th lctrical potntials of th last flow computd to find a cut by picking a cutoff and splitting th vrtics according to whthr thir potntials ar abov or blow th cutoff. Th algorithm is shown in Figur 6. It finds a (1 + )-approximatly minimum s-t cut in tim O m 4/3 8/3 ; applying it to a sparsifir will giv us: Thorm 5.1. For any 0 < < 1/7, w can find a (1 + )-approximat minimum s-t cut in O m + n 4/3 8/3 tim. W not that, in this algorithm, thr is no nd to dal xplicitly with dgs flowing mor than, maintain a st of forbiddn dgs, or avrag th flows from diffrnt stps. W will sparatly study dgs with vry larg flow in our analysis, but th algorithm itslf avoids th complxitis that appard in th improvd flow algorithm dscribd in Sction 4. W furthr not that th updat rul is slightly modifid from th on that appard arlir in th papr. This is don to guarant that th ffctiv rsistanc incrass substantially whn som dg flows mor than, without having to xplicitly cut it. Our prvious rul allowd th wight (but not rsistanc) of an dg to constitut a vry small fraction of th total wight; in this cas, a significant multiplicativ incras in th wight of an dg may not produc a substantial chang in th ffctiv rsistanc of th graph. 1

23 Input : A graph G =(V,E) with capacitis {u }, and a targt flow valu F Output: Acut(S, V \ S) Initializ w 0 1 for all dgs, 3m 1/3 /3, N 5 8/3 m 1/3 ln m, andδ. for i := 1,...,N do Find an approximat lctrical flow f i 1 and potntials φ using Thorm.3 on G with rsistancs r i 1 µ i 1 wi 1 w i w i 1 = wi 1 u, targt flow valu F,andparamtrδ. + cong f i 1()w i 1 + m µi 1 for ach E Scal and translat φ so that φ s =1and φ t =0 Lt S x = {v V φ v >x} St S to b th st S x that minimizs (S x,v \ S x ) If th capacity of (S x,v \ S x ) is lss than F/(1 7), rturn (S x,v \ S x ). nd rturn fail Figur 6: A dual algorithm for finding an s-t cut 5.1 An Ovrviw of th Analysis To analyz this algorithm, w will track th total wight placd on th dgs crossing som minimum cut. Th basic obsrvation for our analysis is that th sam amount of nt flow must b snt across vry cut, so dgs in small cuts will tnd to hav highr congstion than dgs in larg cuts. Sinc our algorithm incrass th wight of an dg according to its congstion, this will caus our algorithm to concntrat a largr and largr fraction of th total wight on th small cuts of th graph. This will continu until almost all of th wight is concntratd on approximatly minimum cuts. Of cours, th dgs crossing a minimum cut will also cross many othr (likly much largr) cuts, so w crtainly can t hop to obtain a graph in which no dg crossing a larg cut has nonngligibl wight. In ordr to formaliz th abov argumnt, w will thus nd som good way to masur th xtnt to which th wight is concntratd on approximatly minimum cuts. In Sction 5., w will show how to us ffctiv rsistanc to formulat such a notion. In particular, w will show that if w can mak th ffctiv rsistanc larg nough thn w can find a cut of small capacity. In Sction 5.3, w will us an argumnt lik th on dscribd abov to show that th rsistancs producd by th algorithm in Figur 6 convrg aftr N = O m 1/3 8/3 stps to on that mts such a bound. 5. Cuts, Elctrical Potntials, and Effctiv Rsistanc During th algorithm, w scal and translat th potntials of th approximat lctrical flow so that φ s =1and φ t =0. W thn produc a cut by choosing x [0, 1] and dividing th graph into th sts S = {v V φ v >x} and V \ S = {v V φ v x}. Th following lmma uppr bounds th capacity of th rsulting cut in trms of th lctrical potntials and dg capacitis. Lmma 5.. Lt φ b as abov. Thn thr is a cut S x of capacity at most φ u φ v u (u,v). (14) (u,v) E

24 Proof. Considr choosing x [0, 1] uniformly at random. Th probability that an dg (u, v) is cut is prcisly φ(u) φ(v). So, th xpctd capacity of th dgs in a random cut is givn by (14), and so thr is a cut of capacity at most (14). Now, suppos that on has a fixd total amount of rsistanc µ to distribut ovr th dgs of a cut of siz F. It is not difficult to s that th maximum possibl ffctiv rsistanc btwn s and t in such a cas is µ, and that this is achivd whn on puts a rsistanc of µ F F on ach of th dgs. This suggsts th following lmma, which bounds th quantity in Lmma 5. in trms of th ffctiv rsistanc and th total rsistanc (appropriatly wightd whn th dgs hav non-unit capacitis): Lmma 5.3. Lt µ = u r, and lt th ffctiv s-t rsistanc of G with dg rsistancs givn by r b R ff (r). Lt φ b th potntials of th lctrical s-t flow, scald to hav potntial drop 1 btwn s and t. Thn µ φ()u R ff (r). E If, φ is an approximat lctrical potntial rturnd by th algorithm of Thorm.3 whn run with paramtr δ 1/3, r-scald to hav potntial diffrnc 1 btwn s and t, thn µ φ()u (1 + δ) R ff (r). E Proof. By Fact.4, th rscald tru lctrical potntials corrspond to a flow of valu 1/R ff (r) and φ() 1 = r R ff (r). So, w can apply th Cauchy-Schwarz inquality to prov φ() φ()u u r r µ = R ff (r). By parts a and c of Thorm.3, aftr w rscal φ to hav potntial drop 1 btwn s and t, it will hav nrgy φ() 1+δ 1 r 1 δ R ff (r) (1 + 3δ) 1 R ff (r), for δ 1/3. Th rst of th analysis follows from anothr application of Cauchy-Schwarz. 5.3 Th Proof that th Dual Algorithm Finds an Approximatly Minimum Cut W ll show that if F F thn within N =5 8/3 m 1/3 ln m itrations, th algorithm in Figur 6 will produc a st of rsistancs r i such that R ff (r i ) (1 7) µi (F ). (15) 3

25 Onc such a st of rsistancs has bn obtaind, Lmmas 5. and 5.3 tll us that th bst potntial cut of φ will hav capacity at most 1+δ F F. Th algorithm will thn rturn this cut. Lt C b th st of dgs crossing som minimum cut in our graph. Lt u C = F dnot th capacity of th dgs in C. W will kp track of two quantitis: th wightd gomtric man of th wights of th dgs in C, 1/uC ν i = w i u, and th total wight µ i = C w i = r i u of th dgs of th ntir graph. Clarly ν i max C w i. In particular, ν i µ i for all i. Our proof that th ffctiv rsistanc cannot rmain larg for too many itrations will b similar to our analysis of th flow algorithm in Sction 4. W suppos to th contrary that Rff i st (1 7) µi (F ) for ach 1 i N. W will show that, undr this assumption: 1. Th total wight µ i dosn t gt too larg ovr th cours of th algorithm [Lmma 5.4].. Th quantity ν i incrass significantly in any itration in which no dg has congstion mor than [Lmma 5.5]. Sinc µ i dosn t gt too larg, and ν i µ i, this will not happn too many tims. 3. Th ffctiv rsistanc incrass significantly in any itration in which som dg has congstion mor than [Lmma 5.6]. Sincµ i dos not gt too larg, and th ffctiv rsistanc is assumd to b boundd in trms of th total wight µ i, this cannot happn too many tims. Th combind bounds from and 3 will b lss than N, which will yild a contradiction. Lmma 5.4. For ach i N such that R ff (r i ) (1 7) µi F, (1 ) µ i+1 µ i xp. So, if for all i N w hav R ff (r i ) (1 7) µi, thn F (1 ) µ N µ 0 xp N. (16) 4