1 Solving Flow Problems using Multiplicative Weights

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1 15-850: Advancd Algorithms CMU, Spring 2017 Lctur #13: Max Flow using Elctrical Flow Fbruary 22, 2017 Lcturr: Anupam Gupta Scrib: Jnnifr Iglsias, Xu An Chuang In today s lctur w will us lctrical flows algorithms to find approximat max-flows in (unitcapacity) undirctd graphs in Õ(m4/3 /poly() tim. As mntiond on th blog, this approach can b xtndd to all undirctd graphs, and th runtim can b improvd to Õ(mn1/3 /poly(). At th tim this rsult was announcd (in 2011), it was th fastst algorithm for th problm. 1 Solving Flow Problms using Multiplicativ Wights W shall summariz th last points of th prvious lctur, which ar also mntiond hr. Rmmbr that w dfind K asy to b: K = {f f P 0, P f P = F } whr w hav a variabl f P for vry s, t path, P. Constraints ar of th form f := f P 1 E P : P Hncforth w will us th shorthand f := P : P f P (th flow on dg ). W hav an oracl which givn wights p for th dgs, rturns a flow f K such that ( ) E p f E W dfin th width of th oracl as th smallst ρ such that p max f ρ. (13.1) W saw that using th multiplicativ wights (MW) algorithm, w find a (1 + )-approximat max flow ˆf i.., a flow of valu F that has ˆf 1 + using O( ρ log m ) calls to th oracl. Th high 2 lvl ida is that w snd flow of F along th shortst path, whr lngth of a path P is dfind as P p, updat th wights and thn rpat. During th prvious lctur, w saw that using shortst-path routing, you can gt ρ = F. Sinc w mf can us Dijkstra s O(m + n log n) to implmnt th oracl, this givs an Õ( ) tim algorithm. 2 Rlaxd Oracl: For th rst of this sction, w ar going to rlax th rquirmnts for th oracl, so that w mrly want th flow to satisfy th capacity constraints approximatly: p + ( ) E p f (1 + ) E This will b usful in rducing th width from F down to Õ(m1/2 ) and vn lowr. 2 Rviw of Elctrical Ntworks Givn an undirctd graph, w can considr it to b an lctrical circuit as shown in Figur 13.1: ach dg of th original graph rprsnts a rsistor, and w connct (say, a 1-volt) battry btwn s to t. This causs lctrical currnt to flow from s (th nod with highr potntial) to t. 1

2 2.1 Elctrical Flows How do w figur out what this lctrical flow is going to b? W us th following laws w know to hold about lctrical flows. φ s = 1 φ t = 0 s t + - Figur 13.1: Th currnts on th wirs would produc an lctric flow (whr all th wirs within th graph hav rsistanc 1). Thorm 13.1 (Ohm s Law). If = (u, v) is an dg, th lctrical flow f uv on this dg is th ratio btwn th diffrnc in potntial (or voltag) and th rsistanc of th dg, whr r is th rsistanc of dg. Thorm 13.2 (Kirchoff s Voltag Law). Along a cycl, th dirctd potntial changs along th cycl sum to 0. Thorm 13.3 (Kirchoff s Currnt Law). Th sum of th currnts ntring a nod is th sam as th sum of th currnts laving a nod; i.., thr is flow-consrvation at th nods. Ths laws giv us a st of linar constraints on th lctrical flow valus f, and solving this systm of linar constraints tlls us what th lctrical flows f ar. (For an xampl, s Wikipdia.) Th Laplacian It turns out that w can writ ths linar constraints obtaind abov as follows, if w introduc a convnint matrix, calld th graph Laplacian. Givn an undirctd graph on n nods and m dgs, w dfin th Laplacian matrix to b a n n matrix, L. L uv dg(u) 1 if u = v if (u, v) E othrwis Anothr way to writ this matrix is L uv = ( u v ) T ( u v ). In a gnral graph G, w dfin th Laplacian to b: L(G) = (u,v) E L uv 2

3 whr A is th adjacncy matrix. If w tak th 6-nod graph in Figur 13.1, th rsulting Laplacian L uv is givn blow. s t u v w x s t u L uv = v w From th prviously mntiond Ohm s and Kirchoff s, w gt: If w snd on unit of currnt from s to t, th voltag vctor φ = (φ v ) v V is obtaind by solving th linar systm Lφ = ( s t ) Onc w hav φ, w can us Ohm s law to gt th currnt flowing on ach dg. So now it rmains to find th voltags for ach of th vrtics. If w want to snd F units of flow from s to t, w solv th systm Lφ = F ( s t ), and lt f uv = φu φv r uv. How do w solv th linar systm Lφ = b? W can us Gaussian limination, of cours, but thr ar fastr mthods: w ll discuss this in Sction Elctrical Flows Minimiz Enrgy Burn Hr s anothr usful way of charactrizing this flow. If w st voltags at s and t, this causs som amount I of flow to go btwn s to t. But how do ths I units of flow split up? This flow happns to b th on that minimizs th total nrgy dissipatd by th flow (subjct to th flow valu bing I). Indd, for a flow f, th nrgy burn on dg is fuvr 2 uv = (φu φv)2 r uv, and th total nrgy burn is E(f) = f 2 r E Th lctrical flow producd happns to b arg min fis an s-t flow of valu I {E(f)}. Not that if G has unit rsistanc on all dgs, thn E(f) = (u,v) E (φ u φ v) 2 r uv = φ T Lφ Solving Linar Systms How do w solv th linar systm Lx = b fast? For th cas whn L is a Laplacian matrix, w can do things much fastr than Gaussian limination ssntially do it in tim nar-linar in th numbr of non-zros of th matrix L. Thorm 13.4 ([KMP10, KMP14]). Suppos w ar givn a linar systm Lx = b for th cas whr L is a Laplacian matrix, with solution x. Thn w can find a vctor ˆx in tim O(m log 2 n log(1/) such that th rror z := Lˆx b satisfis z Lz ( x L x). Som history: Spilman and Tng [ST04, ST14] gav an algorithm that taks tim about O(m 2 log n log log n ). Koutis, Millr, and Png [KMP10] improvd this to O(m log 2 n); th currnt bst tim is O(m log n). This can b convrtd to what w nd: 3

4 Thorm 13.5 ([CKM + 11]). Thr is an algorithm givn a linar systm Lx = b (for L bing a Laplacian matrix), outputs in O( m log R δ ) tim a flow f that satisfis E(f) (1 + δ)e( f), whr f is th min-nrgy flow, and R is th ratio btwn th largst and smallst rsistancs in th ntwork. For th rst of this lctur w will assum that givn a linar systm Lx = b, w can comput th corrsponding min-nrgy flow xactly in tim Õ(m). Th argumnt can b xtndd to incorporat th rrors, tc., fairly asily. 3 Obtaining an Õ(m3/2 ) tim Flow Algorithm W will assum that th flow instanc is fasibl, i.., that thr is som flow f which is in K and satisfis all th dg constraints. Rcall, our oracl taks as input p m = {x [0, 1] m : x = 1}. W show how to implmnt an oracl that satisfis th wakr rquirmnt ( ) and that has width O( m/). Dfin rsistancs r = p + m for all dgs. comput currnts f by solving th linar systm Lφ = F ( s t ) and rturn th rsultant flow f. This ida of stting th rsistanc to b p plus a small rror trm is usful in controlling th width in non-lctrical flows too. Thorm If f is a flow with valu F and f is th minimum nrgy flow (which is rturnd by th oracl), thn 1. E p f (1 + ) E p, 2. max f O( m/). Proof. Rcall that flow f satisfis all th constraints, and that f is th minimum nrgy flow. Thn E(f ) = (f ) 2 r r = (p + ) = (1 + ). m Hr w us that p = 1. But sinc f is th flow K that minimizs th nrgy, E(f) E(f ) 1 +. By Cauchy-Schwarz, w hav that: p f ( p f 2 )( p ) This provs th first part of th thorm. For th scond part, look at th nrgy burnd on : this is f 2 r f 2 m. Th total nrgy burnd is mor than th nrgy on that singl dg, so w gt that f 2 m E(f) 1 +, and hnc This provs th thorm. (1 + )m f 4

5 Using this oracl with th MW framwork givs an algorithm which runs in tim Õ(m3/2 5/2 ). Indd, w hav to run ρ log m itrations, whr th width ρ is O( m/), and ach itration taks 2 Õ(m) tim du to Thorm And this runtim is tight; s,.g., th xampl hr. Unfortunatly, this runtim of O(m 3/2 ) is not that imprssiv: in th 1970s, Karzanov [Kar73], and Evn and Tarjan [ET75] showd how to find maximum flows xactly in unit-capacity graphs in tim O(m min(m 1/2, n 2/3 )). And similar runtims wr givn for th capacitatd problm by Goldbrg and Rao in th lat 1990s. Thankfully, w can tak th lctrical flows ida vn furthr, as w show in th nxt sction. 4 A Fastr Algorithm Our targt is to find an oracl with width Th two main idas ar: ρ = m1/3 log m. 1. W find lctrical flows, but if any dg has mor than ρ flow, thn w kill that dg (st its rsistanc to ). W show that w don t kill too many dgs lss than F dgs. 2. Each tim w kill an dg, w will show that th ffctiv rsistanc btwn s and t incrass by a lot ach tim an dg is killd. A coupl obsrvations and assumptions: 1. W assum that F ρ 2. Instad of using th multiplicativ wights procss as a black box, w will xplicitly maintain dg wights w t. W us th notation W t := wt. Now w will nd to dfin th ffctiv rsistanc btwn nods u and v. This is th rsistanc offrd by th whol ntwork to lctrical flows btwn u and v. Thr ar many ways of formalizing this, w ll us th on that is most usful in this contxt. Dfinition 13.7 (Effctiv Rsistanc). Th ffctiv rsistanc btwn s and t, dnotd R st ff is th nrgy burn if w snd 1 unit of lctrical currnt from s to t. Sinc w only considr th ffctiv rsistanc btwn s and t, w simply writ R ff. Lmma 13.8 ([CKM + 11]). Considr an lctrical ntwork with dg rsistancs r. 1. (Rayligh Monotonicity) If w chang th rsistancs to r r for all thn R ff R ff. 2. Suppos f is an s-t lctrical flow, suppos is an dg such that f 2 r βe(f). If w st r =, thn R ff ( R ff 1 β ). W ll skip th proof of this (simpl) lmma. Lt s giv our algorithm. W start off with wights w 0 = 1 for all E. At stp t of th algorithm: Find th min-nrgy flow f t of valu F with rspct to dg rsistancs r t = w t + m W t. 5

6 If thr is an dg with f t > ρ, dlt, rcomput th flow f t as in th abov stp. Els updat th wights w t+1 Stop aftr T = ρ log m 2 w t (1 + ρ f t ). itrations, and output ˆf = 1 T t f t. W want to argu lik for Thorm 13.6, but not that th procss dlts dgs along th way, which w nd to tak car of. Claim Suppos 1/10. If w dlt at most F dgs from th graph, th following hold: 1. th flow f t at stp t has nrgy E(f t ) (1 + 3)W t. 2. wt f t (1 + 3)W t 2W t. 3. If ˆf K is th flow vntually rturnd, thn ˆf (1 + O()). Proof. Rmmbr thr xists a flow f of valu F that rspcts all capacitis. Dlting F dgs 1 mans thr xists a capacity-rspcting flow of valu at last (1 )F. Scaling up by (1 ), thr xists a flow f 1 of valu F that uss ach dg to xtnt (1 ). Th nrgy of this flow according to rsistancs r t is at most E(f ) = r(f t ) 2 1 (1 ) 2 r t W t (1 ) 2 (1 + 3)W t, for small nough. Sinc w find th minimum nrgy flow, E(f t ) E(f ) W t (1 + 3). For th scond part, wf t t ( w)( t w(f t ) t 2 ) W t W t (1 + 3) (1 + 3)W t 2W t. Th last stp is vry loos, but it will suffic for our purposs. To calculat th congstion of th final flow, obsrv that vn though th algorithm abov xplicitly maintains wights, w can just appal dirctly to th MW algorithm guarant. Th ida is simpl: dfin p t = wt W, and thn th flow f t satisfis t p t f t for prcisly th p t valus that MW would rturn if w gav it th flows f 0, f 1,..., f t 1. Using th MW guarants, th avrag flow ˆf uss any dg to at most (1 + 3) +. Finally, all ths calculations assumd w did not dlt too many dgs. Lt us show that is indd th cas. Claim W dlt at most F dgs. Proof. Th proof tracks two things, th total wight W t and th s-t ffctiv rsistanc R ff. First th wight: w start at W 0 = m. Whn w do an updat, W t+1 = ( w t 1 + ) ρ f t = W t + w ρ f t t W t + ρ (2W t ) (From Claim 13.9) 6

7 Hnc w gt that for T = ρ ln m 2, W T W 0 Now for th s-t ffctiv rsistanc R ff. ( ) T ( ) 2 T m xp = m xp ρ ρ ( 2 ln m Initially, sinc w snd F flow, thr is som dg with at last F/m flow on it, and hnc with nrgy burn (F/m) 2. So R ff at th start is at last (F/m) 2 1/m 2. Evry tim w do an updat, th wights incras, and hnc R ff dos not dcras. (This is why w argud about th wights w t xplicitly, and not just th probabilitis p t.) Evry tim w dlt an dg, it has flow at last ρ, and hnc nrgy burn at last (ρ 2 )w t (ρ 2 ) m W t. Th total nrgy is at most 2W t from Claim This mans it was burning at last β = ρ2 fraction of th total nrgy. Hnc ( Rff nw Rold ff ρ 2 ) (1 ρ2 ) Rold ff xp if w us 1 1 x x/2 whn x [0, 1/4]. For th final ffctiv rsistanc, not that w snd F flow with total nrgy burn 2W T ; sinc th nrgy dpnds on th squar of th flow, w hav R final ff 2W T 2W T. F 2 Obsrv that all ths calculations dpndd on us not dlting mor than F dgs. So lt s prov that this is indd th cas. If D dgs ar dltd in th T stps, thn w gt ( ) ( ) Rff 0 xp D ρ2 R final 2 ln m ff 2W T xp. Taking logs and simplifying, w gt that ρ 2 D = D ρ 2 ln(3 ) + 2 ln m ( ) O(ln m)(1 + ) m 1/3 F. So, D is small nough as dsird, and w don t rmov too many dgs. To nd, lt us not that th analysis is tight; s,.g., th scond xampl hr. Rfrncs [CKM + 11] Paul Christiano, Jonathan A. Klnr, Alksandr Madry, Danil A. Spilman, and Shang-Hua Tng. Elctrical flows, laplacian systms, and fastr approximation of maximum flow in undirctd graphs. In STOC 11, pags , Nw York, NY, USA, ACM. 13.5, 13.8 ). [ET75] S. Evn and R.E. Tarjan. Ntwork flow and tsting graph connctivity. SIAM Journal of Computing, 4: ,

8 [Kar73] A.V. Karzanov. Tochnaya otsnka algoritma nakhozhdniya maksimal nogo potoka, primnnnogo k zadach o prdstavitlyakh. Voprosy Kibrntiki. Trudy Sminara po Kombinatorno Matmatik, Titl transl.: An xact stimat of an algorithm for finding a maximum flow, applid to th problm on rprsntativs, In: Issus of Cybrntics. Proc. of th Sminar on Combinatorial Mathmatics. 3 [KMP10] [KMP14] Ioannis Koutis, Gary L. Millr, and Richard Png. Approaching optimality for solving sdd linar systms. In FOCS 10, pags , Washington, DC, USA, IEEE Computr Socity. 13.4, Ioannis Koutis, Gary L. Millr, and Richard Png. Approaching optimality for solving SDD linar systms. SIAM J. Comput., 43(1): , [ST04] Danil A. Spilman and Shang-Hua Tng. Narly-linar tim algorithms for graph partitioning, graph sparsification, and solving linar systms. In STOC 04, pags 81 90, [ST14] Danil A. Spilman and Shang-Hua Tng. Narly linar tim algorithms for prconditioning and solving symmtric, diagonally dominant linar systms. SIAM J. Matrix Analysis Applications, 35(3): ,

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