18 Extensions of Maximum Flow

Transcription

1 Who are you?" aid Lunkwill, riing angrily from hi ea. Wha do you wan?" I am Majikhie!" announced he older one. And I demand ha I am Vroomfondel!" houed he younger one. Majikhie urned on Vroomfondel. I alrigh," he explained angrily, you don need o demand ha." Alrigh!" bawled Vroomfondel banging on an nearby dek. I am Vroomfondel, and ha i no a demand, ha i a olid fac! Wha we demand i olid fac!" No we don!" exclaimed Majikhie in irriaion. Tha i preciely wha we don demand!" Scarcely pauing for breah, Vroomfondel houed, We don demand olid fac! Wha we demand i a oal abence of olid fac. I demand ha I may or may no be Vroomfondel!" Dougla Adam, The Hichhiker Guide o he Galaxy (199) 18 Exenion of Maximum Flow 18.1 Maximum Flow wih Edge Demand Now uppoe each direced edge e in ha boh a capaciy c(e) and a demand d(e) c(e), and we wan a flow f of maximum value ha aifie d(e) f (e) c(e) a every edge e. We call a flow ha aifie hee conrain a feaible flow. In our original eing, where d(e) = 0 for every edge e, he zero flow i feaible; however, in hi more general eing, even deermining wheher a feaible flow exi i a nonrivial ak. Perhap he eaie way o find a feaible flow (or deermine ha none exi) i o reduce he problem o a andard maximum flow problem, a follow. The inpu coni of a direced graph G = (V, E), node and, demand funcion d : E IR, and capaciy funcion c : E IR. Le D denoe he um of all edge demand in G: D := d(u v). u v E We conruc a new graph G = (V, E ) from G by adding new ource and arge verice and, adding edge from o each verex in V, adding edge from each verex in V o, and finally adding an edge from o. We alo define a new capaciy funcion c : E IR a follow: For each verex v V, we e c ( v) = u V d(u v) and c (v ) = w V d(v w). For each edge u v E, we e c (u v) = c(u v) d(u v). Finally, we e c ( ) =. Inuiively, we conruc G by replacing any edge u v in G wih hree edge: an edge u v wih capaciy c(u v) d(u v), an edge v wih capaciy d(u v), and an edge u wih capaciy d(u v). If hi conrucion produce muliple edge from o he ame verex v (or o from he ame verex v), we merge hem ino a ingle edge wih he ame oal capaciy. In G, he oal capaciy ou of and he oal capaciy ino are boh equal o D. We call a flow wih value exacly D a auraing flow, ince i aurae all he edge leaving or enering. If G ha a auraing flow, i mu be a maximum flow, o we can find i uing any max-flow algorihm. Lemma 1. G ha a feaible (, )-flow if and only if G ha a auraing (, )-flow. 1

2 ' ' A flow nework G wih demand and capaciie (wrien d.. c), and he ranformed nework G. Proof: Le f : E IR be a feaible (, )-flow in he original graph G. Conider he following funcion f : E IR: f (u v) = f (u v) d(u v) f ( v) = d(u v) u V f (v ) = d(u w) w V f ( ) = f for all u v E for all v V for all v V We eaily verify ha f i a auraing (, )-flow in G. The admiibiliy of f implie ha f (e) d(e) for every edge e E, o f (e) 0 everywhere. Admiibiliy alo implie f (e) c(e) for every edge e E, o f (e) c (e) everywhere. Tediou algebra implie ha f (u v) = f (v w) u V w V for every verex v V (including and ). Thu, f i a legal (, )-flow, and every edge ou of or ino i clearly auraed. Inuiively, f diver d(u v) uni of flow from u direcly o he new arge, and injec he ame amoun of flow ino v direcly from he new ource. The ame ediou algebra implie ha for any auraing (, )-flow f : E IR for G, he funcion f = f E + d i a feaible (, )-flow in G. Thu, we can compue a feaible (, )-flow for G, if one exi, by earching for a maximum (, )- flow in G and checking ha i i auraing. Once we ve found a feaible (, )-flow in G, we can ranform i ino a maximum flow uing an augmening-pah algorihm, bu wih one mall change. To enure ha every flow we conider i feaible, we mu redefine he reidual capaciy of an edge a follow: c(u v) f (u v) if u v E, c f (u v) = f (v u) d(v u) if v u E, 0 oherwie. Oherwie, he algorihm i unchanged. If we ue he Diniz/Edmond-Karp fa-pipe algorihm, we ge an overall running ime of O(VE 2 ). 2

3 8/8 11/11 4/4 / 2/ / / 0/1 /1 ' 0/ 0/ 0/ ' 0/ 0/ / / / / / 11/ 2/0.. /..20 /.. 1/1..1 /2..1 4/ /4.. /.. /..20 A auraing flow f in G, he correponding feaible flow f in G, and he correponding reidual nework G f Node Supplie and Demand Anoher ueful varian o conider allow flow o be injeced or exraced from he flow nework a verice oher han or. Le x : (V \ {, }) IR be an exce funcion decribing how much flow i o be injeced (or exraced if he value i negaive) a each verex. We now wan a maximum flow ha aifie he varian balance condiion f (u v) f (v w) = x(v) u V w V for every node v excep and, or prove ha no uch flow exi. A above, call uch a funcion f a feaible flow. A for flow wih edge demand, he only real difficuly in finding a maximum flow under hee modified conrain i finding a feaible flow (if one exi). We can reduce hi problem o a andard max-flow problem, ju a we did for edge demand. To implify he ranformaion, le u aume wihou lo of generaliy ha he oal exce in he nework i zero: v x(v) = 0. If he oal exce i poiive, we add an infinie capaciy edge, where i a new arge node, and e x() = v x(v). Similarly, if he oal exce i negaive, we add an infinie capaciy edge, where i a new ource node, and e x() = v x(v). In boh cae, every feaible flow in he modified graph correpond o a feaible flow in he original graph. A before, we modify G o obain a new graph G by adding a new ource, a new arge, an infinie-capaciy edge from he old arge o he old ource, and everal edge from and o. Specifically, for each verex v, if x(v) > 0, we add a new edge v wih capaciy x(v), and if x(v) < 0, we add an edge v wih capaciy x(v). A before, we call an (, )-flow in G auraing if every edge leaving or enering i auraed; any auraing flow i a maximum flow. I i eay o check ha auraing flow in G are in direc correpondence wih feaible flow in G; we leave deail a an exercie (hin, hin).

4 Similar reducion allow u o olve everal oher varian of he maximum flow problem uing he ame pah-augmenaion algorihm. For example, we could aociae capaciie and lower bound wih he verice inead of (or in addiion o) he edge. We could aociae a range of excee wih every node, inead of a ingle exce value. We can aociae a co c(e) wih each edge, and hen ak for he maximum-value flow f whoe oal co e c(e) f (e) i a mall a poible. We could even apply all of hee exenion a once: upper bound, lower bound, and co funcion for he flow hrough each edge, ino each verex, and ou of each verex. 18. Minimum-Co Flow To be wrien 18.4 Maximum-Weigh Maching Recall from he previou lecure ha we can find a maximum-cardinaliy maching in any biparie graph in O(V E) ime by reducion o he andard maximum flow problem. Now uppoe he inpu graph ha weighed edge, and we wan o find he maching wih maximum oal weigh. Given a biparie graph G = (U W, E) and a non-negaive weigh funcion w : E IR, he goal i o compue a maching M whoe oal weigh w(m) = uw M w(uw) i a large a poible. Max-weigh maching can be found direcly uing andard max-flow algorihm 1, bu we can modify he algorihm for maximum-cardinaliy maching decribed above. I will be helpful o reinerpre he behavior of our earlier algorihm direcly in erm of he original biparie graph inead of he derived flow nework. Our algorihm mainain a maching M, which i iniially empy. We ay ha a verex i mached if i i an endpoin of an edge in M. A each ieraion, we find an alernaing pah π ha ar and end a unmached verice and alernae beween edge in E \ M and edge in M. Equivalenly, le G M be he direced graph obained by oriening every edge in M from W o U, and every edge in E \ M from U o W. An alernaing pah i ju a direced pah in G M beween wo unmached verice. Any alernaing pah ha odd lengh and ha exacly one more edge in E \ M han in M. The ieraion end by eing M M π, hereby increaing he number of edge in M by one. The max-flow/min-cu heorem implie ha when here are no more alernaing pah, M i a maximum maching. A maching M wih edge, an alernaing pah π, and he augmened maching M π wih edge. If he edge of G are weighed, we only need o make wo change o he algorihm. Fir, inead of looking for an arbirary alernaing pah a each ieraion, we look for he alernaing pah π uch ha 1 However, max-flow algorihm can be modified o compue maximum weighed flow, where every edge ha boh a capaciy and a weigh, and he goal i o maximize u v w(u v)f (u v). 4

5 M π ha large weigh. Suppoe we weigh he edge in he reidual graph G M a follow: w (u w) = w(uw) w (w u) = w(uw) for all uw M for all uw M We now have w(m π) = w(m) w (π). Thu, he correc augmening pah π mu be he direced pah in G M wih minimum oal reidual weigh w (π). Second, becaue he maching wih he maximum weigh may no be he maching wih he maximum cardinaliy, we reurn he heavie maching conidered in any ieraion of he algorihm. 2 2 A maximum-weigh maching i no necearily a maximum-cardinaliy maching. Before we deermine he running ime of he algorihm, we need o check ha i acually find he maximum-weigh maching. Afer all, i a greedy algorihm, and greedy algorihm don work unle you prove hem ino ubmiion! Le M i denoe he maximum-weigh maching in G wih exacly i edge. In paricular, M 0 =, and he global maximum-weigh maching i equal o M i for ome i. (The figure above how M 1 and M 2 for he ame graph.) Le G i denoe he direced reidual graph for M i, le w i denoe he reidual weigh funcion for M i a defined above, and le π i denoe he direced pah in G i uch ha w i (π i ) i minimized. To implify he proof, I will aume ha here i a unique maximum-weigh maching M i of any paricular ize; hi aumpion can be enforced by applying a conien ie-breaking rule. Wih hi aumpion in place, he correcne of our algorihm follow inducively from he following lemma. Lemma 2. If G conain a maching wih i + 1 edge, hen M i+1 = M i π i. Proof: I will prove he equivalen aemen M i+1 M i = π i 1. To implify noaion, call an edge in M i+1 M i red if i i an edge in M i+1, and blue if i i an edge in M i. The graph M i+1 M i ha maximum degree 2, and herefore coni of pairwie dijoin pah and cycle, each of which alernae beween red and blue edge. Since G i biparie, every cycle mu have even lengh. The number of edge in M i+1 M i i odd; pecifically, M i+1 M i ha 2i + 1 2k edge, where k i he number of edge ha are in boh maching. Thu, M i+1 M i conain an odd number of pah of odd lengh, ome number of pah of even lengh, and ome number of cycle of even lengh. Le γ be a cycle in M i+1 M i. Becaue γ ha an equal number of edge from each maching, M i γ i anoher maching wih i edge. The oal weigh of hi maching i exacly w(m i ) w i (γ), which mu be le han w(m i ), o w i (γ) mu be poiive. On he oher hand, M i+1 γ i a maching wih i + 1 edge whoe oal weigh i w(m i+1 ) + w i (γ) < w(m i+1 ), o w i (γ) mu be negaive! We conclude ha no uch cycle γ exi; M i+1 M i coni enirely of dijoin pah. Exacly he ame reaoning implie ha no pah in M i+1 M i ha an even number of edge. Finally, ince he number of red edge in M i+1 M i i one more han he number of blue edge, he number of pah ha ar wih a red edge i exacly one more han he number of pah ha ar wih a blue edge. The ame reaoning a above implie ha M i+1 M i doe no conain a blue-fir pah, becaue we can pair i up wih a red-fir pah. We conclude ha M i+1 M i coni of a ingle alernaing pah π whoe fir edge i red. Since w(m i+1 ) = w(m i ) w i (π), he pah π mu be he one wih minimum weigh w i (π).

6 We can find he alernaing pah π i uing a ingle-ource hore pah algorihm. Modify he reidual graph G i by adding zero-weigh edge from a new ource verex o every unmached node in U, and from every unmached node in W o a new arge verex, exacly a in ou unweighed maching algorihm. Then π i i he hore pah from o in hi modified graph. Since M i i he maximum-weigh maching wih i verice, G i ha no negaive cycle, o hi hore pah i well-defined. We can compue he hore pah in G i in O(V E) ime uing Shimbel algorihm, o he overall running ime our algorihm i O(V 2 E). The reidual graph G i ha negaive-weigh edge, o we can peed up he algorihm by replacing Shimbel algorihm wih Dijkra. However, we can ue a varian of Johnon all-pair hore pah algorihm o improve he running ime o O(VE + V 2 log V). Le d i (v) denoe he diance from o v in he reidual graph G i, uing he diance funcion w i. Le w i denoe he modified diance funcion w i (u v) = d i 1 (u) + w i (u v) d i 1 (v). A we argued in he dicuion of Johnon algorihm, hore pah wih repec o w i are ill hore pah wih repec o w i. Moreover, w i (u v) > 0 for every edge u v in G i : If u v i an edge in G i 1, hen w i (u v) = w i 1 (u v) and d i 1 (v) d i 1 (u) + w i 1 (u v). If u v i no in G i 1, hen w i (u v) = w i 1 (v u) and v u i an edge in he hore pah π i 1, o d i 1 (u) = d i 1 (v) + w i 1 (v u). Le d i (v) denoe he hore pah diance from o v wih repec o he diance funcion w i. Becaue w i i poiive everywhere, we can quickly compue d i (v) for all v uing Dijkra algorihm. Thi give u boh he hore alernaing pah π i and he diance d i (v) = d i (v) + d i 1 (v) needed for he nex ieraion. Exercie 1. Suppoe we are given a direced graph G = (V, E), wo verice an, and a capaciy funcion c : V IR +. A flow f i feaible if he oal flow ino every verex v i a mo c(v): f (u v) c(v) for every verex v. u Decribe and analyze an efficien algorihm o compue a feaible flow of maximum value. 2. Suppoe we are given an n n grid, ome of whoe cell are marked; he grid i repreened by an array M[1.. n, 1.. n] of boolean, where M[i, j] = TRUE if and only if cell (i, j) i marked. A monoone pah hrough he grid ar a he op-lef cell, move only righ or down a each ep, and end a he boom-righ cell. Our goal i o cover he marked cell wih a few monoone pah a poible. Greedily covering he marked cell in a grid wih four monoone pah.

7 (a) Decribe an algorihm o find a monoone pah ha cover he large number of marked cell. (b) There i a naural greedy heuriic o find a mall cover by monoone pah: If here are any marked cell, find a monoone pah π ha cover he large number of marked cell, unmark any cell covered by π hoe marked cell, and recure. Show ha hi algorihm doe no alway compue an opimal oluion. (c) Decribe and analyze an efficien algorihm o compue he malle e of monoone pah ha cover every marked cell. c Copyrigh 2009 Jeff Erickon. Releaed under a Creaive Common Aribuion-NonCommercial-ShareAlike.0 Licene (hp://creaivecommon.org/licene/by-nc-a/.0/). Free diribuion i rongly encouraged; commercial diribuion i exprely forbidden. See hp:// for he mo recen reviion.