6. Inroducion... 6. The primal-dual algorihmn... 6 6. Remark on he primal-dual algorihmn... 7 6. A primal-dual algorihmn for he hore pah problem... 8... 9 6.6 A primal-dual algorihmn for he weighed maching problem (a kech)... 6. Inroducion - Background Primal-dual algorihm are baed on complemenary lackne. They were originally developed for nework problem [Danzig, Ford, Fulkeron 96] They provide a general mehod o derive "pecialized" algorihm for combinaorial opimizaion problem, exac and approximae. Baic idea Sar wih an LP in andard form (P) min z = c T x Ax = b! (w.o.l.g.) x! The aociaed dual LP i (D) max w = π T b π T A " c T π unrericed Complemenary lackne yield
6. Inroducion - x S P, π S D are opimal <=> π i (a T i x - b i ) = for all i (hi hold ince Ax = b) (c j - π T A j )x j = for all j (6.) So: (6.) = i he only condiion for opimaliy Primal-dual algorihm Given π S D, find x S P uch ha x and π fulfill (6.) We earch for uch an x S P olving an auxiliary problem, called he rericed primal (RP), deermined by he given dual feaible oluion π S D. If no uch x exi, we ue informaion from he dual (DRP) of he rericed primal (RP) in order o conruc a "beer" dual oluion π S D. We ierae hi proce unil we (hopefully) find an opimal pair x, π 6. Inroducion - primal problem P dual problem D RP DRP π x? π improving π Remark: hi i eenially a dual algorihm, ince we have a dual feaible oluion π in every ep and obain an opimal primal feaible oluion x only a erminaion. I i neverhele called primal-dual becaue of he role of he complemenary lackne condiion.
6. The primal-dual algorihmn 6- Conrucing a dual feaible ar oluion π All c j! => π = i dual feaible, a π T A " c T Some c j < Ue a rick: Inroduce anoher primal variable x n+! Inroduce anoher primal conrain x + x +... + x n+ = b m+ wih b m+! n M (M from Lemma.) and c m+ = Lemma. => hi conrain doe no change S P The dual problem hen i max w = π T b + π m+ b m+ π T A j + π m+ " c j j =,...,n π m+ " π i unrericed, i =,...,m A feaible oluion of hi dual LP i given by 6. The primal-dual algorihmn 6- π i = i =,...,m π m+ = min j c j < (ince a lea one c j < ) => a dual feaible oluion can be conruced quie eaily (much impler han a primal wih he Two-Phaemehod) The Rericed Primal (RP) Aume ha we have a dual feaible oluion π of (D) To fulfill (6.), e J := { j π T A j = c j } We call J he e of admiible column (6.) => x S P i opimal <=> x j = for all j J So we are looking for an x wih # j! J A j x j = b x!, x j = for all j J Thi earch i a pure feaibiliy problem, which we will olve wih Phae I of he implex algorihm. The Phae I problem i called he Rericed Primal (RP):
6. The primal-dual algorihmn 6-!"#! $ " #$% $% # &'()* +, % #+$ #+ & $ % # $ - # # $ %. / / /. " $ +! ' +, $ + $ ' +, $ % #! ' # $ %. / / /. " () may delee hee x j We can olve (RP) wih he implex algorihm. (RP) earche for a feaible oluion of (P) wihou he column A j wih j J. The arificial variable define he iniial bai of (RP). If ξ op =, hen each arificial variable i and x i a feaible oluion of (RP) => x i an opimal oluion of (P) If ξ op >, hen here i no feaible oluion of (RP) => we inveigae he dual LP of (RP) The dual (DRP) of he rericed primal (DRP) read 6. The primal-dual algorihmn 6- max w = π T b (6.).. π T A j j J (6.) π i i =,..., m (6.) π i unrericed i =,..., m (6.) (DRP) Le π be an opimal oluion of (DRP) (i exi becaue of Srong Dualiy) Idea: combine π wih he original dual oluion π o π* := π + θπ (6.6) where θ i choen uch ha π* ay feaible in (D) and he dual objecive funcion of (D) ricly increae Conequence for he dual objecive funcion of (D): (π*) T b = π T b + θ(π ) T b = ξ op > a (RP) and (DRP) are a primal dual pair Hence θ > i required for a ric increae of he dual objecive funcion Conequence for dual feaibiliy in (D)
6. The primal-dual algorihmn 6- dual feaibiliy mean (π*) T A j = π T A j + θ(π ) T A j " c j for j =,..., n hi i no problem if (π ) T A j " (hi hold for all j J ince π S DRP ) There are cae (π ) T A j " for all j =,..., n => θ can be made arbirarily large => he dual objecive funcion of (D) i unbounded Theorem. => (P) ha no feaible oluion (π ) T A j > for ome j J Then we obain a conrain for θ:! T A j + "(! ) T A j! c j > We ummarize o θ c j π T A j (π ) T A j 6. The primal-dual algorihmn 6-6 6. Theorem (Infeaibiliy of (P) in he primal-dual algorihm) If ξ op > in (RP) and (π ) T A j " for all j =,..., n w.r.. he opimal oluion π of (DRP), hen (P) ha no feaible oluion. Proof: clear from he above! 6. Theorem (Improvemen of he dual oluion in he primal-dual algorihm) If ξ op > in (RP) and (π ) T A j > for ome j J, hen!! "# $%&! " # " $ % & # '$ ( % & # # # '( '$ ( % & # ) ) $ '**+( i he large θ, uch ha π* := π + θπ i dual feaible. Then w* := (π*) T b = π T b + θ (π ) T b > w (= π T b) Proof: clear from he above! The primal-dual algorihm Algorihm (Primal-Dual)
6. The primal-dual algorihmn 6-7 Inpu Primal LP (P) in andard form Aociaed dual LP (D) wih feaible oluion π (poibly conruced by he above rick) Oupu A erminaion : an opimal oluion or a meage ha (P) ha no feaible oluion Terminaion can be guaraneed by ani-cycling rule Mehod repea Conruc (RP) by compuing J := { j π T A j = c j } a call Phae I wih co vecor ξ = # x i for (RP) if ξ op > hen call dual Simplex for (DRP) and ake he compued opimal oluion π if (π ) T A j " for all j =,..., n hen reurn "(P) ha no feaible oluion" ele compue θ according o (6.7) 6. The primal-dual algorihmn 6-8 e π := π + θ π unil ξ op = reurn oluion x of (RP)
6. Remark on he primal-dual algorihmn 7- () Rear: The baic opimal oluion of he previou (RP) i a baic feaible oluion for he new (RP) 6. Theorem (Keeping admiible baic column) Every admiible column of he opimal bai of (RP) remain admiible a he ar of he nex ieraion of he primal-dual algorihm Proof Le A j be an admiible column of he opimal bai of (RP) Definiion of admiible column => A j i a column of A, i.e., doe no belong o an arificial variable reduced co of a baic column i, π i a dual opimal oluion of (RP)! " #! " "! "! $# % $ % " "!! $# % $ % "!! " " # $ # $ Then (π*) T A j = π T A j + θ (π ) T A j = π T A j + = π T A j = c j ince A j i an admiible column w.r.. π => A j remain admiible w.r.. π*! 6. Remark on he primal-dual algorihmn 7- An opimal bai of (RP) i compoed of admiible column => ay admiible becaue of Theorem 6. column of arificial variable => ay in he new (RP) => () () (RP) can be olved wih he revied implex algorihm hi follow from Theorem 6.. We only need o updae he e J for he non-baic column () Terminaion can be achieved by ani-cycling rule 6. Theorem (Terminaion of he primal-dual algorihm) The primal-dual algorihm olve (P) in finiely many ep Proof a a Inerpre (RP) a a equence of pivo of variable x,..., xm, x,..., x n (poible ince x j = for j J and hu can be inerpreed a a non-baic variable) => (RP) ravere a equence of baic feaible oluion of (I A) Claim: The objecive funcion decreae monoonically along ha equence (no necearily ricly)
6. Remark on he primal-dual algorihmn 7- hi i clear wihin he repea-loop, becaue hen he algorihm i ju he ordinary (revied) implex algorihm. conider now a new enry ino he repea-loop => we compue θ Le r be he index a which he minimum i aained in he compuaion of θ Sub-Claim: column r i admiible and ha negaive reduced co in he new (RP)!! " " # $ #! " # $ $ % %!! " " # $ #! " # $ $ & $!! " # $!! " " # $ "!! " " # $ # & $ => A r i admiible w.r.. π* => r admiible in he new (RP) in he new (RP) column r ha reduced co (ee Proof of Theorem 6. in hi ubecion) - (π ) T A r < a (π ) T A r > by definiion of θ Sub-Claim => when enering he repea-loop, we may chooe column r a pivo column in he ene of he ordinary implex algorihm wih monoonically decreaing co Claim => adaping he lexicographic rule o he equence of baic oluion of (I A) yield erminaion! 6. A primal-dual algorihmn for he hore pah problem 8- Deriving (P), (D), (RP), (DRP) We conider he formulaion of (SP) from Secion. (P) min c T f Af = b (A = verex-edge-incidence marix) f! where he row of verex i deleed The dual LP i (D) max π - π π i - π j " c ij for all edge (i, j) E(G) π i unrericed π = (correpond o deleed row ) The e of admiible column i IJ = { (i,j) E π i - π j = c ij }
6. A primal-dual algorihmn for he hore pah problem 8- (RP) hen i min ξ = # i=,...,n- x i a x a + Af =... row! for all edge (i,j) E(G) = for all edge (i,j) IJ a x i! for i =,..., n- The aociaed dual (DRP) i max w = π π i - π j " for all edge (i, j) IJ π i " for i =,..., n- (obained from he column of he x i a ) π = Inerpreaion of he primal-dual algorihm 6. A primal-dual algorihmn for he hore pah problem 8- () ξ op = in (RP) <=> here i a pah from o uing only edge from IJ. Each uch pah i an opimal oluion of (P), i.e., a hore,-pah Proof "=>" Le ξ op = => every baic opimal oluion of (RP) i an,-pah wih = for all edge (i,j) IJ => hi pah ue only edge from IJ. "<=" every,-pah conaining only edge from IJ i feaible in (RP) and ha ξ = => hi pah i opimal for (P) becaue of he primal-dual mehod (i aifie complemenary lackne)! () If here i no pah from o wih edge only from IJ, hen π wih π i := i opimal for (DRP) Proof can be reached from i via edge from IJ or i = oherwie
6. A primal-dual algorihmn for he hore pah problem 8- π i feaible for (DRP) π i " and π = hold by definiion Aume ha π i - π j " i violaed for edge (a,b) IJ π ha only value and => π a = and π b = Definiion of π => can be reached from b via edge from IJ (a,b) IJ => can be reached from a via edge from IJ => π a =, a conradicion π i opimal for (DRP) The objecive funcion i max w = π Conrain π " => every π wih π = i opimal => π i opimal! () For ξ op > and π defined in () we obain θ = min { c ij - (π i - π j ) (i, j) IJ, π i - π j = } Proof Le ξ op > and π be defined a in (), o in paricular opimal for (DRP). (6.7) implie 6. A primal-dual algorihmn for he hore pah problem 8-!! " #$%! " #$ " % & ' #$ &% ' & ' #$ # &#( $' )*( &% ' & ' #$ + ( $ (π ) T A ij = π i - π j > <=> π i = and π j = => (π ) T A ij = π i - π j =! () The primal-dual algorihm reduce (SP) o a equence of reachabiliy problem Can be reached from i via edge from IJ? or., afer invering he orienaion of all edge, Which verice can be reached from via edge from IJ? Proof Follow from ()-()! 6. Example Inpu daa
6. A primal-dual algorihmn for he hore pah problem 8-6 c ij c ij! => π = i feaible in (D) Ieraion! IJ = Ø! θ = min { c ij - (π i - π j ) (i, j) IJ, π i - π j = } = for edge (,) => π* = π + θ π = (,...,) T + (,,,,,) T = (,,,,,) T Ieraion 6. A primal-dual algorihmn for he hore pah problem 8-7!! θ = min { - ( - ), - ( - ), - ( - ) } = for edge (,) => π* = π + θ π = (,,,,,) T + (,,,,,) T = (,,,,,) T Ieraion!! θ = min { - ( - ), - ( - ) } = for edge (,) and (,) => π* = π + θ π = (,,,,,) T + (,,,,,) T = (,,,,,) T Ieraion
6. A primal-dual algorihmn for he hore pah problem 8-8!! θ = min { - ( - ), - ( - ) } = for edge (,) => π* = π + θ π = (,,,,,) T + (,,,,,) T = (6,,,,,) T Ieraion π 6 opimum reached π = (6,,,,,) T i dual opimal = vecor of hore diance o Deailed inerpreaion of he differen ep () Define W a 6. A primal-dual algorihmn for he hore pah problem 8-9 W := { i V can be reached from i via edge from IJ } = { i V π i = } π i remain unchanged a oon a i W, ince π i = aferward () When an edge (i, j) ener IJ, i ay in IJ, becaue π i and π j change by he ame amoun => π i - π j ay he ame () i W => π i = lengh of a hore pah from i o (inducive proof) In every ieraion of he algorihm, one add hoe verice from V - W o W ha are cloe o (inducive Proof) Conequence The primal-dual algorihm for (SP) wih c! i eenially Dijkra' algorihm, a in he chord model in Secion.
9- Deriving (P), (D), (RP), (DRP) Primal LP (P) and dual LP (D) We conider he formulaion of he ranporaion problem from Secion. (P) min # i,j c ij.. # j = for all i (pick up upply from verex i) # i = b j for all j (deliver demand b j o verex j )! for all i, j wih (w.o.l.g.) # i = # j b j Inroduce dual variable, β j for he wo group of conrain β j # j = for all i # i = b j for all j The dual (D) hen i (D) max # i + # j b j β j.. + β j " c ij for all i, j, β j unconrained A feaible oluion of (D) i 9- = for all i β j = min i c ij for all j doe no require ha all c ij! ) Rericed primal (RP) The e of admiible column i IJ = { (i,j) E + β j = c ij } Rericed primal (RP) a min ξ = # i=,...,m+n x i a # j + x i = for i =,..., m a # i + x m+j = bj for j =,..., n! for all edge (i,j) IJ = for all edge (i,j) IJ a x i! for i =,..., m+n a We modify (RP) by ubiuing he arificial variable x i in he objecive funcion and obain (wih fij = for all edge (i,j) IJ)
9- ξ =! i +! j b j -! (i,j)! IJ conan => minimizing ξ <=> maximizing # (i,j)! IJ Deleing he arificial variable hen yield (becaue of x i a! ) (RP') max # (i,j)! IJ # j " for i =,..., m # i " b j for j =,..., n! for all edge (i,j) IJ = for all edge (i,j) IJ => (RP') correpond o a max-flow problem in he graph G of admiible edge 9- a! b a I! J a m!! b n = capaciie admiible edge The primal-dual algorihm yield: f i opimal in (P) <=> he maximum flow value fulfill v(f) = # i = # j b j The dual (DRP) of (RP) Inroduce dual variable u i, v j for he wo group of conrain a u i # j + x i = for i =,..., m v j # i + x m+j a = bj for j =,..., n The dual of (RP) i (DRP) max w = # i u i + # j b j v j..
9- u i + v j " for all (i,j) IJ u i, v j " u i, v j unrericed 6.6 Lemma (Opimal oluion of (DRP)) Le ξ op > in (RP) and le f be a maximum,-flow in G. Le I* " I be he e of verice ha can be reached from in G f (i.e. here i a flow augmening pah from o hee verice). Le J* " J be he e of verice ha can be reached from in G f (i.e. here i a flow augmening pah from o hee verice). Then := if i I* := - if i I* β j := - if j J* β j := if j J* i an opimal oluion of (DRP). Proof From ADM I we know ha X := {} # I* # J* i a cu of minimum capaciy of G, and ha every max-flow algorihm compue uch a cu. Hence he e I* and J* can be deermined efficienly. 9-6 We analyze hi cu of G I* no edge from I* o J-J* auraed edge - - J* auraed edge flow = on he edge from I-I* o J* () here i no edge (i,j) from I* o J-J* oherwie j J* becaue of he infinie capaciy of (i,j) () = for all edge (i,j) from I-I* o J* oherwie (j,i) i a backward edge in G f, implying i I* () edge (,i) from o I-I* are auraed oherwie i I* () he edge (j,) from J* o are auraed
9-7 oherwie here i a flow augmening,-pah () he flow value i v(f) = # i! I-I* + # j! J* b j ince: v(f) = ne ouflow ou of X = {} # I* # J* ino V(G) - X => () follow from () - () (A) and β j are feaible for (DRP) Show ha + β j " If + β j > => = and β j = => i I* and j J-J* => a conradicion o () (B) and β j are opimal for (DRP) The objecive funcion value for and β j i w = # i + # j b j β j = # i! I* - # i! I-I* - # j! J* b j + # j! J-J* b i Becaue of (DRP') and () ξ op = # i + # j b j - v(f) = # i + # j b j - ( # i! I-I* + # j! J* b j ) = w Weak dualiy => and β j are opimal! 9-8 Updaing he dual oluion Le and β j be dual oluion of (D) and le and β j be he opimal oluion of (DRP) If ξ op >, hen here are cae in he primal-dual algorihm (Theorem 6.) Cae : + β j " for all (i,j) IJ => (P) i infeaible by Theorem 6. hi canno happen a (P) ha a feaible oluion, e.g. = (/# k a k ) b j Cae : + β j > for ome (i,j) IJ So hi cae i he andard cae. (6.7) yield!! " #$%! " #$ " % & ' #$ &% ' & ' #$ # &#( $' )*( &% ' & ' #$ + ( $! "#$!! "# " $ " " % # $ " % % # # &"& #' '(& $ " % % # ) ( $! "#$!! "# " $ " " % # % # " & ' # ( $ We ummarize
9-9 6.7 Lemma (Updaing he dual oluion) Le ξ op > in (RP) and le f be a maximum,-flow in G. Le I* " I be he e of verice ha can be reached from in G f Le J* " J be he e of verice ha can be reached from in G f Then!! " #$%! " #$ " % # " & $ & # # ' ( $ ) $ and he new dual oluion i obained a * = + θ if i I* * = - θ if i I* β j * = β j - θ if j J* β j * = β j + θ if i J* Every opimal flow of he old (RP') ay feaible in he new (RP'). Proof The new dual oluion i obained a π* := π + θ π Hence he value of * and β j * follow from he value of θ and Lemma 6.6 I remain o how ha he opimal flow ay feaible. Thi follow already from Theorem 6., ince he 9- opimal flow i a baic feaible oluion of (RP'). We give a direc proof below. Claim: edge (i,j) wih poiive flow ay admiible in he new (RP) > => edge (i,j) i admiible in he old (RP) => + β j = c ij analyi of he cu of G (Proof of Lemma 6.6) => wo poible cae Cae : i I* and j J* => * + β j * = + θ + β j - θ = + β j = c ij Cae : i I-I* and j J-J* => * + β j * = - θ + β j + θ = + β j = c ij! Der primal-dual algorihm for he ranporaion problem Algorihm alpha-bea Inpu Inance of he ranporaion problem, i.e., number >, b j >, and c ij wih # i = # j b j Oupu A minimum co ranporaion plan Mehod
9- Deermine a feaible oluion of (D) by := for all i β j := min i c ij for all j repea conruc graph G of admiible edge from IJ := { (i,j) E + β j = c ij } compue a maximum,-flow f in G // warmar wih he flow from he previou ieraion i poible if v(f) < # i hen e I* := { i I here i a flow augmening,i-pah in G f } e J* := { j J here i a flow augmening,j-pah in G f } e!! "# $%&! " #$ " % # " & $ ' # # ' ( $ ) $ and ake a new dual oluion := + θ if i I* := - θ if i I* β j := β j - θ if j J* β j := β j + θ if i J* unil v(f) = # i 9- reurn flow f Inerpreaion For he ranporaion problem, he paradigm of he primal-dual algorihm lead o wo need loop of reachabiliy problem. The loop "combinaorialize" co and capaciie. Tranporaion Problem combinaorialize co Max-Flow-Problem combinaorialize capaciie Reachabiliy Problem find a flow augmening pah 6.8 Example Inpu daa
9- b j die c ij Marix Calculaing he dual oluion β j = min i c ij mark he admiible edge (i,j) Ieraion Conruc graph G of admiible edge and compue a maximum,-flow in G and he e I* and J* 9- G f I* J* flow value v(f) = 9 < # i = => updae he dual oluion Updaing he dual oluion Deermine θ β j!! "# $%&! " #$ " % # " & $ ' # # ' ( $ ) $ c ij marix
9- I* = { }, J* = { } edge (,) -> ( - - ) / = / edge (,) -> ( - - ) / = / => θ = / edge (,) -> ( - - ) / = Compuing he new dual oluion := + θ if i I* := - θ if i I* β j := β j - θ if j J* β j := β j + θ if i J* So α = / α = -/ α = -/ β = / β = / β = / β = / Ieraion Conruc graph G of admiible edge 9-6! # $ $! " " " " " / -/ -/ mark admiible edge G Compue a maximum,-flow in G and he e I* and J*
9-7 f I* J* f I* J* previou flow new flow flow value v(f) = < # i = => updae he dual oluion Updaing he dual oluion Deermine θ 9-8!! "# $%&! " #$ " % # " & $ ' # # ' ( $ ) $! # $ $! " " " " "! i / -/ -/ I* = {, }, J* = {,, } edge (,) -> ( - / - /) / = / edge (,) -> ( + / - /) / = / => θ = / Compue he new dual oluion := + θ if i I* := - θ if i I* β j := β j - θ if j J* β j := β j + θ if i J* So α = α = α = - β = β = β = β = Ieraion Conruc graph G of admiible edge
9-9 G β j - mark admiible edge Compue a maximum,-flow in G and he e I* and J* 9- f I* f J* previou flow new flow flow value v(f) = = # i = => have conruced an opimal oluion Runime of he algorihm Le w.o.l.g. m " n => G ha O(n) verice and O(n ) edge The primal objecive of (RP') increae wih every flow augmening pah and i bounded from above by # i. => oal runime for flow augmenaion i (# i ) O(breadh fir earch) = (# i )O(n )
9- All oher compuaion (conrucing G, θ, he new dual oluion) are in O(n ) and happen a mo (# i ) ime => Toal runime of he primal-dual algorihm i (# i )O(n ) => he primal-dual algorihm i only peudo-polynomial I can be improved by Capaciy Scaling for he and b j (ee ADM I, Secion 6., for capaciy caling) => eenially only log(max {, b j }) many max-flow problem wih runime O(n ) An inereing pecial cae he aignmen problem, which i defined by = b j = (hen n = m) => # i = n => runime O(n ) The alpha-bea algorihm wa fir developed for maximum weighed maching in biparie graph (Paul Kuhn 9) and i known a Hungarian mehod, ee e.g. A. Schrijver, Combinaorial Opimizaion: Polyhedra and Efficiency Volume, Chaper 7. 6.6 A primal-dual algorihmn for he weighed maching problem (a kech) - Goal of hi ecion Skech of a primal dual algorihm for weighed (perfec) maching. Thi cloe he gap from ADM I, Secion 7. Maching Le G be an undireced graph. A maching of G i a e M " E(G) of edge uch ha no edge of M hare an endpoin a maching M i called perfec if every verex of G i inciden o an edge of M a graph wih a perfec maching (red edge)
6.6 A primal-dual algorihmn for he weighed maching problem (a kech) - Maximum weigh maching problem (MWMP) Inance Undireced graph G, edge weigh c(e) Tak Find a maching M wih maximum weigh c(m)!!"" # # "!!#" Minimum weigh perfec maching problem (MWPMP) Inance Undireced graph G, edge weigh c(e) Tak Find a perfec maching M wih minimum weigh c(m) or repor ha here i no perfec maching 6.8 Lemma (Equivalence of maching problem) 6.6 A primal-dual algorihmn for he weighed maching problem (a kech) - MWMP and MWPMP are equivalen in he ene ha here i a imple ranformaion from one problem o he oher uch ha one can conruc from an opimal oluion of one problem an opimal oluion of he oher. Proof "=>" Le (G,c) be an inance of he minimum weigh perfec maching problem. Chooe K large enough o ha c'(e) := K - c(e) > for all edge e, and only maximum cardinaliy maching of G have maximum weigh w.r.. c'. ( K := + # e! M c(e) uffice) Le M be an opimal oluion of he maximum weigh maching problem for (G, c') M ha maximum cardinaliy => M i a perfec maching for (G,c) or here i no perfec maching in G If M i perfec, hen c'(m) = Kn/ - # e! M c(e). So M ha maximum weigh w.r.. c' iff M ha minimum weigh w.r.. c. "<=" Le (G,c) be an inance of he maximum weigh maching problem. Add V(G) many new verice o G and o many edge ha he new graph G' i complee. Se c'(e) := - c(e) if e E(G) and c'(e) := if e i a new edge. Le M' be an opimal oluion of he minimum weigh perfec maching problem for (G', c')
6.6 A primal-dual algorihmn for he weighed maching problem (a kech) - => M := M' $ E(G) i an opimal oluion of he maximum weigh maching problem! The primal-dual algorihm for he minimum weigh perfec maching problem Primal LP (P) There i no obviou LP-formulaion. Der following heorem wa one of he pioneering reul of Edmond. Given an inance (G,c) wih G = (V, E), he conider he LP (P) min # e! E c(e)x e x(δ(v)) = for all v V x(δ(s))! for all odd verex e S of G x! Here x(δ(s)) := # e! δ(s) x e 6.9 Theorem (Maching polyope Theorem, Edmond 96) Le (G,c) be an inance of he minimum weigh perfec maching problem. Then: () G ha a perfec maching <=> (P) ha a feaible oluion () In hi cae he minimum weigh of a perfec maching of G i equal o he opimal value of (P). Proof 6.6 A primal-dual algorihmn for he weighed maching problem (a kech) - The primal-dual algorihm conruc an opimal oluion x of (P) (if here i one) ha i a perfec maching of G. => Theorem 6.9 and Lemma. imply ha all baic feaible oluion of (P) correpond o perfec maching.! Dual LP (D) We do no ranform (P) ino andard form. Der primal-dual algorihm can be adaped alo o oher form of (P). Le U be he e of all odd verex e of G. The dual LP of (P) i (D) max #(y v : v V) + #(Y S : S U) y v + y w + #(Y S : e S U) " c(e) for all edge e = vw E Y S! for all S U y v unrericed Complemenary lackne condiion x e > => c(e) - (y v + y w + #(Y S : e S U)) =
6.6 A primal-dual algorihmn for he weighed maching problem (a kech) -6 Y S > => x(δ(s)) = If x i he incidence vecor of a perfec maching M, hen he complemenary lackne condiion are equivalen o e M => c(e) - (y v + y w + #(Y S : e S U)) = (6.8) Y S > => M $ δ(s) = (6.9) Remark on he primal-dual algorihm (6.8) define "admiible" edge for he maching ha we look for in he (RP) (6.9) correpond o M $ δ(v) =, if S ha been hrunk o a peudo-node v (a bloom) => a oluion of (RP) correpond eenially o earching a perfec (o a maximal) maching in he graph of admiible edge in which all e S wih Y S > are hrunk => we can compue uch a oluion wih he algorihm for a maximum cardinaliy maching (e.g. he one from ADM I). An opimal oluion of (DRP) can (imilar o he ranporaion problem) be obained direcly from he be maching in (RP), bu hi i more complicaed han for he ranporaion problem. Alogeher, he minimum weigh perfec maching problem reduce o a equence of maximum cardinaliy 6.6 A primal-dual algorihmn for he weighed maching problem (a kech) -7 maching problem. The algorihm can be implemened o run in O(n m) ime. In paricular, a mo n variable Y S > hroughou he algorihm. There are improvemen for dene graph (ha work only on pare ube of he edge) For deail ee Chaper. in W. J. Cook, W. H. Cunningham, W. R. Pulleyblank and A. Schrijver Combinaorial Opimizaion Wiley 998