1•f""'91), ~"";) f (J-2);, 10j pyC)71.?niY) I il 11 <.------..

1 ·. •1 'h Q

<----1-~~----+.. -71-"02 -= (i)

~l

0 i1Ji~

~ ----i.. 'CY)

. . . . ---\0) j i -z ~? ::: tJ ~~r

Q = 0 ll:. (l

"'1~ i.t~'2J,">1/~ 0 °fv>iol) "! 1'>v-l 55~i ""-( 0} \)

r·------·---·-··---.--~

\/3 ~ 'j >1}Q>J °W 0_1 +~ ?!1 <.~ J -L ( 0 J ! )JJ 1->o \NJ,Lv1~+ ::.

-t t~-'2.~1i~ );~ j-~TV~~~<l.

~Jo l t-i-z..~~L~ Q1-ö~1>~

~ "'197) ~

)~r') ~ ~'>-1<1. ·vi,ni t ~-'-'ß~ 1

/'i/ -:i '1J :i t?2

N ~ ~n u»>j)-t. „J'1YJ 1 ~ r;. 6Jl,~f1

·~s /~ ~ ~

I /\~ f

"'t~ __ / ·-=-=-~--r--------

L.q\ f<tl. • 1~ ~r·"j~>1 ~-a ~ff„~1 -J J r~-· )! 011 ] '°)

~ "'Ylf'°~ i.y ~r„~ (~(} '·.~ '6'~ \.j(?·LI?

~a!007royyt t~ Y11 ~"''1±_. ~ ~o1_p1J ~~~ ~'11 a~

_.J~11'"~0 ~ (011~~ r~~Y>U 11~ t'6"'11 007-0~

dJ ~~J. S'S~_:i 1 ~~ ~Y\

~ ~0fY) ~"'1os J"'ho ~""lo"1~ ~~.>o-"'1 ·1~f r~:t. 9~1 o\5 :: .l~n 'i'i0 : t~~"o

l.'~~o~~.:i ""1,.,")~1j l~h-\ ~S~=l:_ ·t'$ -15~~~1p>\\r. j§

'd5 r~±_ (}~-1""1'" ~(L 1 JI

}-i'

.'-....,/

~___,;/

7..,q ~+ 0 ~ fe.; ~(';1...4., ~ +~ ( e - la.) qi,,, kor~ k u~„~ C.11

·Et„_ !zv.._,i.....,·1;..Av +.e.ds ~7 ~7kc.re-; 2.-<l„J T e1~+

(1) JJ ~~r~~~~e.i. ~"L·~ ~~ ·.

~°' (~) ~ UQ .\f <9 t- [ 01 T t

1/o,p({~tlei~ ~Q_uQ''< k~V] J,'f r~u ~t<k

°'-t:S1cv) o

r t!.., ~si €,3 e

'';>'' !( Ge.~ ·1~ l<"{~ ~~6./.-

($) i'=" ~ ~~ ~~ ~~rk-1 T

~ot f c. ( G) ::: 0 ~w- G > 1- tci ( +~~_! l.J lve-h- ~.r~-

wE'~ s~ ~~ !}_ ~t T )

lv~(\ ~ ~~)fr~ f~S.~ ~ : 1 T

vJ<~ (~) ~ Z 5 f~ (&)Je - Z S flt (G-<:~) J.9 ~f~t('!.) C> q~J-(s) T

T ~ T

::: - Z~+(&) f~(<Y)Je t Z S {~Ce-r~)de l.\

6 0 Q_f & (_ l) l~

~ i\J.e-H- o ~ ~>_c Q.l{;;> s. = N-eflo G-.., {t s:s ;\,{ t-

~"'(,· ~1v~!'_,_}l__v.._~ 1.<.t1'). ~-fr:. sr ~=-~·ts:_ \?a-~1.«

G-t~~ ~ 'Drif~r~ G':::(\J>A)1

<;/:r; V, uGI) r°" > · iJk~·~l- r

-6(_~l-(_<\_\-vt(\Rv- ~(l - r,{.c..d~

lcAcp<'r'Lvt .Je""-<. T4 C:J' <.ver)

+ '/4 C0, T~]

Vc.i_WL ( ~ )

~ 6.2. J),-~l(~ V>. Sk~(}~ ~

:,eü4.11i~ 'l},_~-~d»~ 1tt~~i- skh~ &..i

cl:~L:~1t ~q ~ ~ o1.e T ~~s. \,( tA.r rw,_ ~'1 rk:~ _ o, 1 '.) c ~ -··.

~t!>L<ft,1-.v T~.!>c;; 5 t 3. (0)

T Ht...r, P.rw.o, ~'lt. --~ 1~,s le~-~

+ei..ss 4,~ 1-, ~;4~>9

T

) k k~ i:- t..~~ *" l ~) Tr-~ f "fl<Aa.Lev-.

3~ c ~) A

U.c, J /..(q

f l.-i cti '\ {.( t-e. k_ tz. ~1 +,e,.\S

I (:j ) G

re4.~ "t~„~

~1 f (_

11.R,.P,1,,.:;' \l..o /,,;,,, ""~J {1'-VJ 1>., l ... k "' "'"'3 ~ ? .• J ~0.1<1-t ~~ J.~k~Ce l{'""-„~~\~ +.e...~! ~>;. .tk.l·~·tf

r)f'0Ö<Nw.'>c.~ i~J~ tl.{ ~ ~ ~o-f kl€r~ IJIA." t.{l«~-~l~.J.-( ~" ~-) ~ r u cj l<..Qe; ~ Aptro l(\ IM~ ~-tvv> ß_Jlu ~

11

V~rM~~f,,~ 1' ~Q )

Time expanded networks

Observation. Discrete flows over time in G correspond to static flows in time-expanded networks G T (f or constant r

0)

G V

f

Example: T = 5.

1 waiting in a node 1 .... ········ s •••••••••• „ •••• „„„ ......................... „„ .....•.......•••.•••

).\ . t \,

0 - 4

0=1

0=0; l<orre ric.. G 9:. 'tk 1J,{.'.;.(,l

k1t..k.. 1v,19~ V<JtO ;, li ~. (Ck 4.h("t ~ol 'ro..

A static flow in the time expanded network

T=5

s

w

1/1=12

t

1

2

7

• • • •

[ wi ;; / J 1 yVin

I .... t<I y Ir~ ~~ yVJ Y:~tf'<J "l ~·ii ~ ~ ~Irr") t I~ 1 n~ "~? ~~ ( 1 . 0 "'1/Q'/1? \Y(t) ~r <a1r 07>n Ov>,1~ <l\~Y1 ~)Q!·\sl) tQ-"1'?

\1'7) + 9 h.l X ~'i I HCL\y' ~ \"')'\ \>~ if""t~lS J cC 7> <L \: lo a r~ so·~ i.. -= J )< t~ J ' l-~ ';<. "";) (l

\~0rri -r~t 2 ö ··t'e-0Jwv_iYJOff"J0 J!/11 °OJ~r~ (c

[ T J..<c-y ~ ~" 'lf~ &'°'"'1 L~ ~ ~~"L.oJ 'i.1 \Y3'3 J

0 -=-(/\)/ °" = V l>y)

(

~~ ~ }i'l)O'°~ d('fV ., "':1,l"'O')lJ!l ;<..IT)-h!l?) ::::

V

\t~ b

(X) m~l\·_J_ -li~ "'1 t~JQ?i t / 5 2 ·1·p

~(,.,;

~ : ::; ( st1)-+-9 ni x ('ro.:t i~; -"-!rry ~1-~'<>ls ~01Crl~ (~

:~I ~!J

L'8Stl] ~~,""O'-f --ys ~J:f x~~ 4j. ~~4>1 Q'_:!· f1-,f-"'dll G-n""l1t~~dt' <i'':)

Qb,''x.o"'l ~d<>fY>?J -1Yl'I)

9 ;„~ t"'S ~ 1i !:>'~df r-'n-1.Q"'~oiJ J_ ~ "'i

'1>LJ~~~ ~.vaj ·~ ~!~-~ J. ~ ö~~ y '

~ fl) ~1~">J -srrrJ.J. X-;i") -2_' l) 'b(J r~ ~ d i. ''.~c!_, 1->i( ~/r>S'~±_ o''J', J S •

-1. 9 "°'! ~S~i. ~LJ'fi -j'$ K'o~ \J

-1.. r~< VJib~ f,'Wi s~ro+ o'·'"'t'o'-1~1 -}'~ x-....,

1 6·-/r-i ~ c.t ?{V, °1Jjj ~, 11 )j)dx-u>oc,

•

)~ \ ... Y)1~~~ : '4'\'i,?.1:_ ru~~,,.,i-o"'l.f °icH1~-'° ö~>'tl\

_s-5l

[ o) T- rf' l t

~~~ &:tpt.t..ll ?ML..- ~~ ~ s)

\:o J.'»s~ +eurs rr-~~~ 1-cv ~-). T

11-t ~ k~ t- a. v. koL!.!~f

Example

A path decomposition of x

Paths flow values 2 t 1 s

2 s

w w

Je-~ -:;-~.s~ '} ~~ ttl 52..--..~ ~[_,;cl<tve,-.e (.<tw:>~C<..,k -t{d- ~J

lviü/ 1\~A& Lwt >~ 4~ ') rAO( ' 11' { ~ w,

~ ~ e, ~J-rc; k.- ~A 1),J r„ ~ cs ·1S~ (~o~

1·v. ~ 2,..rJ crc:F •.1 ... J 1J.& ... -..e.. „.,. f'~ 1; _ ~ l k"1)11nc ,·t:ft

• L<C(fCtc~b ei- LJU-uu~ '.

\et l (0) ~ 2_ X'p :::c X(\_ < U.<l.

'P:ct_E:'P

·Ah~ ~1 1 ot J~ µ.e~1t.eA-< ~.ru) rl0i ~ r k ~tc-k

T~I;.~ 'Uv- ~1 T-tr ~Q.d<+ tv?'"'o(

~~ ~ \ &.eri '?k <it . ..Q w le~ L oeJ~~ T ..eu He-~ ~ ~~----------·---·~,~~-~~~---....

V~(_~) ':'. L (' -lp) x'? =

'PE]->

~ kQ~ {l:w ok, ~ .Q ~ k h'~ 2 -l "' - T. V n ( ... ) C., Aq ~ ,,

c.GA

~' wtill\-c.w-\. (€ow 1

Ltv ~4. V~ ?~c..J ~CA-it-4.,.s

::c o ~ ~q_,xi ~itA rh.... T~~r l &-v ~ ?h_1-e·Jl

W~ 4~1-w_ Jy 1.t~"'-.1·1~ sr{ t~·>~ l.t-t·J ~1 hV)·'2~-t T • -·~--= o•-=------

V ? ~ ,">V\A'bt 11f- ? -t (!: ,>) ?-7 ~ IM-J po t11:vt.,

Ko.sk11

i)f J/~ 41tl VJ.rn k lf(v) E [ 0 1 cXJ [ ~r \J E- V l«i i Tr(~) == 0) 1f (J) ~ T ·

v 1~ w~~ 1>1> J.eb·11.H~il,. ~"'R c.....f ~ k.-~h TrR~ t:. U°"t_ -~~

X ( 9) ~ := { ve: V 1 nl") ~ 0 ~

X(&\ V-X(~) ~-~ >if r]<....'t~

rk Q1.1)erf

D1~ +J?~ tc.J \..t.J ~J~,j_, Viefl..e_ V{)>9.H.~ ~~ x1 CX2 c_ __ c)k

TI(~),,._o

1\(v)><O

\T (VI) ::.1_

lf(t) ~ 1

CAp c ~) ; ==- ) [ ) . 0 ~ '' 1 L_ '«°'x T/Cw )- f((v - rlt > J • '-"--a.

'&;!) d; e_

OL i~ Jf.,.,.

>[),{,(;ff. 1\-).

--1~

"4.c, )(1 ''; ~ 1.u i, f "z"' LtJL_

iu..0 ~ +Rv ~l ~ lt:ec... k_ ~ "ICUM c-e..c J~,

\ _ _/

'\ ~Qua.., l<e_

~-" rli

kt11.t1Ak1 w~k~oe 0.. 1't..,_ \Q1.<iJ./ 1'.)~

(;J lENKA .. Tt_;, ~ ~ ot("'-· ~rf.- +.-e..ss .( u-...eA j..e-4 J/"- <;Jt - _(Q"'J~

1<At.t ~~ ~1 k0i~f T ~e+

"------~~{) __ ~ ·-· c~ei_~.l-.----········-·---··•·•n••-···········-···-----·······"·''''

() d • '• J ' ~ ~ t ( x,. ) 1>ÖV) ~~ :,J~uJQ L 01 T [ 1-V?·,J tl.u~tA ~ G i;.Q~~ ~9 ' l.,Q)

lev ottL~ \c i .srx") ~trr L) _ ,

0 ~ ·-+- -~ l --------·---·- ---- - --··- - „. „ .„., tl1 t>2.

J :~ ' t;I ~ ~ ~ i ll { Tr(w) I cv) w )1 E,' I~ .,. c X'J s

'

'i::euu-~~ ~iv lL~[,J~ J'-f(,() J,''C /r,.,, 1 ) •

~11'-'lt>vcJQ [e~_ 1 > ~~] -~~ ka.4w

'M.c...')( t 11(w)-1f(v') -Tq, 0} . U..<f ===- C~p (f:>) tJ

lv.t<1('icl<<J k.tC...)(it...,~

b Tu.'k1..~~ 2 t;,,f]..~·,J-~~ "'-' o j:~~.s- 0i. ~'+~<;~~-"'l~~ :-2..t. (w;&) lCtA.~i_:

', _I

G-10

Cllp ( r) 1 ~v) -) IT(v) - iTC~) - -Ca. = o-o-o -;> ~... l1; lr~~ J [4_ ke..·>ur.,,,, s.9„;J~

(s, w) -) 1-0 -3 -" l&u1,, ~L-,,,-.}

(vJ w) -) 1 -6 - 1 4> ke...., ~il-o..j (v,t.) -") "$, -o -s -? ~ '< t?.e.J, ~ J

(w, t:J J-1 ~„ -o -) 2.5 = 6

~ ~ ~~ ly '-'l.Ot '-vt ~„ S4, +~~ . tl;' v ~a.{ ~1r~~l e ole.~-41ei-e JZ)<fC v) (J) ()~ Cl'~

D<\\A\I\ f~\-- vcl~ (J) '.::. .ex~ ( f) T)

~ vtuvt, t Q{~ ol(t.t€itt,i.til.u ~e-Li_ 4-J) ~1+ Wer~ -rr(v)

~ ~ exf. ( v) ll(v))

V

~ 2'. ( z V C{E-:i)-(v)

Tf(v}-'L~

f tCJ)l ~ u .....

T[(v)

-2 J fCVJ!) Cl(;~+(v) 0

0(. ~(v, w)

kr1"k„ \,-~e Q~ t;\.lck_

~t 'kctV\k ~L-C(Pt kt

(

ff (w) -Tllt

f {CJ)Jy 0

1T[w)-~

f f C.V <lj o:. = (yJw) Tr(\/)

ifC.v)

- f tC!) l! ) 0

b .- ll

IAA,cu( j T/ (w) - //(v) - "[et ) 6 J · V:C\

~~ ~ ~·J k C>ph·~&:G.-..1/ J-0 (4)Q.v( wc4>~~~ Tlutl-e.J;. t k~lfkrtrl '«~ ~ ~ ~-l.<4i dy~q_~,-J.t.Q,,,t !)- s:'~'-<rH ~

"1A J-\- V J2 v.Q_ ( ~ ) :: c_Ot f> lt )

~-·~lni~t.i·~ ~ ~r9.~t_~t°->~--±~;; -l %_ E"l-~ ~ MJ~

<k.J:~.- 1T(v) : -= L~~ JCC~ lc.~k,. lJQ~ t>o..t s. i.tf;frß V lk.t.

~;~J>0r~p4 (G +(t,.s>)><. =: ~x ~;>. k~kt c°'

1~~:._.;r--J :.~,~i.~-;~~· ~:;~~l ~ ~~-- . . . ··--·- . ------+----···- ______ 16_ __ ~-J.-·---·---·......:..__. __ _

if ( s) ~ 0 ( c:; X R_ f l..QJ k... ""'4 "11 ._\ ~ l,• v4, '{ 'r k ..a ) T\ (t. )>,. T ( >""t 1' ~l "" 1 ~ G X -<:., - <;) ltJ_,j

~ t L ~v.~ l ~ k„\ ~ J < T

-=o G' )<, · ·ho-.l ~~ l\.k vtvt i7k&

~rhpf""-~ 1 IY'Oi_ X: k°"~- ~;{,(;u._~ )

lf(w) ~ T](V) + TGL F 0 ·J:v.C-vu~ 1-J ~I lut'~!~~

=o j W-<>j ?0 ''"' ?[°'; ~~Lt 'j "t.u._ X ~; ~ Q c T'; U. .,,_J__

(J_) duv .e..J.->k Te.._si .e„..,.Je~ t; l~ ~ f '""- v VLa. Q 'Ze;.{ 1T (v) cti;

( ~"" <1.µ ~ ~ lv.€~, Jw.~ Q, ~ r1.u 'f~ ~S 1'~ V ~ h ~ i\ (V)

~k°""'-t.c.J > ~Q.J :~O!..( 11(v)) ::: L XP :- Xq_ ~ U.~ )

'P~Ot o&

\11) ~ .e_h-k z~~> rtJJ,_ ?o ~;_>i:.t \{) 19~ 1Tt\v)

( ~t ~Cl ( Tf(t,;>) ::: 2 ;(p ~ )(Cl '"" Le~ ) 'j>7o_

--' - - ~!~

t \JJ~j_ 'P 1~ ~;~.9 r'°'P\_ Gx \Mi+ La;"'~ -rrc v) , t<r>~() 1-1„ Jl t}_~ ~"'~"'.s. 'P0 ~'f4 ~~s.~ -=P ~clx 1.vo.til~~.„J~ s1'-{J. 1'1..( G><'

-=::v ?c>,"l ..,.'7oCv1sJ [j.t ?7WL ~:::/·i~ La.'~r 11-t Gx

'~() 4J r lt; SF cO 't.<.t t fYr/,.,· "" J_; f c.1. (Je..,,,_ X

t~sJ 'k........L~\~ ? 9 ~ .f>11;s.d- l.v ver 1T( 1.J)

~v ( &~ ~~ . e,,) 17M~ Q~ l ~~.,J ß.,,_ff ..._1 ~ sclti<:kt-)

lk4q~fu LJ.e~ ?„ + <i<..:::ckl.; c..'b ~ l4_ ... ~ ~ 'P0 1

c .,... tv t > "' ... ,_.ft ,r,-- N\ _,,. --f

Di'e-;. .--.s+ w12~ ·it.. ~x ~i~ U.'"'~ ( u~ie.,) l~)

TC ( f ) - -z: C ?0' ) < · Tf. ( t ) -r vc iv) - 11 tO ~ i1 C w)

LJibi~r9 JCV'!N 1 ~c.ss il(w) L..:."'~· ~~ l<:i.:.~~ UJ.e~ e.i:.i IJ tl\_ Gx ,J 0

(ke1'Lte, Ce:/~~ if;i, v..ell's~-~ i..vt- )

(

=o \A.l,(,f ~r l.!crtvo...'< b L<~"!k....

-·

'b. 2- SATi: ( ~r-J - t="\.Qkeneq ~~ ) . (1) \2v MJtf.; /.-~ tq ~ l~ ~ ;.e.r f

.e{~ c;JJwiJl Jyi;,. s,t- t}~~rl ~~ &....1lc.ri·-c~t T.

(2"> 'Yr~ L~r~q lu>'f~ oe~„1.i1erJ ~,J J,,~ ~Rt.tv.~ ~ >i~ ~';~ Wt il-< - c~+- '+}Ä.L1.ei. ><.

(3) ~ 0ph'kJ2e iy~~14 u,~ +~,~ nj ~~.[}.. W?Jd.vk~+

.~ ~,)_ LlÖ'k.t~ i ()~~. ~-~l"t.t'-'\ f'-( k\,(_() ~ ~s. ~· .,______ ---------------~~--6\---,~-~~--~--~--~~-.--...-----=-~_,.,.,.~

h.i..~i +:&nv ?~1.v1. Jl., L? f er !M,Qt<:.R...- (?) ( 'Du..~ rc.r1.-vt.~·1e ~ (D) GU!'Gk~-E: TJ \IT! \ ~-~r Jct~J v~(\) ~ 0PT( 'P) 1

" J v~(f) ~ Of'T (3>)