Cominoril Opimizion Prolm : oluion. Suppo impl unir rp mor n on minimum pnnin r. Cn Prim lorim (or Krukl lorim) u o in ll o m? Explin wy or wy no, n iv n xmpl. Soluion. Y, Prim lorim (or Krukl lorim) n u o in ll minimum pnnin r. O our, o lorim only prou inl minimum pnnin r wn i i run, o w w rlly mn i vry minimum pnnin r i poil rul o o o lorim; y mkin ppropri oi urin lorim, ny priulr minimum pnnin r n oun. W in wi lmm wo proo i nilly m proo o Torm. in Ppimiriou n Siliz. Lmm. L G = (V, E) onn rp wi n wi (u, v) or {u, v} E. L (V, T ) minimum pnnin r o G, n l (V, F ) pnnin or u F T. L U V vrx o onn omponn o (V, F ), n l U E o o G vin xly on npoin in U. (T U i ll ounry o U.) Tn mon in U vin minimum wi, r i on i in T. Proo. Suppo or k o onriion o minimum-wi in U onin no o T. L {u, v} U minimum-wi. A {u, v} o T ; i r (uniqu) yl. Bu i yl onin l on vrx in U n l on vrx no in U (nmly, u n v), i mu onin nor {u, v } U irn rom {u, v}, n {u, v } mu in T. By umpion, (u, v) < (u, v ), o i w rmov {u, v } w oin nw pnnin r (V, T ), wi T = T { {u, v} } \ { {u, v } }, wi rily mllr wi n (V, T ). Bu i onri (V, T ) i minimum pnnin r. E irion o Prim lorim (or Krukl lorim) oni o in n o pnnin or, iniilly vin no, unil pnnin or om pnnin r. Tror, pnnin or rul r k irion o ir o lorim xly k. Clim. Fix minimum pnnin r (V, T ) o G. For vry 0 k n, r k irion o Prim lorim (or Krukl lorim), i i poil or rulin pnnin or (V, F ) o iy F T. Proo. By inuion on k. Clrly mn i ru or k = 0, u o Prim lorim n Krukl lorim in wi pnnin or vin no, o F = T. Suppo k. By inuion, i i poil, r k irion, or Prim lorim (or Krukl lorim) o prou pnnin or (V, F ) vin F T ; in k < n, w know F T. In k irion o Prim lorim, minimum-wi i l ou o U o vin xly on npoin in U, wr U i vrx o onn omponn o (V, F ) oninin ix vrx v rirrily on innin o lorim. In k irion o Krukl lorim, minimumwi = {u, v} i l ou o o ll wo npoin r in irn onn omponn o (V, F ); i i in U, wr U i vrx o onn omponn o (V, F ) oninin u. In ir, w y lmm ov mon in U vin minimum wi, r i on i in T, o w my l o n in T, n n F := F {} T. Tror, kin k = n in i lim, w i i poil or Prim lorim (or Krukl lorim) o prou r (V, T ). A (V, T ) w n rirry minimum pnnin r, i ow i i poil or o o lorim o in ny minimum pnnin r.
A n xmpl, onir rp own on l low. pnnin r, own prly. I wo minimum I v i on o vrx innin o Prim lorim, n o Prim lorim n Krukl lorim r y lin {, }. In nx irion, o {, } n {, } r ni o l. Slin {, } l o minimum pnnin r own in nr ov, wil lin {, } l o on own on ri.. Explin ow minimum pnnin r lorim n ily u o in mximum pnnin r in rp. Tn xplin wy, i you wn o in lon p wn wo vri in rp, uin i m niqu wi Dijkr lorim o no work. Soluion. A mximum pnnin r n oun y nin ll o wi n n pplyin minimum pnnin r lorim o rulin rp. Alrnivly, i w woul lik o work only wi nonniv wi, w n in lr wi M, rpl vry wi w wi M w, n n pply minimum pnnin r lorim. Ti i quivln o nin ll o wi n n inrin wi y M. Ti ppro work u vry pnnin r on rp wi n vri xly n, o ol wi o vry pnnin r will inr y xly (n )M. T impl ron nin wi o no work o in lon p wi Dijkr lorim i Dijkr lorim rquir ll wi o nonniv. T M w rik on work wi Dijkr lorim ir, u irn p wn ivn pir o vri my oni o irn numr o. Nin wi n n in M o wi will ror irn mulipl o M o ol wi o p, n i will no nrily prrv lon or or p.. Fin mximum low n minimum u in ollowin low nwork. Soluion. T irion o For Fulkron lorim r own low. In iur, pii o r r own irl in ry, no ll r in rn,
n nonzro low lon r r in li lu. T no ll r ivn in orm (rom[i], ow-mu[i]). Ar wi zro low r rwn ri lk rrow, ur r r rwn oul r rrow, n unur r wi nonzro low r rwn wvy li lu rrow. Blow iur i LIST o vri uil up For Fulkron lorim pror rou nwork. W r LIST quu r: no r o ri-n i wn y r ll, n y r rmov rom l-n i (ini y roin m ou) wn y r nn. (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) LIST : \; \, \; \, \, \; ; ;. T no riv ll, o w v oun n umnin p. T vlu o ll ow-mu[] i, o w n umn low y lon i p. To iniy p, w ollow rom ll kwr rom : rom[] =, rom[] =, n rom[] =, o our umnin p i. All o rom ll lon i p wr poiiv (y i no v lin minu in), o low lon vry r in i p will inr y. W ju low n rp llin pro rom innin. (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) LIST : \; \, \; \, \; \; \; \; ;. (, ) Followin rom ll kwr rom, w in umnin p. W umn low lon vry r in i p y ow-mu[] = n rp llin pro rom innin.
(, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) LIST : \; \, \; \, \; \; \; \; \;. (, ) T umnin p r i ; w umn low lon i p y. (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) LIST : \; \, \; \, \; \, \; \; \;. (, ) T umnin p r i ; w umn low lon i p y. (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) LIST : \; \, \; \; \; \; \, \; \;. T umnin p r i ; w umn low lon i p y.
(, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) LIST : \; \, \; \; \, \; \, \; \;. T umnin p r i ; w umn low lon i p y. (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) LIST : \; \, \; \; \, \; \; \;,. T umnin p r i ; w umn low lon i p y. (, ) (, ) (, ) (, ) (, ) (, ) LIST : \; \, \; \; \, \.
Now i no ll, o r i no umnin p, o i low i opiml. T vlu o i low i. A minimum u i rmin y priionin no orin o wr y riv ll in i l irion: W = {,,,,, }, W = {,,, }. W W T piy o i u i um o pii ij o r poin rom no i in W o no j in W : + + + = + + + =, wi qul vlu o mximum low, in orn wi mx-low min-u orm. No lo in mximum low ll r o orwr ro u (rom W o W ) r ur wil ll r o kwr ro u (rom W o W ) v zro low.. Crully ri n lorim or ollowin prolm: Givn impl unir rp G = (V, E), rmin wr G i ipri, n i o iv ipriion (i.., priion o vrx V ino wo U n W u vry on npoin in U n on npoin in W ). Illur oprion o your lorim on wo xmpl, on ipri rp n on non-ipri rp. Prov your lorim i orr in nrl. Soluion. Hr r p o n lorim or i prolm.. Bin wi ll vri unolor. S LIST :=.. Coo n unolor vrx v. Color v r. A v o LIST.. Coo vrx w in LIST, n rmov w rom LIST.. I ny nior o w m olor w, op: rp i no ipri.. For unolor nior x o w: olor x lu i w i r, or olor x r i w i lu; n n x o LIST.. I LIST i nonmpy, o k o p.. I r ill xi unolor vri, o k o p.. W r on. T rp i ipri. L U o r vri, n l W o lu vri; n (U, W ) i ipriion. For xmpl, onir ollowin rp, wi i ipri:
. W in wi ll vri unolor n n mpy LIST.. W oo n unolor vrx, y, n olor i r. W o LIST. Now LIST = {}.. W oo vrx in LIST ; i mu, i only lmn o LIST. W rmov i rom LIST, o now LIST i mpy in.. No nior o m olor, o w kip p.. W olor unolor nior o lu n vri o LIST. T vri r n. Now LIST = {, }.. LIST i nonmpy, o w o k o p.. W oo vrx in LIST, y, n rmov i rom LIST. Now LIST = {}.. No nior o m olor, o w kip p.. W olor unolor nior o r n vri o LIST. Ti i only vrx. Now LIST = {, }.. LIST i nonmpy, o w o k o p.. W oo vrx in LIST, y, n rmov i rom LIST. Now LIST = {}.. No nior o m olor, o w kip p.. No nior o i unolor, o r i noin o o in p.. LIST i nonmpy, o w o k o p.. W oo vrx in LIST, wi mu, n rmov i rom LIST. Now LIST i mpy.. No nior o m olor, o w kip p.. No nior o i unolor, o r i noin o o in p.. LIST i mpy, o w kip p.. Tr ill xi unolor vri, o w o k o p.. W oo n unolor vrx, y, n olor i r. W o LIST. Now LIST = {}.. W oo vrx in LIST, wi mu, n rmov i rom LIST. Now LIST i mpy.. No nior o m olor, o w kip p.. W olor unolor nior o lu n vri o LIST. Ti i only vrx. Now LIST = {}.. LIST i nonmpy, o w o k o p.. W oo vrx in LIST, wi mu, n rmov i rom LIST. Now LIST i mpy.
. No nior o m olor, o w kip p.. W olor unolor nior o r n vri o LIST. T vri r n. Now LIST = {, }.. LIST i nonmpy, o w o k o p.. W oo vrx in LIST, y, n rmov i rom LIST. Now LIST = {}.. No nior o m olor, o w kip p.. No nior o i unolor, o r i noin o o in p.. LIST i nonmpy, o w o k o p.. W oo vrx in LIST, wi mu, n rmov i rom LIST. Now LIST i mpy.. No nior o m olor, o w kip p.. No nior o i unolor, o r i noin o o in p.. LIST i mpy, o w kip p.. All vri v n olor, o w kip p.. T rp i ipri, n U = {,,,, }, W = {,, } i ipriion. For nor xmpl, onir ollowin rp, wi i no ipri:. W in wi ll vri unolor n n mpy LIST.. W oo n unolor vrx, y, n olor i r. W o LIST. Now LIST = {}.. W oo vrx in LIST, wi mu, n rmov i rom LIST. Now LIST i mpy.. No nior o m olor, o w kip p.. W olor unolor nior o lu n vri o LIST. T vri r n. Now LIST = {, }.. LIST i nonmpy, o w o k o p.. W oo vrx in LIST, y, n rmov i rom LIST. Now LIST = {}.. No nior o m olor, o w kip p.
. W olor unolor nior o r n vri o LIST. Ti i only vrx. Now LIST = {, }.. LIST i nonmpy, o w o k o p.. W oo vrx in LIST, y, n rmov i rom LIST. Now LIST = {}.. No nior o m olor, o w kip p.. W olor unolor nior o lu n vri o LIST. Ti i only vrx. Now LIST = {, }.. LIST i nonmpy, o w o k o p.. W oo vrx in LIST, y, n rmov i rom LIST. Now LIST = {}.. W noi on o nior o, nmly, m olor. Tror w rpor rp i no ipri. In orr o prov orrn o i lorim, w ir prov Propoiion rom Sion A. o Ppimiriou n Siliz, n in orr o o w ir prov lmm. Lmm. Evry lo o wlk onin n o yl. Proo. By inuion on ln l o wlk. T or nrl rp i l =. In i, wlk oni o loop (n joinin vrx o il). T loop il i n o yl. No i nno our in impl rp, wi no loop. T or impl rp i l =. In i, wlk mu yl o ln (u rp o no onin loop). For inuiv p, uppo l (or l or nrl rp). I no vrx i rp in wlk (xp ir n l vrx, wi r m), n wlk il i n o yl. Orwi, om vrx v i rp, n w n rk wlk ino wo v v wlk. Sin l i o, ln o on o v v wlk mu o, n y inuion i onin n o yl. Propoiion. A rp i ipri i i no irui o o ln. Proo. (= ) Suppo G i ipri, n l (U, W ) ipriion o vrx. Tn vry on npoin in U n on npoin in W, o vri o wlk lrn wn vri in U n vri in W. Tu wlk in n n m vrx (in priulr, yl) mu v vn ln. So G no yl o o ln. ( =) Suppo G = (V, E) no yl o o ln. W ll onru ipriion (U, W ) o G. Wiou lo o nrliy, w my um G i onn; orwi, w n onru ipriion (U i, W i ) o onn omponn prly n n k U = U i n W = W i. Fix n rirry vrx u V, n or ll v V in (v) o or ln o p rom u o v. [Sin G i onn y umpion, (v) i wll in or ll v V.] L U = { v V : (v) i vn } n W = { v V : (v) i o }. I r i n wn wo vri u, u U, n or p rom u o u, plu rom u o u, plu or p rom u o u yil lo o wlk in U. By prin lmm, u wlk onin n o yl. Bu i onri ypoi G onin no o yl, o r n no u wn vri in U. Likwi, r n no wn vri in W. So (U, W ) i ipriion. Now w li orrn o lorim. Suppo lorim rpor rp i ipri. Ti mu ppn in p, wi nno ppn unil ll vri v n olor n pro y p. I rulin vrx olorin o rp in m olor o o npoin o n, n i woul v n noi in p wn proin on npoin; o i
o no ppn. Tror vry on r npoin n on lu npoin, o (U, W ) i ipriion, n rp i in ipri. On or n, uppo lorim rpor rp i no ipri. i our wn i i noi in p wo jn vri w n x v m olor. T vri on v in p r ir vri in ir onn omponn o in olor. Evry or vrx i in r i i i n vn in rom on o v or lu i i i n o in (wr in mn numr o in or p ). I wo jn vri v m olor, n r i lo o wlk in rp [ in ( =) pr o proo o Propoiion ], o rp onin n o yl, o i i in no ipri.. Fin mximum min in ollowin ipri rp. u v w x y z Soluion. On wy o o i i o ru prolm o o inin mximum low in ollowin nwork: u v w x y z 0
T orrponin mx-low LP i own low. In i LP, vril v no vlu o low, n or r (i, j) in nwork vril ij no low lon r. T ir onrin nor low ln vry no: n oulow vry no mu 0, xp no wr i i v n no wr i i v. T rminin onrin nor pii o r rom n r o. mximiz uj o v + + + + + = v u + w = 0 u + v + w + y = 0 x + z = 0 u + v + w + y = 0 x + z = 0 x + z = 0 u u u u = 0 v v v = 0 w w w w = 0 x x x x = 0 y y y = 0 z z z z = 0 u v w x y z = v u v w x y z ll vril nonniv. An opiml oluion o i LP i v =, w = y = x = v = z =, = = = = = v = w = x = y = z =, n ll or vril 0. Tror, mximum min in i rp i own low. u v w x y z (Ti mximum min i no uniqu.)
. A mll rukin ompny l o iv ruk, n on rin y vn lo o livr. In ollowin l, pii o ruk n iz o lo r o ivn in uni o 000 poun. Dily Truk Cpiy o $00 $00 $00 $0 $00 Lo Wi A B C D E F G T ily o or ruk mu pi i ruk i o u o mk ny livri. Bu o ir loion, lo A n D nno livr y m ruk, nor n lo B n E. Formul n inr prorm o rmin wi lo oul in o ruk in orr o minimiz ol ily o. Soluion. For i {A, B, C, D, E, F, G} n j {,,,, }, l x ij {0, } ini wr lo i i o livr y ruk j. For j {,,,, }, l y j {0, } ini wr ily o i o pi or ruk j. Our ojiv i o minimiz ol ily o, wi i 00y + 00y + 00y + 0y + 00y. W v vrl o onrin. Fir, vry lo mu livr y xly on ruk, o x ij = or ll i {A, B, C, D, E, F, G}. j= T ruk pii nno x, o x Aj + x Bj + x Cj + x Dj + x Ej + x Fj + x Gj W j or ll j {,,,, }, wr W j i piy o ruk j. T ily o mu pi o u ruk, o x ij y j or ll i {A, B, C, D, E, F, G} n ll j {,,,, }, u i y j = 0 n x ij mu lo 0. Finlly, lo A n D nno livr y m ruk, nor n lo B n E, o x Aj + x Dj or ll j {,,,, }, x Bj + x Ej or ll j {,,,, }.
Tror w v ollowin inr prorm. minimiz 00y + 00y + 00y + 0y + 00y uj o x A + x A + x A + x A + x A = x B + x B + x B + x B + x B = x C + x C + x C + x C + x C = x D + x D + x D + x D + x D = x E + x E + x E + x E + x E = x F + x F + x F + x F + x F = x G + x G + x G + x G + x G = x A + x B + x C + x D + x E + x F + x G x A + x B + x C + x D + x E + x F + x G x A + x B + x C + x D + x E + x F + x G x A + x B + x C + x D + x E + x F + x G x A + x B + x C + x D + x E + x F + x G x A y, x A y, x A y, x A y, x A y x B y, x B y, x B y, x B y, x B y x C y, x C y, x C y, x C y, x C y x D y, x D y, x D y, x D y, x D y x E y, x E y, x E y, x E y, x E y x F y, x F y, x F y, x F y, x F y x G y, x G y, x G y, x G y, x G y x A + x D, x B + x E x A + x D, x B + x E x A + x D, x B + x E x A + x D, x B + x E x A + x D, x B + x E x ij {0, } or ll i {A, B, C, D, E, F, G} n ll j {,,,, } y j {0, } or ll j {,,,, }. T ormulion i ll i nry or i prolm. I i inr prorm i olv, i i oun r r i opiml il oluion, vin ol o $0: A, B A, B B B C C C C F F A, F A, F A, F A, E B, D F D, E G D, E G D, E G G D, E C, G C, D, E C, G C, D, E B, G B, D, F A, E, F A, B, G For inn, ir oluion ov x A = x B = x C = x D = x E = x F = x G =, y = y = y = y =, n ll or vril 0.