A comparison of routing sets for robust network design

Transcription

1 A omprison of routing sts for roust ntwork sign Mihl Poss Astrt Dsigning ntwork l to rout st of non-simultnous mn vtors is n importnt prolm rising in tlommunitions. Th prolm n sn two-stg roust progrm whr th rours funtion onsists in hoosing th routing for h mn vtor. Allowing th routing to hng ritrrily s th mn vris yils vry iffiult optimiztion prolm so tht iffrnt susts of missil routings hv n isuss in th litrtur. In this ppr, w ompr thortilly th optiml pity llotion osts for six of ths routing sts: ffin routing, volum routing n its two simplifitions, th routing s on n ritrry -ovr of th unrtinty st, n th routing s on ovr limit y hyprpln. W show tht th two routing sts s on ovrs of th unrtinty st yil th sm optiml osts. W show thn tht th two simplifi volum routings r spil ss of ffin routings. Finlly, ssuming tht th unrtinty st is th on stui y Brtsims n Sim (004), w show tht th optiml ost provi y volum routing is not lss thn th osts provi y th simplifi volum routings. Kywors: Roust optimiztion; Ntwork sign; Routing st; Routing tmplt; Affin routing. Introution Givn grph n st of point-to-point ommoitis with known mn vlus, th trministi ntwork sign prolm ims t instlling nough pity on th rs of th grph so tht th rsulting ntwork is l to rout ll ommoitis. In prti it is howvr vry iffiult to know with prision th xt vlus of th mns t th tim th sign isions r tkn. In th st s, w n stimt st tht ontins most likly vlus for th mn. Th introution of th unrtinty st ls to roust optimiztion prolm. In this ontxt, solution is si to fsil for th prolm if it is fsil for ll mn vtors tht long to th stimt unrtinty st D, s Soystr [8] n Bn-Tl n Nmirovski [9, 0], mong othrs. This rigi frmwork is omputtionlly sy ut it os not llow th mol to rt ginst th unrtinty. To rss this rwk, Bn-Tl t l. [8] introu two-stg roust optimiztion mols tht llows to just sust of th prolm vrils only ftr osrving th tul rliztion of th t. This justing prour is oftn ll rours. This two-stg pproh pplis nturlly to ntwork sign sin first stg pity sign isions r usully m in th long trm whil th routing isions pn on th rliztion of th mn. Hn, th routing isions n sn s th rours. Fr rours is ll ynmi routing in th ontxt of roust ntwork sign prolms. It hs n shown y Chkuri t l. [4] n Gupt t l. [8] tht th roust ntwork sign with ynmi routing is intrtl. Alry iing whthr or not fix pity sign llows for ynmi routing of mns in givn polytop is o-n P-omplt (on irt grphs). It is known lry tht two-stg roust progrmming with ritrry rours is omputtionlly intrtl [8]. For this rson, Bn-Tl t l. [8] limit th rours to ffin funtions of th unrtintis whih mks th prolm trtl. Furthr works y Chn n Zhng [5] n Goh n Sim [7] suggst to xtn th son stg to pi-wis linr funtions of th unrtintis. In ft, onsiring spil typs of rourss h n us lry in th ontxt of ntwork sign. Bn-Amur n Krivin [4, 5] introu th onpt of stti routing: ftr fixing th sign, th routing of ommoity is llow to hng ut only linrly with th vrition of th ommoity. Stti routing n lso sn s singl stg roust progrm whr th st of routings pths togthr with th prntl splitting mong th pths r hosn t th sm tim th sign isions Dprtmnt of Computr Sin, Fulté s Sins, Univrsité Lir Bruxlls, Brussls, Blgium, mposs@ul..

2 r m. Th rsulting st of pths n prntl splitting is oftn ll routing tmplt, whih is us y ll mn vtors in th unrtinty st. Th us of stti routing mks th roust ntwork sign prolm trtl ut it yils mor xpnsiv pity llotions thn th prolm with ynmi routing. Stti routing hs n us y vrious uthors sin its introution y Bn-Amur n Krivin, inluing Altin t l. [], Kostr t l. [9], Oróñz n Zho []. Svrl uthors tri to introu routing shms tht r mor flxil thn stti routing whil still ing omputtionlly sir thn ynmi routing. Bn-Amur [3] ovrs th mn unrtinty st y two (or mor) susts using sprting hyprplns n uss spifi routings tmplts for h sust. Th rsulting optimiztion prolm is N P-hr whn no ssumptions is m on th hyprplns. Sutllà [7] gnrlizs this i to ritrry ovrs of th unrtinty st. Sh llows st of routing tmplts to us onjointly so tht h mn vtor n rout y t lst on of th routing tmplts. Sh lso introus prour tht works in two stps. First, n optiml pity llotion with stti routing is omput. Thn, sh llows llows to rrout prt of th mn vtors oring to son routing tmplt. Bn-Amur n Zotkiwiz [6] introu volum routing, frmwork tht shrs th mn twn two routing tmplts, oring to thrshols. Thy prov tht th rsulting optimiztion prolm is N P- hr n introu two simplifitions. Finlly, pplying th ffin rours from Bn-Tl t l. [8] to roust ntwork sign prolms, Ouorou n Vil [4] introu th onpt of ffin routing. Rntly, Poss n Rk [6, 5] stuy th proprtis of ffin routing, n ompr th ltr to th stti n ynmi routings, oth thortilly n mpirilly. Thy onlu tht ffin routing tns to yil vry goo pproximtions of ynmi routing whil ing omputtionlly trtl. In this ppr, w ompr thortilly th optiml pity llotion osts provi y th ffin routings from Ouorou n Vil [4], th volum routings from Bn-Amur n Zotkiwiz [6], n th routings s on ovrs of th unrtinty st in two susts (Bn-Amur [3] n Sutllà [7]). In th nxt stion, w introu th roust ntwork sign prolm n fin routing st. W mol th roust ntwork sign prolm with th xpliit pnny on th routing st n formliz h of th routing frmworks stui hrin. Our min rsults r stt in Stion.3. In Stion 3, w try to unrstn how goo is th ost of th optiml pity llotion provi y h of th routing sts, n w ompr ths osts mong th iffrnt routing sts. W strt y ompring th osts otin with routings s on ovrs of th unrtinty st in Stion 3.. Thn, in Stion 3.3, w turn to ffin n volum routings. In Stion 4, w prsnt xmpls showing tht it is not possil, in gnrl, to ompr som of ths osts. Finlly, w onlu th ppr in Stion 5. Roust ntwork sign. Prolm formultion Th prolm is fin low for irt grph G = (V, A) n st of ommoitis K. W formliz first th onpt of routing. Thn, w introu th roust ntwork sign prolm. Eh ommoity k K hs sour s(k) V, stintion t(k) V, n mn vlu k 0. A multi-ommoity flow is vtor f R A K + tht stisfis th flow onsrvtion onstrints t h no of th ntwork: δ + (v) f k δ (v) k if v = s(k) f k = k if v = t(k) 0 ls for h v V, () whr δ + (v) n δ (v) rsptivly not th st of outgoing rs n inoming rs t no v. In this work th vlus of th mn vtor r unrtin n long to th los, onvx, n oun st D R K +. W ll suh st n unrtinty st n ny D is ll rliztion of th mn. W not y (D, R A K ) th st of ll funtions from D to R A K. Thn, routing is funtion f (D, R A K ) tht stisfis () for ll rliztions of th mn, tht is f k () k if v = s(k) f k () = k if v = t(k) for ll v V, D, () δ + (v) δ (v) 0 ls n tht is non-ngtiv f k () 0 for ll D. (3)

3 A routing with no furthr rstritions is ll ynmi routing. Hn, th st of ll ynmi routings is th st of ll funtions from D to R A K tht stisfy () n (3): F f (D, R A K ) f stisfis () n (3). (4) In this ppr, w r intrst in using spil kins of routings. This orrspons to using spifi susts F F. Ths susts r sri in th nxt stion. In wht follows, w sri th roust ntwork sign prolm mking xpliit th st of missil routings. A vtor x R A + is ll pity llotion. A pity llotion is si to support th st D if thr xists ynmi routing f F srving D suh tht for vry D th orrsponing multi-ommoity flow f() os not x th pitis sri y x. Similrly, w sy tht (x, f) supports D whn oth th routing f n th pity llotion x r givn. Mor gnrlly, w sy tht (x, F ) supports D whn thr xists routing f F suh tht (x, f) supports D. Givn n unrtinty st D n routing st F F, roust ntwork sign now ims t proviing th ost miniml pity llotion x suh tht (x, F ) supports D: min κ x A RND(F ) s.t. f F (5) f k () x, D (6) k K x 0, whr κ R is th ost for instlling on unit of pity on r A. Noti tht in rl pplitions, ths osts r usully non-ngtiv. W shll not th optiml ost of RND(F ) y opt(f ). Prolm RND(F ) ontins n infinit numr of vrils f() for ll D s wll s n infinit numr of pity onstrints (6). Morovr, th prolm my not vn linr, pning on th onstrints fining st F. Consiring th st of ll routings F, RND(F) is two-stg roust progrm with rours following th mor gnrl frmwork sri y Bn-Tl t l. [8]. Th pity sign hs to fix in th first stg, n osrving mn rliztion D, w r llow to just th routing f() ritrrily in th son stg. In tht s, (5) is rpl y () n (3) so tht RND(F) is linr progrm, yt infinit. Whnvr D is polytop, Poss n Rk [6], mong othrs, show how to provi finit linr progrmming formultion for RN D(F). Th formultion is s on numrting th xtrm points of D, so tht its siz tns to inrs xponntilly with th numr of ommoitis. In ft, th prolm is vry iffiult to solv givn tht only iing whthr givn pity llotion vtor x supports D is on P-omplt for gnrl polytops D, s Chkuri t l. [4] n Gupt t l. [8]. Morovr, th us of ynmi routings suffrs from nothr rwk. It my iffiult in prti to hng ritrrily th routing oring to th mn rliztion. For ths rsons, vrious uthors stuy rstritions on th routings tht n us, introuing iffrnt susts of routings F F. Thir hop is tht opt(f ) provis goo pproximtion of opt(f) whil yiling n sir optimiztion prolm RND(F ). For instn, Frngioni t l. [6] n Poss n Rk [6] show unr vry strong ssumptions on D tht th optiml pity llotions provi y ynmi routings r quivlnt to th ons provi y stti routings n ffin routings, rsptivly, whih r polynomilly solvl whn D hs ompt formultion. In th nxt stion, w prsnt iffrnt hois of F isuss in th litrtur, inluing stti n ffin routings. Thn, w summriz th min ontriutions of this ppr in Stion.3. Not tht if thr xists only on pth from s(k) to t(k) for ommoity k K, thn ll routings oini for tht ommoity. Unlss stt othrwis, in th following w ssum tht for ll k K thr xist t lst two istint pths p, p in G from s(k) to t(k), tht is, two pths tht iffr y on r t lst.. Routings frmworks In th nxt stions, w fin formlly th st of stti routings n th routing sts from Ouorou n Vil [4], Bn-Amur [3], Sutllà [7] n Bn-Amur n Zotkiwiz [6]. 3

4 .. Stti routing Th simplst ltrntiv to ynmi routing hs n introu y Bn-mur [5] n hs n us xtnsivly sin thn, s Altin t l. [, ], Kostr t l. [9], Muhntongsuk t l. [], n Oróñz n Zho []. This frmwork onsirs rstrition on th son stg rours known s stti routing (lso ll olivious routing). Eh omponnt f k : D R A + is for to linr funtion of k : f k () := y k k A, k K, D. (7) Noti tht (7) implis tht th flow for k is not hnging if w prtur th mn for h k. By omining () n (7) it follows tht th multiplirs y R A K + stisfy to y k if v = s(k) y k = if v = t(k) for h v V. (8) 0 ls δ + (v) δ (v) Th flow y is ll routing tmplt sin it is, for vry ommoity, whih pths r us to rout th mn n wht is th prntl splitting mong ths pths. W fin formlly th th st of ll routing tmplts s Y y R A K + y stisfis (8), (9) n th st of ll stti routings s F stt f (D, R A K ) y Y : f k () = y k k A, k K, D. An importnt rsult is tht ompt linr formultion n provi for RND(F stt ) s long s th sription of D is ompt (s Altin t l. [] mong othrs). Hn, th rsulting optimiztion prolm is polynomilly solvl. In th following, w rviw ltrntiv routing sts F tht r lss rstritiv thn stti routings whil not ing s flxil s ynmi routings. Si iffrntly, F stt F F... Covrs of th unrtinty st limit y hyprpln Givn st D, olltion of susts of D forms ovr of D if D is sust of th union of sts in th olltion. Bn-Amur [3] introus th i of ovring th unrtinty st y two (or mor) susts using hyprplns n proposs to us routing tmplt for h sust. This yils th following st of routings: F f (D, R A K ) y, y Y n α R K, β R : f k () = y k k D, α β y k k D, α β A, k K, D. Th finition ov implis tht oth routing tmplts y n y must l to rout mn vtors tht li in th hyprpln, α = β without xing th pity. H provs tht RND(F ) is N P-hr in gnrl n sris simplifition shms, whr α is givn. H furthr works on th frmwork in Bn-Amur n Zotkiwiz [7]...3 Aritrry ovrs of th unrtinty st Sutllà [7] introus th i of using onjointly two routing tmplts. Formlly, sh proposs to us two routing tmplts y n y suh tht h D n srv ithr y y or y y (or oth). This yils th following st of routings: F f (D, R A K ) y, y Y n D, D D, D = D D : f k () = y k k D y k k D A, k K, D. 4

5 Sh mntions tht th omplxity of RND(F ) is unknown. W show in this ppr tht this optimiztion prolm is N P-hr, us it is gnrliztion of RND(F ), prov to N P- hr y Bn-Amur [3]. Th frmwork sri y F hs n inpnntly propos for gnrl roust progrms y Brtsims n Crmnis [] (s lso Brtsims t l. []) whr th uthors propos to ovr th unrtinty sts with k susts n vis inpnnt sts of rours vrils for h of ths susts...4 Volum routings Mor rntly, Bn-Amur n Zotkiwiz [6] introu frmwork tht shrs th mn twn two routing tmplts, oring to thrshols h k for h k K. Formlly, thy us th following st of routings: F V f (D, R A K ) y, y Y, h R K + : f k () = y k min( k, h k ) + y k mx( k h k, 0) A, k K, D. Thy prov tht RND(F V ) is n N P-hr optimiztion prolm. Hn, thy introu simplr frmworks sri low. Dfining k min = min D k n k mx = mx D k, th st of routings oms on of th following F VS F VG f (D, R A K ) y, y Y : f k () = y k k min + y k ( k k min) f (D, R A K ) y, y Y : f k () = y k k min k mx k k mx k + y k min k mx k k min k mx k min A, k K, D, A, k K, D, whih r oth wll-fin whnvr k min < k mx for h k K. Whn k min = k mx for som k K, th k-th omponnt of f F VG is fin y f k () = y k k...5 Affin routings Bn-Tl t l. [8] introu Affin Ajustl Roust Countrprts rstriting th rours to n ffin funtion of th unrtintis. Ourou n Vil [4] pply this frmwork to roust ntwork sign y rstriting f k to n ffin funtion of ll omponnts of giving F ff f (D, R A K ) f 0 R K, y R A K : f k () = f 0k + h K y kh h A, k K, D, f stisfis () n (3). This frmwork hs n ompr thortilly n numrilly to stti n ynmi routings y Poss n Rk [6]. In prtiulr, th uthors show tht ompt formultion n sri for RND(F ff ) s long s D hs ompt sription, gnrlizing th rsult otin for stti routing lry. W point out tht mjor iffrn twn F ff n th routing sri in Stion is tht th formrs r uil up using routing tmplts, so tht it is impliitly ssum tht flow onsrvtion onstrints () n non-ngtivity onstrints (3) r stisfi. In opposition, routings in F ff r uil up using orinry vtors so tht tht stisftion of () n (3) must stt xpliitly..3 Contriutions of this ppr Th ojtiv of this ppr is to ompr opt(f ) mong th routing sts rll in prvious stions. This omprison is rri out in Stion 3. Our min rsults r stt nxt. () Lt D n unrtinty st. It hols tht opt(f ) = opt(f ), opt(f ff ) opt(f VG ) opt(f VS ), n opt(f V ) opt(f VS ) for ny ost vtor κ R A. () Lt D n unrtinty polytop suh tht for h k K, thr xists non-ngtiv numrs 0 k min k mx suh tht k k min, k mx for h xtrm point of D. It hols tht opt(f V ) = opt(f VS ) for ny ost vtor κ R A. 5

6 Th polytop introu y Brtsims n Sim [3], us for roust ntwork sign prolms in [6, 3, 4, 0, 6], stisfis th ssumption of () whn th numr of vitions llow is intgr. W prsnt xmpls in Stion 4 showing tht it is not possil, in gnrl, to orr opt(f ), opt(f V ) n opt(f ff ). 3 Optiml osts Th ojtiv of this stion is to ompr th ost of th optiml pity llotions otin for RND(F ) using iffrnt routing sts F. W prov in Stion 3. tht it lwys hols tht opt(f ) = opt(f ). In Stion 3.3, w prov tht it lwys hols tht opt(f ff ) opt(f VG ) opt(f VS ). W show lso tht unr itionl ssumptions on D, it hols tht opt(f VS ) = opt(f V ), opt(f VG ) opt(f V ) n opt(f ff ) opt(f V ). In Stion 3., w sri th mthoology us hrin to otin th sir rltions. 3. Mthoology Givn two routing sts F n F, w prov tht opt(f ) opt(f ) using two iffrnt pprohs. Th first pproh onsists in ompring irtly th routing sts thmslvs, y showing tht F F. Proving this inlusion is vry strong rsult, whih hols only for losly rlt routing sts. In suh sitution, w sy tht F is spil s of F. Bus it is not lwys possil to ompr irtly th routing sts thmslvs, th son pproh is s on ompring th sts of ll pity llotions tht support D whn onsiring spifi routing st. Ths sts r fin formlly s X (F ) x R A + (x, F ) supports D, (0) for ny routing st F. To ttr unrstn th link twn X (F ) n opt(f ), RND(F ) n quivlntly writtn s min κ x s.t. x X (F ). () A Th son pproh is wkr thn th first on in th sns tht F F implis tht X (F ) X (F ). Hn, it n ppli to mor pirs of routing sts. W prov nxt proprty stisfi y (0). W sy tht st F (D, R A K ) is onvx if th lin sgmnt twn ny two lmnts of F lis in F, tht is, if for ny f, f F n ny 0 λ w hv tht λf + ( λ)f F. Lmm. If F is onvx sust of (D, R A K ) thn X (F ) is onvx sust of R A +. Proof. Consir x, x X (F ). Hn, thr xists f, f F suh tht oth (x, f) n (x, f) support D. Bus (6) is onstitut of linr qutions, w s tht (λx + ( λ)x, λf + ( λ)f) supports D for ll 0 λ. Thrfor, λf + ( λ)f F implis tht λx + ( λ)x X (F ). Th two pprohs r formliz in th rsult low. Proposition. Lt F n F two routing sts. Th following hols:. If F F, thn opt(f ) opt(f ) for ny ost vtor κ R A.. If X (F ) X (F ) thn opt(f ) opt(f ) for ny ost vtor κ R A. 3. If opt(f ) opt(f ) for ny ost vtor κ R A n F is onvx sust of (D, R A K ) thn X (F ) X (F ). Proof. : Follows immitly from th finition of RND(F ). : Follows from th ft tht opt(f ) is th ost of th optiml solution of (). 3: Suppos thr xists x X (F )\X (F ). Applying Lmm, X (F ) is onvx st so tht thr xists hyprpln H R A suh tht H X (F ) is mpty n x H. Thrfor, lt κ th vtor in R A orthogonl to H n pointing towrs th hlf-sp ontining X (F ). By finition of κ, A κ x < A κ x for ll x X (F ). Hn, opt(f ) < opt(f ). In th following stions, w will us Proposition to rlt th optiml pity llotion osts mong th routing sts introu in Stion.. 6

7 3. Routings tht ovr D D D, α = β D D () Covr ssoit with routing f F. () Consiring sts whr y or y stisfis (6). () Th rsulting hyprpln. In this stion w fous on F n F. Figur : Constrution of sprting hyprpln. Thorm. Lt D n unrtinty st. It hols tht X (F ) = X (F ). To prov Thorm, w n th following Lmm. Lmm. Lt D onvx st in R K +. Lt D n D two los n onvx susts of D suh tht D D, D D, n D = D D. Thn thr xists hyprpln H R K suh tht H D D D. Proof. Consir th sts onv(d \D ) n onv(d \D ). Thr xists hyprpln H R K suh tht H sprts onv(d \D ) n onv(d \D ). Sin D D n D D, H D is non mpty n longs to l(d D ) = D D. Proof. of Thorm. : Follows from th ft tht F is th sust of F whr th intrstion of D n D must hyprpln. : Consir pity llotion x n routing f F suh tht (x, f) supports D. Routing f is fin y th ovr D = D D, s Figur (), n th routing tmplts y n y. W shll prov tht f n lwys trnsform to routing ˆf F suh tht (x, ˆf) supports D. Lt D (rsp. D ) fin s th sust of D whr y (rsp. y ) stisfis (6) for pity x. Sin w ssum tht (x, f) supports D, (x, f) stisfis (6) so tht D D n D D, s Figur (). Hn, D = D D. In ition, D n D r onvx. To s tht D is onvx, onsir, D, so tht thy stisfy th inqulitis y k x n y k x for h A n k K. Thn, for ny 0 λ, y k (λ + ( λ) ) = λ y k + ( λ) y k λx + ( λ)x = x, so tht λ + ( λ) D. Th proof is th sm for D. Thn, sin th inqulitis in (6) r not strit, w s sily tht D n D r los. Suppos tht D D. Thn, w n fin routing ˆf F y onsiring hyprpln tht os not intrst D so tht th only routing tmplt us is y, whih provs th rsult. W n pro similrly if D D. Hn, w n ssum tht D D n D D. Thrfor, D, D n D stisfy ll th hypothsis of Lmm. Hn, thr xists hyprpln, α = β suh tht, α = β D D D, s Figur (). Assum w.l.o.g. tht D, α β D n D, α β D. Hn, w n onstrut ˆf F through th hyprpln, α = β, tht is, ˆf () := y k k for h D, α β n f ˆk () := y k k for h D, α β. It follows from Proposition tht th osts of optiml pity llotions r lwys qul. Corollry. Lt D n unrtinty st. It hols tht opt(f ) = opt(f ) for ny ost vtor κ R A. 7

8 Thorm os not imply F = F, sin w my f routings in F with omins D n D tht o not rsult from hyprpln sprtion of D, s for instn Figur (). W rmrk tht Sutllá [7] mntions tht th omplxity of RND(F ) is unknown. Th omplxity of RND(F ) follows irtly from th suffiiny onition of Thorm n th ft tht Bn-Amur [3] provs RND(F ) to N P-hr. Corollry. Th optimiztion prolm RND(F ) is N P-hr. 3.3 Volum n ffin routings In this stion w ompr volum n ffin routings. Bn-Amur n Zotkiwiz [6] mntion tht F VS is spil s of F V, tht is, F VS F V. Th inlusion is sily vrifi for F VS, y hoosing h k = k min. Lmm 3. [6] It hols tht F VS F V. Howvr, w xplin in wht follows tht it is is not tru tht F VG F V. Th routings in F V n only inrs th mount of flow snt on ny r of G for ommoity k whn k riss. Si iffrntly, th flow for ny f F V, fin y f k () = y k min( k, h k ) + y k mx( k h k, 0) A, k K, is non-rsing funtion in. In opposition, routings in F VG n lso rs th flow snt on som of th rs sin ny routing f F V is fin y f k () = y k k min k mx k k mx k + y k min k mx k k min k mx k min A, k K, whih is sum of rsing trm n n inrsing trm. Th vntg of rsing th flow snt on som rs whn th mn for ommoity riss llows to ttr omin iffrnt ommoitis within th vill pity. W provi in Stion 4. n xmpl showing tht, in gnrl, it hols tht F VG F V. Routing sts F VS or F VG r nvrthlss spil ss of ffin routings. Bus ny routing in F VS or F VG is sri y n ffin funtions from D to R A K +, it must lso long to F ff. Th thorm low provs, morovr, tht F VS is spil s of F VG. Thorm. Lt D n unrtinty st. Th following hols:. F VS F VG. Th inlusion is strit if n only if 0 < k min < k mx for t lst on k K.. F VG F ff. Th inlusion is strit if n only if im(d) > or im(d) = n D is orthogonl to on of th oorint xs of R K +. Proof.. Lt th routing tmplts y n y sri ny routing f F VS. In wht follows, w st up routing tmplts y n y tht yil routing f F VG quivlnt to f. W onsir inpnntly h omponnt f k of f. If k mx = 0, thn k = 0 for h D n w n hoos y k n y k ritrrily sin th rsulting f k n f k r lwys null. Similrly, if k min = k mx, thn ny routing in F VS or F VG is uniquly trmin y uniqu routing tmplt for k. Suppos tht 0 < k min < k mx. Thn, w intify f k n f k t k = k min n k = k mx, otining th following tmplts for f F VG : y k y k = y k = y k k min k mx + y k k mx k min k mx, for h A, k K. () To s tht th inlusion is strit, hoos routing f F VG fin y routing tmplts y n y suh tht y k y k k min y k k mx woul stritly lss thn zro so tht y woul not routing tmplt. < 0 for som A. Using gin th intifition from (), w s tht. Lt y n y sri ny routing f F VG. Th omponnts of f 0 n y of th orrsponing ffin routing r otin y grouping th trms of f oring to thir gr in. W otin 8

9 tht y kh = 0 for h k h K, n f 0k = k min k mx k mx k min y kk = k mx k min (y k y k ) ( k mxy k k miny k ), for h A, k K, whih provs th inlusion F VG F ff. Suppos tht im(d) = n tht D is not orthogonl to ny of th oorint xs, n onsir ny routing f F ff n ommoity k K. Th flow for k is givn y f k () = f 0k + h K y kh h. (3) Sin D is not orthogonl to th k-th xis, w n prmtriz D through its orthogonl projtion on th xis. Nmly, thr xists positiv rls λ h R + for h h K\k suh tht h = λ h k for ll D. Hn, (3) oms f 0k + y kk + h K\k λ h y kh k = f 0k + y kk k. Thn, intifying f n f F VG t k = k min n k = k mx, ny ffin routing f is quivlnt to th routing f F VG with y k y k = f 0k = f 0k / k min + y kk / k mx + y kk, for h A, k K, (4) whih provs th inlusion F VG F ff whn im(d) = n D is not orthogonl to ny of th oorint xs. Th quntitis in (4) r non-ngtiv us f stisfis non-ngtivity onstrints from (3). Suppos now tht im(d) = n tht D is orthogonl to th k-th xis. Thrfor, k min = k mx = k for ny D so tht f k () = y k k is onstnt for ny f F VG. Th prolm must ontin mor thn on ommoity us im(d) = n D is orthogonl to th k-th xis. Thus, thr xists h K\k suh tht D is not orthogonl to th h-th xis. Thrfor, w n fin n ffin routing f F ff suh tht f k is not onstnt y hoosing propr y kh 0. Finlly, if im(d) >, D ontins smll ll B of imnsion t lst two. Hn, thr xists pir, B suh tht k = k n h h for som k, h K. As for, ll routings in F VG for ommoity k yil intil flows for n, whil w n fin n ffin routing f F ff yiling iffrnt flows y hoosing propr y kh 0. Th strit inlusion of Thorm.. hs n vrifi numrilly y Bn-Amur n Zotkiwiz [6], whr it is shown tht opt(f VG ) n stritly smllr thn opt(f VS ). W show nxt tht F VS is lwys t lst s ffiint s F V whnvr D stisfis th ssumption low. Givn onvx st D R K +, w not y xt(d) th st of its xtrm points. Assumption. Th unrtinty st D is polytop suh tht for h k K, thr xists nonngtiv numrs 0 k min k mx suh tht k k min, k mx for ll xt(d). Assumption is stisfi y wll-known fmily of unrtinty polytops, s Exmpl. Exmpl. Brtsims n Sim [3] onsir gnrl linr progrms whr th offiints of h linr inqulity long to intrvls suh tht th numr of offiints tking onjointly thir mximum vlu is oun y onstnt Γ. Consiring upwrs vitions only, thir unrtinty st n formliz in R K + s follows D Γ R K + k [ k min, k mx] for h k K, k k min k mx k Γ min Whn Γ is intgr, it is sy to s tht D Γ fulfills Assumption. Morovr, D Γ hs n frquntly us s th unrtinty st for roust ntwork sign prolms, s [6, 3, 4, 0, 6], mong othrs. k K. 9

10 Th proof of th thorm low rquirs th following simpl proprty. For ny x R A + n f F VS, (x, f) supports D (x, f) supports xt(d). (5) Proprty (5) follows irtly from th ft tht ny routing in F VS is linr funtion. Thorm 3. Lt D n unrtinty st tht fulfills Assumption. It hols tht X (F VS ) = X (F V ). Proof. : Follows irtly from th ft tht F VS F V y tking h k = k min. : Consir pity llotion x n routing f F V suh tht (x, f) supports D. W show nxt tht thr xists routing f F VS suh tht f() = f() for h xt(d). Thrfor, (x, f) supports xt(d). Sin (5) is stisfi, w hv tht (x, f) supports D, proving th rsult. Noti tht th flow for ommoity k of ny routing in F V or F VS is funtion tht only pns on th k-th omponnt of D. Thus, lt us fin D k s th orthogonl projtion of D into its k-th omponnt so tht funtions f k n f k r fin on D k. In th following, w show how to onstrut f F VS suh tht f() = f() for h xt(d) inpnntly for h ommoity k K. By Assumption, xt(d k ),. First, suppos tht xt(d k ) = so tht D k = k is singlton. If k = 0, thn w n hoos y k n y k ritrrily sin f k (0) will null nywy. If k > 0, thn w tk y k = f k ( k )/ k n y k = 0. Suppos now tht xt(d k ) =. If k min = 0, thn w n hoos y k ritrrily (sin it will multipli y k min = 0) n w st yk = f k ( k mx)/ k mx. If k min > 0, thn w st yk = f k ( k min )/k min n yk = f k ( k mx)/( k mx k min ) for h k K. Bus of Assumption, ny xt(d) is suh tht k k min, k mx. Thrfor, f k () = f k () for h xt(d), so tht (x, f) supports xt(d). Thorm 3 stts tht whnvr D stisfis Assumption, on shoul not try to us th omplx st of routings F V, sin opt(f V ) will nvr t opt(f VS ). This is of prtiulr intrst us RND(F V ) is N P-hr in gnrl whil Bn-Amur n Zotkiwiz [6] show tht RND(F VS ) is ssntilly of th sm iffiulty s RND(F stt ). 4 Non-omprl routings In this stion, w ompr opt(f ), opt(f V ) n opt(f ff ) for gnrl unrtinty sts. W show tht it is not possil to orr ths osts y prsnting thr xmpls whr on of th osts is stritly lss thn th two othrs. To vis xmpls showing tht F ff my yil mor xpnsiv pity llotions thn F n F V, w shll us th following rsult. Lt k th k-th unit vtor in R K +. Proposition. [6, Proposition 8] Lt D mn polytop. If 0 D n for h k K thr is ɛ k > 0 suh tht ɛ k k D, thn opt(f ff ) = opt(f stt ). Noti tht in our xmpls som of th ommoitis hv uniqu pths from thir sours to thir sinks, so tht ll routings r qul for ths ommoitis. This nls us to prou simpl grphs tht prsnt th proprtis rquir y our xmpls. On n sily xtn ths xmpls to lrgr grphs for whih h ommoity k K hs t lst two iffrnt pths from its sour s(k) to its sink t(k). 4. opt(f V ) n stritly smllr thn opt(f ff ) n opt(f ) Consir th ntwork sign prolm for th grph pit in Figur () with two ommoitis k : n k :. Th unrtinty st D is fin y th xtrm points = (, ), = (, ), 3 = (, 0), 4 = (0, ), n 5 = (0, 0), n th pity unitry osts r th g lls of Figur (). Eg lls from Figur () n Figur () rprsnt optiml pity llotions with ynmi n stti routing, rsptivly. Thy hv osts of 7 n 8, rsptivly. A routing f F tht stisfis th pity from Figur () is pit on Figur () n Figur (), for n, rsptivly. W show nxt tht th optiml pity llotions for F V, F, n F ff r 7, 8 n 8, rsptivly. Th routing f from Figur () n Figur () n xtn to routing in f F V suh tht (x, f) supports D y fixing h k =, h k =, y k = f k ( ), y k = f k ( ) f k ( ), n y k = y k = f k ( ). 0

11 0 () g osts () optiml pity llotion with F V () optiml pity llotion with F or F ff (,) (0,) (,0) (,0) (,0) (0,0) () f( ) () f( ) Figur : Exmpl showing tht opt(f V ) n n stritly smllr thn opt(f ff ) n opt(f ) () g osts () optiml pity llotion with F VG or F ff () optiml pity llotion with F V or F (3,0,0) (0,0,0) (0,3,0) (0,3,0) (0,,0) (0,,0) (0,0,) (0,0,0) (0,0,0) (0,0,) () f( ) () f( ) Figur 3: Exmpl showing tht opt(f VG ) (thus opt(f ff )) n n stritly smllr thn opt(f V ) n opt(f ). Howvr, w xplin nxt why it nnot xtn to routing in F within th pity x from Figur (). W rstrit our ttntion to th sust of D tht onsists of th lin sgmnt D = onv(, ) n show tht f n lry not xtn to routing in F for D. Consir flow f( ) pit in Figur (). This flow uss th routing tmplt y fin s y k = f k ( k )/ k for k = k, k. Similrly, flow f( ) uss th routing tmplt y k = f k ( k )/ k for k = k, k. Thn, w s tht (rsp. ) is th uniqu mn vtor in D tht n rout within th pity x from Figur () using routing tmplt y (rsp. y ). Thrfor, fining D (rsp. D ) s th sust of D tht ontins ll mn vtors tht n rout long routing tmplt y (rsp. y ), w hv tht D D D. This shows tht it is not possil to xtn f to routing in F for D, so tht it is not possil to o so for D ithr. In ft, w hv tht th optiml pity llotion for F is otin whn D is ovr only y itslf, yiling opt(f ) = 8. For F ff, w n pply Proposition (us (0, 0), (, 0), (0, ) D) so tht opt(f ff ) = opt(f stt ) = opt(f VG ) n opt(f ff ) n stritly smllr thn opt(f V ) n opt(f ) Consir th ntwork sign prolm for th grph pit in Figur 3() with thr ommoitis k :, k : n k 3 :. Th unrtinty st D is fin y th xtrm points

12 () g osts () optiml pity llotion with F () optiml pity llotion with F V or F ff (,0,0) (0,0,0) (0,,0) (0,,0) (0,,0) (0,,0) (0,0,) (0,0,0) (0,0,0) (0,0,) () f( ) () f( ) Figur 4: Exmpl showing tht opt(f ) n n stritly smllr thn opt(f ff ) n opt(f V ). = (3,, 0) n = (0, 3, ), n th pity unitry osts r th g lls of Figur 3(). Eg lls from Figur 3() rprsnt n optiml pity llotion for ynmi routing, with ost. A routing f F tht stisfis th pity from Figur 3() is pit on Figur 3() n Figur 3(), for n, rsptivly. This routing n xtn to routing f F VG suh tht (x, f) supports D y stting y k = y k = f k ( )/3, y k = f k ( )/, y k = f k ( )/3 n y k3 = y k3 = f k3 ( )/. Applying Thorm.., f lso longs to F ff. Howvr, f nnot xtn to routing in F V lry us k > k n f k ( ) < f k ( ), tht is, f is not non-rsing funtion. W n show in ition tht, using rsoning similr to th on us in th prvious stion, f nnot xtn to routing in F within th xisting pity. W n s tht n optiml pity llotion using F V or F is lso n optiml pity llotion using F stt, n it rquirs two mor units of pity on n no pity on, s Figur 3(), whih yils totl ost of opt(f ) n stritly smllr thn opt(f ff ) n opt(f V ) Consir th ntwork sign prolm for th grph pit in Figur 4() with thr ommoitis k :, k : n k 3 :. Th unrtinty st D is fin y th xtrm points = (,, 0), = (0,, ), 3 = (, 0, 0), 4 = (0,, 0), 5 = (0, 0, ), 6 = (0, 0, 0) n th pity unitry osts r th g lls of Figur 4() (it is th sm s Figur 3()). Eg lls from Figur 4() n Figur 4() rprsnt optiml pity llotions with ynmi n stti routing, rsptivly. Thy hv osts of 5 n 7, rsptivly. A routing f F tht stisfis th pity from Figur 4() is pit on Figur 4() n Figur 4(), for n, rsptivly. This routing n xtn to routing f F suh tht (x, f) supports D y onsiring th ovr through hyprpln, k = n stting y k = y k = f k ( )/, y k = f k ( ), y k = f k ( ), n y k3 = y k3 = f k3 ( )/. Howvr, f nnot xtn to routing in F V lry us k > k n f k ( ) < f k ( ), tht is, f is not non-rsing funtion. In ft, w hv tht opt(f V ) = opt(f stt ) = 9. Thn, w n pply Proposition (us (, 0, 0), (0,, 0), (0, 0, ), (0, 0, 0) D) to th prolm with F ff, so tht opt(f ff ) = opt(f stt ) = 9. 5 Conluing rmrks This ppr stuis th optiml pity llotion ost provi y roust ntwork sign mols rstrit to us spifi routing sts. Ths routing sts r: ffin routing, volum routing n its two simplifitions, n th routings s on ovrs of th mn unrtinty st. W show tht th routing st s on n ritrry ovr of th unrtinty is quivlnt to th routing st tht uss sprtion hyprpln. W show thn tht th simplifi volum routings r spil

13 ss of ffin routings. Finlly, w show tht th gnrl volum routing is no mor flxil thn its simplifitions whnvr th unrtinty st is th polytop introu y Brtsims n Sim. An importnt hrtristi of ths routing sts is th omplxity of th rsulting ntwork sign prolm. In this rspt, th gnrl volum routings n th routing sts s on ovrs of th unrtinty st l to N P-hr optimiztion prolms. Morovr, whil finit linr progrmming formultion n provi for th roust ntwork sign prolm with ynmi routing unr polyhrl unrtinty (y onsiring only th xtrm points of th mn polytop), no suh formultions r known for th prolms tht us th gnrl volum routings or th routings s on ovrs of th unrtinty st. In this sns, ths two routing sts yil optimiztion prolms tht r omputtionlly vn mor iffiult thn th roust ntwork sign with ynmi routing. In opposition, ffin routing n th two simplifi volum routings l to polynomilly solvl optimiztion prolms, givn tht th unrtinty polytop hs ompt sription. 6 Aknowlgmnts Th uthor is grtful to Christin Rk for numrous ommnts tht hlp in improving th prsnttion of th ppr. Rfrns [] A. Altin, E. Amli, P. Blotti, n M. Ç. Pinr. Provisioning virtul privt ntworks unr trffi unrtinty. Ntworks, 49():00 5, 007. [] A. Altin, H. Ymn, n M. Ç. Pinr. Th roust ntwork loing prolm unr hos mn unrtinty: Formultion, polyhrl nlysis n omputtions. INFORMS Journl on Computing, 3():75 89, 0. [3] W. Bn-Amur. Btwn fully ynmi routing n roust stl routing. In 6th Intrntionl Workshop on Dsign n Rlil Communition Ntworks, 007. DRCN 007, 007. [4] W. Bn-Amur n H. Krivin. Nw onomil virtul privt ntworks. Communitions of th ACM, 46(6):69 73, 003. [5] W. Bn-Amur n H. Krivin. Routing of unrtin mns. Optimiztion n Enginring, 3:83 33, 005. [6] W. Bn-Amur n M. Zotkiwiz. Volum orint routing. In 4th Intrntionl Tlommunitions Ntwork Strtgy n Plnning Symposium (NETWORKS), pgs 7, 00. [7] W. Bn-Amur n M. Zotkiwiz. Roust routing n optiml prtitioning of trffi mn polytop. Intrntionl Trnstions in Oprtionl Rsrh, 8(3): , 0. [8] A. Bn-Tl, A. Goryshko, E. Guslitzr, n A. Nmirovski. Ajustl roust solutions of unrtin linr progrms. Mthmtil Progrmming, 99():35 376, 004. [9] A. Bn-Tl n A. Nmirovski. Roust solutions of linr progrmming prolms ontmint with unrtin t. Mthmtil Progrmming, 88:4 44, 000. [0] A. Bn-Tl n A. Nmirovski. Roust optimiztion mthoology n pplitions. Mthmtil Progrmming, 9: , 00. [] D. Brtsims n C. Crmnis. Finit ptility in multistg linr optimiztion. Automti Control, IEEE Trnstions on, 55():75 766, 00. [] D. Brtsims, V. Goyl, n A. Sun. Chrtriztion of th powr of finit ptility in multi-stg stohsti n ptiv optimiztion. To ppr in Mthmtis of Oprtions Rsrh., 0. [3] D. Brtsims n M. Sim. Th Pri of Roustnss. Oprtions Rsrh, 5():35 53, Jn 004. [4] C. Chkuri, G. Oriolo, M. G. Sutllà, n F. B. Shphr. Hrnss of roust ntwork sign. Ntworks, 50():50 54,

14 [5] X. Chn n Y. Zhng. Unrtin Linr Progrms: Extn Affinly Ajustl Roust Countrprts. Oprtions Rsrh, 57(6):469 48, 009. [6] A. Frngioni, F. Psli, n M. G. Sutllà. Stti n ynmi routing unr isjoint ominnt xtrm mns. Oprtions Rsrh Lttrs, 39():36 39, 0. [7] J. Goh n M. Sim. Distriutionlly roust optimiztion n its trtl pproximtions. Oprtions Rsrh, 58(4-Prt-):90 97, 00. [8] A. Gupt, J. Klinrg, A. Kumr, R. Rstogi, n B. Ynr. Provisioning virtul privt ntwork: ntwork sign prolm for multiommoity flow. In STOC 0: Proings of th thirty-thir nnul ACM symposium on Thory of omputing, pgs , 00. [9] A. M. C. A. Kostr, M. Kutshk, n C. Rk. Towrs roust ntwork sign using intgr linr progrmming thniqus. In Proings of th NGI 00, Pris, Frn, Jun 00. Nxt Gnrtion Intrnt. [0] A.M.C.A. Kostr, M. Kutshk, n C. Rk. Ajustl roust ntwork sign: Mols, inqulitis, n omputtions, 00. [] S. Muhntongsuk, F. Oronz, n J. Liu. Roust solutions for ntwork sign unr trnsporttion ost n mn unrtinty. Journl of th Oprtions Rsrh Soity, 59:55 56, 008. [] F. Oróñz n J. Zho. Roust pity xpnsion of ntwork flows. Ntworks, 50():36 45, 007. [3] A. Ouorou. Affin ision ruls for trtl pproximtions to roust pity plnning in tlommunitions. Tlk t th Intrntionl Ntwork Optimiztion Confrn, 0. [4] A. Ouorou n J.-P. Vil. A mol for roust pity plnning for tlommunitions ntworks unr mn unrtinty. In 6th Intrntionl Workshop on Dsign n Rlil Communition Ntworks, 007. DRCN 007, pgs 4, 007. [5] M. Poss n C. Rk. Affin rours for th roust ntwork sign prolm: twn stti n ynmi routing. In Proings of INOC 0, Hmurg, Grmny, numr 6703 in LNCS, pgs Springr. [6] M. Poss n C. Rk. Affin rours for th roust ntwork sign prolm: twn stti n ynmi routing. ZIB Rport -03, Zus Institut Brlin, Frury 0. [7] M. G. Sutllà. On improving optiml olivious routing. Oprtions Rsrh Lttrs, 37(3):97 00, 009. [8] A. L. Soystr. Convx progrmming with st-inlusiv onstrints n pplitions to inxt linr progrmming. Oprtions Rsrh, :54 57,