Graph Algorithms and Combinatorial Optimization Presenters: Benjamin Ferrell and K. Alex Mills May 7th, 2014

Transcription

1 Grp Aloritms n Comintoril Optimiztion Dr. R. Cnrskrn Prsntrs: Bnjmin Frrll n K. Alx Mills My 7t, 0 Mtroi Intrstion In ts ltur nots, w mk us o som unonvntionl nottion or st union n irn to kp tins lnr. In prtiulr, lt A + B := A B, n A B := A \ B, n lt A + := A {} n A := A \ {}. W will lwys not sts y pitl romn lttrs (.. A, B, C) n lmnts o sts y lowr-s romn lttrs (..,, ) to nsur t prior nottion is unmiuous. W us t inpnnt st ormultion o mtroi. A init mtroi M = (E, I) is pir onsistin o roun st E n mily o susts o E, I E, ll t inpnnt sts o M wit t ollowin two proprtis: M. (Hrity Axiom) Susts o n inpnnt sts r inpnnt, i.. F I, F F = F I. M. (Exn Axiom) I A, B I, n A > B, tn tr is n lmnt o A B wi wn to B orms n inpnnt st. i.., A > B = A B : B + I. Mtrois v t proprty tt ty r mnl to usin ry loritm. Mny importnt prolms su s inin spnnin trs n ormult n solv s mtrois. In nrl sttin, tis is t prolm o inin mximum-siz inpnnt st in mtroi. Tt is, ivn mtroi M = (E, I), in F I su tt F is mximum. Givn two mtrois in ovr t sm roun st, w r intrst in inin mximum-siz st wi is inpnnt in ot mtrois. Tis is known s t mtroi intrstion prolm. As w will sow, it s polynomil tim loritm wi prous orrt solution. Furtrmor, t loritm w sri uss umntin pt tniqus wi r similr to tos us in loritms or mximum-low n mtin. Formlly, our prolm is in s ollows: Givn two mtrois, M = (E, I ), n M = (E, I ), in F I I su tt F is mximum. Krol Grps T loritm w sri mks us o iprtit rp onstrution u to S. Krol []. W will spn som tim motivtin its onstrution n xminin t proprtis o tis rp. Givn M n M, in ovr roun st E, suppos w r ivn som st I I I. In t ours o t loritm, w woul lik to lmnts rom E I to I in orr to otin som I o stritly lrr siz. W n lwys rily lmnts rom E I to I s lon s t rsult is inpnnt in ot mtrois, i.. i tr is n E I, su tt I + I I. Howvr, wn w nnot umnt I in ry mnnr, w must v som mnism or rmovin som lmnts rom I n rplin tm wit otr lmnts in su wy tt t siz o I inrss. W o tis y inin pt in t Krol rp. T pross w us is similr to inin n umntin pt in ntwork lows (wr w in n s t pt wi stritly inrss t mount o low w n sn) n in mtin (wr w in n ltrntin pt twn two r nos wi stritly inrss t siz o t mtin). To in t Krol rp ivn n inpnnt st I I I, w in irt iprtit rp G I = (I, E I, D I ) wit isjoint sts I n E I, n st D I, wr or i I n j E I, (i, j) D I I i + j I, n (j, i) D I I i + j I. Tt is to sy, or i I n j X I, i onnts to j vi irt i, rplin i wit j in I mintins inpnn in mtroi M. Likwis, j onnts to i i rplin i wit j mintins inpnn in M.

2 Potntil Pitll: B rul to not tt t orinttion o t os not init wi lmnt is to I n wi is rmov rom I. Rrlss o t orinttion o (i, j), t lmnt rom I is lwys rmov, n t lmnt rom E I is lwys. To unrstn ow umntin pts in t Krol rp inrs t siz o our inpnnt st, onsir ny pt in G I wi strts in E I n ns in E I. Sin t rp is iprtit, vry pir o jnt nos must on opposit sis o t rp. Lt Q = j, i, j, i,...j n, i n, j n+ t squn o nos on su pt, n lt Q j = {j,..., j n+ }, Q i = {i,..., i n }. Sin pt Q strts n ns in E I, vry no in Q j must in E I, n vry no in Q i must in st I, so w v Q j E I, n Q i E. I Q is simpl pt, (i.. no no is visit mor tn on) tn w v tt Q j = Q i +. Suppos w tn wr to orm I = I Q i + Q j, tn lrly, I = I +. Tis is tru or ny simpl pt rom E I k to E I. So y tkin simpl pt rom E I k to E I w n orm I wit stritly lrr siz. Howvr, w lso wnt I to inpnnt in ot mtrois. Tis n iv y sltin pt Q rom E I to E I wit t ollowin tr proprtis,. (Strt point) Q strts rom no in st X = {x E I I + x I }.. (En point) Q ns t no in st X = {x E I I + x I }.. (Sortst pt) Q is sortst simpl pt rom X to X. Any pt vin ts proprtis is ll n umntin pt in G I. T proo tt ny su umntin pt n us to yil st wi is inpnnt in ot M n M is non-trivil. W will lv tis proo until t n o t ltur nots. Mtroi Intrstion Aloritm T loritm or mtroi intrstion strts wit I = n rily s lmnts wi mintin mmrsip in I I. Wn no mor lmnts n, t loritm orms Krol rp G I, in wi it ins n umntin pt Q ( sortst pt rom X to X ). T loritm tn umnts I y Q, n in rily s lmnts until no mor lmnts n. T loritm ontinus in tis mnnr until no umntin pt n oun. Mor spiilly:. St I.. In ry mnnr, to I no z E I or wi I + z I I. Do tis until no otr z mtin ts ritri n oun.. Form Krol rp G I = (I, E I, D I ) s in ov. Lt X = {z E I I +z I }, X = {z E I I + z I }.. Fin sortst simpl pt Q rom X to X in G I. I non xists, stop n rturn I.. Aumnt I usin Q. Tt is, I I Q i + Q j. Goto stp n ontinu. Noti tt stp is not stritly n. Witout it, t loritm woul in pts o lnt zro onsistin o nos in X X. Howvr, inluin stp vois t wstul ronstrution o nw Krol rps vry tim su n lmnt is to I. Bus stp is inlu, y stp vry pt s t lst s, so w mk us o umntin pts in tis loritm only wn w n to. W provi proo o orrtnss in t nxt stion. T rst o tis stion is onrn wit iiny n trmintion o our loritm. Eiiny is lr s ollows. W strt wit I = n lrly I E t ny point, ivin totl o E = n itrtions. In t worst s, w orm t rp G I t stp. Givn I, w n to onsir I E I = O(n ) potntil s. But, ow o w know wtr to inlu ivn twn i

3 n j? To o so, w v to l to tst n s i I i + j is inpnnt in M n run sprt tst to s i it is inpnnt in M. Bus t siz o I n I oul xponntil in n, w ssum tt mtroi orls M n M r ivn. A mtroi orl or mtroi M = (E, I) is prorm wi n rspon to quris onsistin o susts U E. T qustion t orl nswrs is: is U I? As onrt xmpl, onsir t mtroi M G in on rp G = (V, E), wr t inpnnt sts r yl-r susts o E. For ny sust o s, E w n omput wtr or not E ontins yl, n nswr ys or no. Su prorm woul onstitut n orl or M G. Bk to t Krol rp, it soul lr tt w n to qury ot orls on or o G I. Spiilly, or (i, j), wit i I n j E I, w sk, is I i+j I? W n onstrut t rp in t most O(n ) su quris. I ot M n M v orls wit polynomil runnin tims, t runnin tim o t mtroi intrstion loritm will polynomil. Mor spiilly, i T (M ) n T (M ) r runnin tims or t orls or M n M, tn t mtroi intrstion loritm runs in tim O(n mx[t (M ), T (M )]). In orr to lt, w must l to tt wn no X X pt xists in G I. To o tis iintly, w n o t ollowin: In G I, onnt spil no r to vry no in X. Run rt-irst sr rom r, otinin rt-irst sr tr T. I T osn t r X, tn tr is no X X pt. Els, t sortst pt rom X to X n oun in T. Brt-irst sr runs in O( V + E ) tim, wi is omint y t O(n ) rp onstrution in stp. Exmpl Common Ll Spnnin Tr T ollowin is n xmpl o t mtroi intrstion loritm in tion. Our urrnt mtrois M = (V, I ) n M = (V, I ) r orst mtrois s on rps G n rsptivly. Tt is, I (rsptivly I )is t st o ll yl-r s in G (rsptivly, ). T rps r sown in Fiur. Essntilly, w will lookin or lrst ommon spnnin tr. Wn t loritm strts, our intrstin inpnnt st I is mpty. Any lmnt I I is tn rily to I su tt I rmins inpnnt in I n I. Wn no nw lmnt n to I, t loritm tn srs or n umntin pt to inrs I. Fiur sows t urrnt snrio. W rily os lmnts,, n to to I n now nnot ny mor lmnts. Noti tt in w n itr or to I n still rmin inpnnt in I. Howvr, or G tis will us I to los inpnn in I. Tis is wr t umntin pt oms in to lp rrrn tins to wr w n potntilly mor lmnts. W onstrut t Krol rp, G I. Alon wit G I, two sts (X n X ) r in. G I is sown in Fiur wit lmnts in X olor r n lmnts in X olor rn. At tis point sortst pt rom n lmnt in X to n lmnt in X is slt. T loritm tn upts I y rmovin ll y I n in ll z V I tt r lon t slt pt. Fiur ilits n ritrrily slt sortst pt (,, ). I will tn upt to I + + n inrs in siz y on. Fiur ilits t lmnts in t upt I. A nw Krol rp D I (Fiur ) is rt s on tis st. Sin tr r no lmnts in itr X or X, tis inits tr is no umntin pt to inrs I. T loritm trmints n rturns I = {,,, }. T nxt xmpl provis s snrio monstrtin t nssity o oosin sortst X X pt. Fiur 6 sows rps G n wit ommon inpnnt st ilit. T irt iprtit rp is onstrut n inst o oosin sortst pt, non-sortst pt is osn inst. Fiur 7 sows t slt pt n t rsultin umnttion. Noti tt t ommon st

4 G Fiur : Initil rps wr I = G Fiur : Hilit s in G n orm ommon inpnnt st. I = {,, }. is no lonr inpnnt. I w osn sortst pt su s in Fiur 8, t rsultin umnttion woul v yil lrr ommon inpnnt st. Aloritm Corrtnss To prov t orrtnss o t mtroi intrstion loritm, w n to sow two tins.. I is o mximum siz t t n o t loritm.. I I I t t n o t loritm. W n rin ts sttmnts mor ormlly into Torm n Torm. Noti tt ts sttmnts r slitly rminisnt o t umntin pt torms or mx-low. Torm. I tr is no X X pt in G I, tn I is o mximum siz. Torm. I tr is n X X pt, Q = j, i, j, i,..., j n, i n, j n+ wi is n umntin pt, tn I Q i + Q j I I. To prov Torm w will n t onpts o t rnk n sis o ny st U E. Givn st U E, wr E is t s st o mtroi M, w in t rnk r M (U) s t mximum siz inpnnt sust o U. I U is inpnnt tn r M (U) = U, otrwis w n rmov lmnts rom U wi us it to pnnt. Rmovin t wst numr o su lmnts yils mximum siz U U, wr U is inpnnt in M, n tror r M (U) = U. A sis o st U E, not B U, is

5 I E I D I Fiur : D I sows pts rom X (r) to X (rn). G Fiur : Hilit s in G n orm nw ommon inpnnt st tr in lmnts n n rmovin. I = {,,, }. I E I D I Fiur : No X X pt in D I.

6 G Fiur 6: Nssity o sortst pt I V I G I G Fiur 7: Non-sortst pt umnttion yils non-inpnnt st. mximum siz inpnnt sust o U. Not tt t sis o ivn sust n not uniqu, ut mtroi tory tlls us tt ll ss must v t sm rinlity. Tt is, or ny two ss B U, B U o st U, B U = B U = r M (U). Arm wit ts onpts, w n inlly iv t ollowin proo. Ts proos wr oriinlly xprss in mor onis orm, owvr rt r s n tkn to itionl tils to i t onus rr, wo my not v prviously sn mtroi tory ppli. W v rwn rom ll t rrns it t t n o tis stion, n s our proo primrily on tos in [, ]. Proo (Torm ). Givn tt tr is no X X pt, w n to sow tt I is o mximum siz. First, noti tt ivn ny J I I, n ny U E, w v J = J U + J (E U) r M (U) + r M (E U). () Wi sys rouly tt t portions o J on itr si o t prtition (U, E U) nnot x t mximum inpnnt sts on itr si o t ut. Tis is rsonly intuitiv, ut w must iv orml proo. To prov () not tt sin ny sust o J is inpnnt, tis must lso ol or J U J. So, or ny M sis o t suprst U w must v J U B U = r M (U). Similrly, sin J (E U) J, or ny M sis o E U, w must v J (E U) B E U = r M (E U). Tkin t sum o ts inqulitis yils (). W will sow tt i tr is no X X pt, tt w v qulity in (), wi implis tt w nnot inrs I ny urtr. To sow qulity or I w must prou suitl st U. Lt U t st o lmnts in E wi n r X. Mor ormlly, lt U = {z E tr is pt in G I rom z to X }. Clrly, ny no in X must in U, so X U. Furtrmor, sin tr is no pt rom X to X, X U =. W lim tt r M (U) = I U n r M (E U) = I (E U), wi provs tt w v I w oul inrs I wil qulity ols or (), tt woul imply tt () is invli, sin I woul x t rit-n si. But w just prov () so w must not l to inrs it urtr. 6

7 I V I G I G Fiur 8: Sortst pt umnttion yils lrr ommon inpnnt st. qulity in (). W will only sow tt r M (U) = I U, t proo tt r M (E U) = I (E U) is similr. W will sow I U = r M (U) y ontrition. Suppos tt I U < r M (U). Sin I U is inpnnt in M, w n xtn I U to sis o U, or wi I U < B U. But tis mns tt tr is suprst o I U wi is inpnnt in M, nmly B U. By t xn xiom, w must v som x B U (I U) su tt (I U) + x I. Not tt tis urnts tt x B U I. Sin X U =, it must tt I + x / I, (y inition o X ). Finlly, sin (I U) + x ws sown to inpnnt in M, n I + x is not, w v tt I U I; tror (I U) + x I. () W will sow tt () implis tt tr must som y I U su tt I y + x I, ut irst w will sow ow tis omplts t proo. Not tt sin I y + x I, w must v tt (y, x) D I, n tis is irt rom I to E I sin w v inpnn in M or I y + x. But w lry know tt tr s pt rom x to X, us x B U implis tt x U. Tror, sin tr is som no y wi is not in U, yt wi n r X (y wy o pt trou x), w otin ontrition. It rmins to sow tt () implis t xistn o y I U su tt I y + x I. W onsir two ss s ollows. Suppos tt (I U)+x = I. Sin ot sis v t sm siz, yt w som x / I to (I U), w must v tt (I U) + x = I y + x or som y I U. Suppos now inst tt (I U) + x < I. Sin ot sts r inpnnt in M w n pply t xn xiom. Tror w must v x I ((I U) + x) = I U su tt (I U) + x + x I. W n rpt tis y inution until w otin I y + x I or som y I U, just s or. Tis omplts t proo. To prov Torm w will n t onpt o t spn o st S E. T spn o ny st S E is in s spn M (S) = {x x E, r M (S + x) = r M (S)}. T spn is tos lmnts wi, wn to S os not us t rnk o S + x to ir rom tt o S. Tt is to sy, t mximum siz inpnnt st o S + x is t sm s tt o S. W will us t t tt, i x / spn M (S), tn S + x I. W will lso n t ollowin lmm. Lmm. Lt M = (E, I) mtroi. Lt I I, n lt j, i,..., j k, i k squn o istint lmnts o E su tt () j m / I, i m I or m k; () I + j m / I, (I + j m ) i m I or m k; () I + j m i l / I or i m < l k. Lt I = I {i,...i k } + {j,..., j k }. Tn I I n spn M (I ) = spn M (I). 7

8 Wi rouly sys tt or n umntin pt Q = j, i,..., j n, i n, (tt is, t oriinl umntin pt Q xluin j n+ ), tt umntin y Q yils n inpnnt st in I. Tis will om lr on w pply t lmm to prov Torm. Not tt onition () is quivlnt to rquirin Q to sortst pt, sin i I j m + i l wr inpnnt wit m < l, tis woul imply tt sortut xists in t Krol rp. W will prov Lmm t t n o tis stion. For now, w will sow ow it n us to prov Torm. Proo (Torm ). Consir t umntin X X pt Q = j, i,..., j n, i n, j n+ wi xists y t ssumption o Torm. W n to k t onitions o Lmm in orr to pply it. Sin Q strts in E I, n G I is iprtit, w v tt t squn j, i,..., j n, i n stisis onition (). Wn w onsir mtroi M, onition () ollows sin w v tt I + j m / I (otrwis tr is sortr X X pt), n tt I j m + i m I, sin (j m, i m ) is in t Krol rp. Conition () ollows sin Q is sortst pt. Tror, y Lmm, w v tt I = I {i,..., i n } + {j,..., j n } I, n urtrmor tt spn M (I ) = spn M (I). Sin j m+ X, n tror I + j m+ I, w know tt j m+ / spn M (I). Tror, w t tt j m+ / spn M (I ) = spn M (I), so w v tt I + j m+ I. It s sy to s tt t sm umntin pt yils similr squn i, j,..., i n, j n+ or wi t onitions o t lmm ol wit rr to mtroi M. Tt is, j m+ / I, ut i m I, or ny m n, stsiyin (); w know tt I + j m+ / I, yt (I + j m+ ) i m I or i m n, sin s (i m, j m+ ) r in G I, stisyin (); onition () ollows sin Q is sortst pt, s or. Lmm n tror ppli s or, n symmtri rumnt us to sow j / spn(i {i,...i n } + {j,..., j n }), implyin tt I {i,...i n } + {j,...j n } + j I. Tkn totr wit t prvious rsult, w v tt I Q i + Q j I I, s sir. As onsqun o vin orrt loritm or mtroi intrstion, w n sow t ollowin unmntl mx-min rltion ovr t intrstion o mtrois. Torm. T siz o t mximum inpnnt st ommon to M n M is qul to t siz o t mximum inpnnt sts on itr si o iprtition o E. Formlly, mx I = min [r M (U) + r M (E U)]. I I I U X Proo. Our proo rlis on t xistn o orrt loritm wi n prou n I n U su tt qulity ols. T mtroi intrstion loritm is su n loritm, s vin y Torms n. Furtrmor, w sow in t proo o Torm tt ols, so no otr I n x t siz o tt oun y our loritm. O ours, to l to pply Lmm in t proo o Torm, it must sown to orrt. W will sow tis t y inution in t ollowin proo. Proo (Lmm ). W prorm inution on k, l t lnt o t squn j, i,..., j k, i k. As our s s, onsir t sitution wn k =. W v squn j, i, wr t onitions o t Lmm ol. T lmm stts tt I = I j + i. By onition (), w v tt I I. W n to sow tt spn M (I ) = spn M (I). Suppos tt spn M (I ) spn M (I), n tt w v som spn M (I) or wi / spn M (I ). Tis mns tt I must sis o I + i +, ut I + orms sis o lrr siz. Sin sis o sust o X must v t sm siz, tis is ontrition. Similrly, i spn M (I ) ut / spn M (I) tn I orms sis o I + i +, n w t I + s sis o lrr siz, yilin ontrition. So in itr s, spn M (I) = spn M (I ). For t inutiv s, suppos k n t lmm ols or ll vlus smllr tn k. Fix I = I Q i + Q j or t rminr o t proo. W will pply t lmm to smllr siz squns in two wys. First, pply it to st I n t squn j, i,..., j k, i k, yilin tt I j k + i k I, n spn M (I j k + i k ) = spn M (I). () 8

9 (Not tt I j k + i k is just I umnt y t pt o lnt k wi xlus t lst pir (j k, i k )). W lso pply t lmm to st I i k n t sm squn (j, i,..., j k, i k ) yilin tt I j k I, n spn M (I j k ) = spn M (I i k ). () Not tt j k / spn M (I i k ), sin y onition (), I i k + j k is inpnnt. Sin t spns in () r t sm, j k / spn M (I j k ), so w v tt I j k + j k = I I, wi is t irst onsqun o t lmm tt w n to sow. Applyin (), it s lr tt in orr to sow tt spn M (I ) = spn M (I), it suis to sow spn M (I ) = spn M (I j k + i k ). To o tis, w pply inution wr k = to t st I n t squn i k, j k, yilin tt spn M (I ) = spn M (I j k + i k ). In orr to o tis, w must stisy t onitions o t Lmm. Conitions () n () r t only two onitions o onsqun, us o t smll siz o t squn. Clrly, () is stisi, sin i k / I, n j k I. To s tt () is stisi, w must v tt I j k + i k I, wi ollows sily rom (), n tt I + i k / I, wi lso ollows rom (), sin i k I, n I spn M (I) = spn M (I j k + i k ). Tror, ll t onitions r stisi, n w v tt spn M (I) = spn M (I ), provin t lmm. Rrns [] Cnr Ckuri. Ltur 6 on //00: Unrossin s proo o mtroi polytop intrlity, s xn proprtis. ttp://ourss.nr.illinois.u/s98s/sp00/, 00. [] Cnr Ckuri. Ltur 7 on /6/00: Mtroi intrstion. ttp://ourss.nr.illinois.u /s98s/sp00/, 00. [] Willim J. Cook, Willim H. Cunninm, Willim R. Pullylnk, n Alxnr Srijvr. Comintoril Optimiztion. Jon Wily & Sons, In., Nw York, NY, USA, 998. [] Stin Krol. A omintoril s or som optiml mtroi intrstion loritms. Tnil Rport STAN-CS-7-6, Computr Sin Dprtmnt, Stnor Univrsity, Stnor, Cliorni, 97. [] Jnos P. Mtroi intrstion. Ltur 7. ttp://.pl./p-8766-n.tml, 0. [6] Alxnr Srijvr. T ry loritm n t inpnnt st polytop. In Comintoril Optimiztion: Polyr n Eiiny, volum B. Sprinr, 00. [7] Alxnr Srijvr. Mtroi intrstion. In Comintoril Optimiztion: Polyr n Eiiny, volum B. Sprinr, 00. 9