Approximation Algorithms

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1 Prsnttion or us with th txtook, Alorithm Dsin n Applitions, y M. T. Goorih n R. Tmssi, Wily, 2015 Approximtion Alorithms 1 Bik Tour Suppos you i to ri iyl roun Irln you will strt in Dulin th ol is to visit Cork, Glwy, Limrik, n Blst or rturnin to Dulin Wht is th st itinrry? how n you minimiz th numr o kilomtrs yt mk sur you visit ll th itis? 1

2 Optiml Tour I thr r only 5 itis it s not too hr to iur out th optiml tour th shortst pth is most likly loop ny pth tht rosss ovr itsl will lonr thn pth tht trvls in i irl Exhustiv Srh: Too Mny Tours Thr is prolm with th xhustiv srh strty th numr o possil tours o mp with n itis is (n 1)! / 2 n! (pronoun n toril ) is th prout n (n 1) (n 2) Th numr o tours rows inrily quikly s w itis to th mp #itis #tours , , ,440 Th numr o tours or 25 itis: 310,224,200,866,619,719,680,000 2

3 Th Trvlin Slsmn Computr sintists ll th prolm o inin n optiml pth twn n points th trvlin slsmn prolm (TSP) Th TSP is mous prolm irst pos y Irish mthmtiin W. R. Hmilton in th 19th ntury intnsly stui in oprtions rsrh n othr rs sin 1930 This tour o 13,500 US itis ws nrt y n vn lorithm tht us svrl triks to limit th numr o possil tours Rquir 5 CPU-yrs Dlin with Hr Comintoril Prolms (CP) 1. For mny Comintoril Prolms (CP), not ll instns r hr to solv. Som spil ss n solv sily,., rph olorin is hr, ut trs n two-olor sily. 2. For smll prolms, it my possil to solv thm usin ktrkin srh or rnh-noun lorithms. 3. Look or oo-nouh pproximt solutions. 4. Us rnom lorithms to in solutions with hih proility. 3

4 Bin Pkin Bin Pkin Prolm Th ins (pity 1) Itms to pk Bin Pkin Bin Pkin Prolm Optiml Pkin Optiml solution: C* = 4 4

5 First Fit Pkin Alorithm Bin Pkin Prolm Optiml solution: C* = 4 First Fit Pkin Alorithm (FF): For h itm s i, in th irst in (rom lt to riht) whih n hol s i..1 FF solution: C = C C * 2 Bst Fit Pkin Alorithm Bin Pkin Prolm Bst Fit Pkin Alorithm (BF): For h itm s i, in th most-ull in whih n hol s i..1 BF solution: C = C C * 2 5

6 First Fit Drsin Pkin Alorithm Bin Pkin Prolm First Fit Drsin (FFD): First sort th itms in rsin orr, thn pply FF FFD solution: C = C* = 4 C C * 2 Bst Fit Drsin Pkin Alorithm Bin Pkin Prolm Bst Fit Drsin (BFD): First sort th itms in rsin orr, thn pply BF BFD solution: C = C* = 4 C C * 2 6

7 Nxt Fit Pkin Alorithm Bin Pkin Prolm C* = 4 Nxt Fit Pkin Alorithm: I th urrnt in nnot hol s i, strt nw in (oo or onlin pplition) C= 6 Bin Pkin Approximtion Alorithms: Not lwys optiml solution, ut with som prormn urnt (, no worst thn twi th optiml) Evn thouh w on t know wht th optiml solution is!!! 7

8 Approximtion Trminoloy Givn n instn I o n optimiztion prolm, lt C * = OPT(I) th ost o n optiml solution, n lt C th ost o th solution o n pproximtion lorithm. Th lorithm hs n pproximtion rtio o ρ(n) i, or ll solutions mx(c/c *,C * /C) ρ(n). W sy tht n pproximtion lorithm with n pproximtion rtion o ρ(n) is ρ(n)- pproximtion lorithm. Approximtion Trminoloy (2) An pproximtion shm is n pproximtion lorithm tht tks n instn n n ε > 0, n prous (1+ε) pproximtion (ρ(n) = 1+ε). I n pproximtion shm runs in polynomil tim in th siz its input whn ivn ny ε, w sy it is polynomil-tim pproximtion shm. Not tht th runnin tim oul still inrs rpily s ε rss. 8

9 Approximt Bin Pkin Alorithms: C* = OPT(I), C is th pproximt ost 1 B 1 B 2 B 3 B C-1 B C s(b 1 )+s(b 2 ) > 1 s(b 2 )+s(b 3 ) > 1 s i = s(b i ): totl siz o itms in B i Lt s 1 + s s m = S Thn C* S From 2S > C 1 W hv 2C* 2S C s(b C-1 )+s(b C ) > 1 s(b 1 ) > 0, s(b C ) > 0 2(s(B 1 )+s(b 2 )+ + s(b C )) > C 1 Or 2S > C 1 Approximt Bin Pkin Alorithms Thorm First-Fit: C 17/10C* + 2 First-Fit Drsin: C 11/9C* + 4 9

10 Vrtx Covr Prolm Lt G=(V, E). Th sust S o V tht mts vry o E is ll th vrtx ovr. Th Vrtx Covr prolm is to in vrtx ovr o th minimum siz. It is hr COP. 19 Exmpls o vrtx ovr 20 10

11 Approximtin Vrtx Covr Th ollowin lorithm is 2-pproximtion lorithm or th vrtx-ovr optimiztion prolm: Choos n rom G. A oth s npoints (sy u n v) to th vrtx ovr. Rmov ll s inint to u n v. Rpt whil thr r s lt. APPROX_VERTEX_COVER(G) 1 C 2 E' E( G) 3 whil E' 4 o lt ( u, v ) n ritrry o E' 5 C C { u, v} 6 rmov rom E' vry inint on ithr u or v 7 rturn C 11

12 Approx. solution Optiml solution C = 6, C* = 3 Complxity: O(E) Thorm C*: th siz o optiml solution C: th siz o pproximt solution A: th st o s slt in stp 4 APPROX_VERTEX_COVER hs rtio oun o 2. Proo. Lt A th st o slt s. C 2 A A C * C 2C * Whn on is slt, 2 vrtis r into C. No two s in A shr ommon npoint u to stp 6. 12

13 Vrtx Covr: Gry Alorithm 2 I: Kp inin vrtx whih ovrs th mximum numr o s. Gry Alorithm 2: 1. Fin vrtx v with mximum r. 2. A v to th solution n rmov v n ll its inint s rom th rph. 3. Rpt until ll th s r ovr. How oo is this lorithm? 25 Vrtx Covr: Gry Alorithm 2 OPT = 6, ll r vrtis. SOL = 11, i w r unluky in rkin tis. First w miht hoos ll th rn vrtis. Thn w miht hoos ll th lu vrtis. An thn w miht hoos ll th orn vrtis

14 Vrtx Covr: Gry Alorithm 2 k! vrtis o r k Not onstnt tor pproximtion lorithm! Gnrlizin th xmpl! k!/k vrtis o r k k!/(k-1) vrtis o r k-1 k! vrtis o r 1 OPT = k!, ll top vrtis. SOL = k! (1/k + 1/(k-1) + 1/(k-2) + + 1) k! lo(k), ll ottom vrtis. 27 Th Trvlin Slsmn Prolm with th Trinl Inqulity Th osts in th trvlin slsmn prolm o not n to rprsnt istns (.., thy my thins lik irlin rs). I th osts o rprsnt omtri istns, thy oy th trinl inqulity or ll vrtis u, v, n w: (u,w) (u,v) + (v,w) Th prolm rmins vry hr

15 Approximtin Trvlin Slsmn Prolm with th Trinl Inqulity Th ollowin is 2-pproximtion o TSP with th trinl inqulity: omput minimum spnnin tr T o th wiht liqu orr th vrtis orin to prorr wlk on T rturn this orr s th TSP tour Euliin Trvlin Slsmn Prolm APPROX-TSP-TOUR(G, W) 1 slt vrtx r Є V[G] to root. 2 omput MST or G rom root r usin Prim Al. 3 L = list o vrtis in prorr wlk o tht MST. 4 rturn th Hmiltonin yl H in th orr L

16 root Euliin Trvlin Slsmn Prolm h MST h Pr-Orr wlk Hmiltonin Cyl h h 31 Pr-orr Solution Trvlin slsmn prolm This is polynomil-tim 2-pproximtion lorithm. (Why?) Bus: APPROX-TSP-TOUR is O(V 2 ) C(MST) C(H*) Optiml C(W) = 2C(MST) C(W) 2C(H*) C(H) C(W) C(H) 2C(H*) H*: optiml soln W: Prorr wlk H: pprox soln & trinl inqulity 32 16

17 St Covr SET COVER: Givn st U o lmnts, olltion S 1, S 2,..., S m o susts o U, n n intr k, os thr xist olltion o k o ths sts whos union is qul to U? Smpl pplition. m vill pis o sotwr. St U o n pilitis tht w woul lik our systm to hv. Th i th pi o sotwr provis th st S i U o pilitis. Gol: hiv ll n pilitis usin wst pis o sotwr. Ex: U = { 1, 2, 3, 4, 5, 6, 7 } k = 2 S 1 = {3, 7} S 4 = {2, 4} S 2 = {3, 4, 5, 6} S 5 = {5} S 3 = {1} S 6 = {1, 2, 6, 7} 33 St Covr (Gry Alorithm) OPT-SET-COVER: Givn olltion o m sts, in th smllst numr o thm whos union is th sm s th whol olltion o m sts? OPT-SET-COVER is hr COP. Gry pproh prous n O(lo n)-pproximtion lorithm

18 Gry St Covr Anlysis Consir th momnt in our lorithm whn st S j is to C, n lt k th numr o prviously unovr lmnts in S j. W py totl hr o 1 to this st to C, so w hr h prviously unovr lmnt i o S j hr o (i) = 1/k. Thus, th totl siz o our ovr is qul to th totl hrs m. To prov n pproximtion oun, w will onsir th hrs m to th lmnts in h sust S j tht lons to n optiml ovr, C. So, suppos tht S j lons to C. Lt us writ S j = {x 1, x 2,..., x nj } so tht S j s lmnts r list in th orr in whih thy r ovr y our lorithm. 35 Gry St Covr Anlysis, ont

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