10.1 Approximation Algorithms

Transcription

1 Approxmato Algorthms Let us exame a problem, where we are gve A groud set U wth m elemets A collecto of subsets of the groud set = {,, } s.t. t s a cover of U: = U The am s to fd a subcover, = U, cotag as few subsets as possble Ths problem s ow as the Mmum et Cover mc) Oe of the oldest ad most studed combatoral optmzato problems 29 The correspodg decso problem Gve: a groud set U, cover ad a atural umber Questo: Does U have a subcover s.t. '? Theorem The decso verso of mmum set cover problem s NP-complete. Proof. Obvously mc NP: Let us guess from the gve cover a subcover ' cotag subsets ad verfy determstcally polyomal tme that we really have a subcover.

2 292 Polyomal tme reducto VC mp mc s easy to gve. Let G, be a stace of the vertex cover whch G = V, E). We choose the mappg f: fv, E), ) = E, V E,, where V E s the collecto of edges coected to the odes of G. I other words, for each v V has a correspodg set { e E e = v, w) }. Clearly f s computable polyomal tme ad s a reducto

3 294 Hece, mc s a tractable problem we do ot ow of a polyomal tme algorthm for solvg t Therefore, we attempt to fd a polyomal tme algorthm that does ot ecessarly gve the best possble optmal) soluto, but ca be show always to be at most a fucto of the put legth worse tha the optmal soluto uch a algorthm s called a approxmato algorthm Let us deote by Opt the cost of the soluto gve by a optmal algorthm ad App that of the soluto gve by a approxmato algorthm 295 ce mc s a mmzato problem, App/Opt The closer to ths rato s, the better the soluto produced approxmates the optmal soluto From a approxmato algorthm oe requres that the fracto s bouded by a fucto of the legth of the put App ) Opt ) s the approxmato rato of the algorthm The algorthm s called a )-approxmato algorthm At the best the approxmato rato does ot deped at all o the legth of the put, but s costat 3

4 296 Let us exame the followg algorthm for vertex cover We wll show that t s a 2-approxmato algorthm for the problem Iput: A udrected graph G = V, E) Output: Vertex cover C. C ; 2. E' E; 3. whle E do a. Let u, v) be ay edge of the set E'; b.cc{ u, v}; c. Remove from E all edges coected to odes u ad v; 4. od; 5. retur C; 297 electo of the frst radom edge: b, c) b c d a e f g 4

5 298 We remove other edges coected wth odes b ad c b c d a e f g 299 The ext radom choce : e, f) ad Removal of other edges coected wth ts odes b c d a e f g 5

6 300 The oly remag choce d, g) We ed up wth a cover of 6 odes, whle the optmal oe has 3 odes e.g., b, d, e) b c d a e f g 30 Theorem 0. The above gve algorthm s polyomal tme 2- approxmato algorthm for vertex cover. Proof. The tme complexty of the algorthm, usg adjacecy lst represetato for the graph, s OV + E), ad thus uses a polyomal tme. The set of odes C retured by the algorthm obvously s a vertex cover for the edges of G, because odes are serted to C the loop of row 3 utl all edges have bee covered. Let A be the set of edges chose by algorthm row 3a. I order to cover the edges of A ay vertex cover partcular also the optmal vertex cover has to cota at least oe of the eds of each edge A. 6

7 302 Because the ed pots of the edges A are dstct by the desg of the algorthm, A s a lower boud for the sze of ay vertex cover. I partcular, Opt A. The above algorthm always selects row 3a a edge whose ether ed pot s yet the set C. Hece, App = C = 2 A. Combg the above equatos yelds App = 2 A 2 Opt, ad therefore App/Opt Also set cover has a smple greedy approxmato algorthm Nether ths or ay other polyomal tme determstc algorthm ca atta a costat approxmato rato Iput: Groud set U ad ts cover Output: et cover C. XU; C ; 2. whle X do a. select s.t. X s maxmzed; b.xx\'; c. CC{ }; 3. od; 4. retur C; 7

8 Greedy: 4 subsets

9 306 Optmal: 3 subsets The greedy algorthm ca qute easy to mplemet to ru polyomal tme the legth of the put U ad The loop row 2 s executed at most m U, ) tmes ad the body of the loop tself ca be mplemeted to requre tme O U ) Altogether the tme requremet thus s O U m U, )) It s also possble to gve a lear tme mplemetato for the greedy approxmato algorthm for set cover The collecto C retured by the algorthm s obvously a set cover, because the loop of row 2 s executed utl there are o more elemets to cover 9

10 308 I order to relate the cost of the set cover retured by the greedy algorthm, we set cost to each of the chose sets Let be the set selected by the greedy algorthm at roud We dstrbute the cost of evely amog all those elemets t that ow become covered for the frst tme Let c u deote the cost assged o elemet u U Each elemet gets assged a cost oly oce, the frst tme t s covered by some set If u s frst covered by the set, the cost assged to t s: c u \ 2 ) 309 Each set selected by the greedy algorthm s assged cost so that App C O the other had, the cost of the optmal cover C* s Because each u U belogs to at least oe C*, we have c u c c u ' C* u ' ' C* u ' c u uu uu Combg the above gve yelds App c u ' C * u ' u 0

11 30 Let H) deote the -th harmoc umber H ) j j 2 We defe H0) = 0 Next we show that for ay t holds H ' ) The, by the prevous equalty, App H ' ) c u u ' ' C* C* Hmax{ ' : ' }) Opt H max{ ' : ' }) 3 Lemma For each t holds c H ' ) u u ' Proof. Let be arbtrary ad =, 2,, C. Furthermore, let = \ 2 ) be the umber of those elemets of that have ot yet bee covered whe the greedy algorthm has chose sets, 2,, to the set cover. Let 0 = '. Let be the smallest dex s.t. = 0;.e., every elemet of belogs to at least oe of the sets, 2,,. The ad, =, 2,,, covers elemets for the frst tme.

12 2 32 Now ce s chose greedly, t must cover at least as may elemets as the set ' or otherwse ' should have bee selected). Hece, Whch further yelds. ) \ ) ' u u c ) \ ) \ '. ) u u c 33 because j. Moreover, sce the other terms the sum cacel each other out., ) ' j j u u j c ), ) ) ) 0 j j H H H H j j

13 34 We have chose = 0 ad defed H0) = 0. Therefore, further = H 0 ) H0) = H 0 ) = H ' ) ad we have proved the lemma. For the harmoc umber H) t holds l < H) l + From the above results t follows: Theorem For the greedy algorthm of the set cover problem t holds that App Opt H max{ ' : ' } l U 35 I some applcatos max { ' : } s a small costat The the soluto retured by the greedy algorthm s oly a small costat away from the optmal oe I partcular, f subsets ' have a upper boud d for ther sze, App/Opt Hd) E.g., whe the odes of the graph of vertex cover have maxmum degree 3, the the soluto retured by the greedy set cover algorthm s at most H3) = /6 < 2 tmes as large as the optmal cover 3

14 36 Fege, 996: o polyomal-tme algorthm ca approxmate mc wth ) l m, for ay > 0, uless NP DTIME loglog ) Hece, t s ot possble to fd a approxmato algorthm for mc that would be sgfcatly better tha the greedy oe laví, 996: A more exact upper boud for the approxmato rato of the greedy algorthm s l m l l m + ) I fact ths s also a lower boud for the approxmato rato of the greedy algorthm l m l l m + ) s thus the asymptotcally exact approxmato rato of the greedy algorthm 37 Course Recap All computatoal problems caot be solved algorthmcally A determstc fte automato DFA) has a uque mmal automato. The mmal automato ca be costructed a straghtforward maer Nodetermstc fte automata NFA) do ot recogze more laguages tha DFAs A laguage s regular t ca be recogzed wth a fte automato t ca be descrbed wth a regular expresso t ca be geerated wth a rght-lear grammar 4

15 38 trgs of a regular laguage ca be ''pumped'' There are sesble formal laguages that are ot regular Cotext-free laguages are a proper superset of regular laguages A laguage s cotext-free f ad oly f t ca be recogzed wth a pushdow automato PDA) By the Church-Turg thess ay problem solvable o a computer ca also be solved usg a Turg mache TM) Varats of Turg maches cludg odetermstc Turg maches have equal recogto power to the stadard sgletape mache 39 The ''effcecy'' of dfferet maches vares Laguages geerated by urestrcted grammars are equvalet to those recogzed by Turg maches A total Turg mache decder) halts o every put A formal laguage s Turg-recogzable TR), f t ca be recogzed wth a TM ad Turg-decdable, f t has a decder A ad B decdable, A B ad A B are decdable A, B TR A B),AB)TR A decdable A, TR A TR, ot decdable TR 5

16 320 D = { c { 0, }* c LM c ) } TR E.g., A DFA, E DFA, ad EQ DFA are decdable laguages U = { M, w w LM) } TR, ot decdable = { M, w w LM) } TR H = { M, w Mw) } TR, ot decdable = { M, w Mw) } TR Chomsy herarchy: fte regular cotext-free cotext-sestve laguages geerated by urestrcted grammars = TR) 32 NE = { M M s a TM ad LM) } TR, ot decdable REG = { M M s a TM ad LM) s regular } s ot decdable Rce s theorem: All otrval sematc propertes of Turg maches are udecdable Lear bouded automato LRA) caot use more wor space tha that already requred by put A LRA s decdable, whle E LRA s udecdable A * s reducble to B, B *, deoted A m B, f there exsts a computable fucto f: * * s.t. x A fx) B x * 6

17 322 A { 0, }* s TR-complete, f.atr ad 2.B m Afor all B TR Laguage U s TR-complete I tme complexty aalyss of Turg maches oe exames the worst case wth puts o certa legth Relatg growth rates of fuctos: O,, o, The umber of tapes does ot have a sgfcat mpact o the effcecy of a Turg mache The effcecy dfferece of a determstc ad odetermstc TM, o the other had, s expoetal 323 P = 0 DTIME + ) EXPTIME = 0 DTIME2 ) P cludes laguages that ca be decded tme that s polyomal the legth of the put E.g., fdg a drected path a graph, PATH, ad decdg whether two umbers are relatvely prme are examples of problems P The correspodg odetermstc composte classes are NP ad NEXPTIME For stace, CLIQUE ad UBET-UM are problems NP Problems belogg to P are solvable practce 7

18 324 P NP. I addto NP cotas problems for whch o polyomal tme algorthm s ow A * s polyomal tme reducble to B, B *, deoted A mp B, f there exsts a polyomal tme computable fucto f: * * s.t. x A fx) B x * A { 0, }* s NP-complete, f.anp ad 2.B mp Afor all B NP All problems NP are polyomal tme reducble to a NPcomplete problem. If ay NP-complete problem s P, the P = NP 325 howg that A NP s NP-complete:. elect a smlar problem B that s ow to be NP-complete 2. Gve a polyomal tme reducto f: B mp A; by Theorem 7.36 also A s NP-complete NP-complete problems: AT, CAT, 3AT, VC, I, CLIQUE, Hamltoa path, TP, ubset-sum, ad mc PPACE = 0 DPACE ) NPPACE = 0 NPACE ) PPACE P NP EXPTIME NPPACE EXPPACE NEXPTIME NEXPPACE 8

19 326 TQBF s PPACE-complete o are the asymptotc versos of chess ad GO L = DPACElog ) NL = NPACElog ) PATH NL s NL-complete L =?= NL NL P Oe ca try to approxmate a tractable problem ca effcetly Vertex cover has a effcet 2-approxmato algorthm Mmum set cover has a effcet l m + )-approxmato algorthm 327 THE END Keep md that programmg vdeo games also requres uderstadg computatoally demadg problems 9