Review Answers for E&CE 700T02

Transcription

1 Review Aswers for E&CE 700T0

2 . Deermie he curre soluio, all possible direcios, ad sepsizes wheher improvig or o for he simple able below: 4 b ma c B N B N ^0 0 0 curre sol =, = Ch for * Ch for improvig direcio Ch obj f is 0, lambda = /5 ^ /5 0 0 /5 B N N B,4 are basis variables * OTHER DIRECTION IS NOT IMROVING BUT OBTAINS LAMBDA =/ WITH X,X BASIS VARIABLE

3 . Compue opimal soluio o : ma z z such ha -z z <= zz<=6 z<=4 z,z>=0 Sol: z=4, z=

4 .Compue opimal soluio o: ma 45 such ha <= <= 4<= 45<= 5<= as. == 5=0.5

5 4. Fid he miimal cos spaig ree for he graph below where values o edges represe he coss As. 8 see bold black edges, Noe: oher ses of edges are also possible

6 5. Fid he maimum flow from s o of he graph below. Each forward s o arc Has capaciy of uless labeled oherwise. s,, As. Ma Flow =5

7 6.Fid maimum flow for give flows ad capaciies labelled o he graph below flow, capaciy. As. Ma flow = 9,4 S 5,5,6 0,, T 4,,5,5,,7 6,4,6

8 7. Give he iiial machig o he graph, provide he H ree Ad he e machig, usig he maimum machig algorihm. use curvy lies o ideify he e machig o he graph. X y5 6 y y y y4 y5 y6 ah o flip is 4-y6--y-6-y5 5 y y y4 y y6 4

9 8. Usig shores pah algorihm used durig lecures, ideify all Shores pahs from ode u o all oher odes i he graph. Specifically Defie S={u}, S={u, }, S={u,, } hrough he shores pah algorihm a b Sages. 5 7 f 4 u g c 4 6 e d S={u}={u,a}={u,a,e}={u,a,e,f,d,b}={u,a,e,f,d,b,c}={u,a,e,f,d,b,c,g}

10 9. You are give a se of M asks, =,,,M ad N processors, =,, N. each ask mus be assiged o oe processor ad each rocessor ca be assiged a mos wo asks. Assume Represes he proposiio ha ask is assiged o processor use biary variables, a Formulae he followig cosrai: b Formulae he followig cosrai:, i i j, i,, j j, c Give a formulaio of he objecive fucio which is o miimize The umber of processors beig used. You may defie ew cosrais Or variables as required

11 ,,,,,,, j i j i j i,,,,,,,,,,, j i i j i d b a c b a d c b a d c b a d c b a j i j i a b

12 c We have oly biary variables: we iroduce a ew biary variable: p which isif processor has a leas oe ask assiged o i, else i is zero., ow we ca creae he objecive fucio Mi opioally, we could add,, N p N p p p however his is o ecessary sice our objecive fucio is creaig his for us.,

13 0. Give a good I formulaio for he followig problem: The problem is he maimum cardialiy ode packig problem O he graph below cosraied wih he followig iequaliy: i = 0 bi i i= 7 4, b = {,5,, }

14 i 0 = bi i i= 7 5 5y 5y C C C C 4 8 y y = {,}, is a face. = {,}, is a face = {,4}, is a face = {,,4}, we do o kow if i is a face or o. Faces : 4, b = {,5,, } 9 y y y y 4, 8 4, le y , 8 4 = 0 0 Is a kapsack iequaliy So we ca geerae faces or Sroger iequaliies o improve The formulaio We ca use graph o erac ode packig Faces/iequaliies i 4 6 0, 5 7, ec, clique faces lifig lemma for odd cycles 4 5,

15 . Describe how you would solve he followig problem i geeral: Fid he miimum eecuio ime for processor o eecue a se of ask defied usig a direced acyclic graph, where arcs represe daa rasfers bewee asks. Assume oe ask a a ime ca be eecued o each processor, asks ca be eecued i parallel oe differe processors, ad here is o commuicaio delay. all asks have he same duraio=>maimum machig o complemeary ask graph

16 . Reformulae he followig opimizaio problem, so ha oe ca impleme i wih a L solver which oly suppors mi or ma objecive fucios o he mima objecive below Mi A b Ma a i j j i, j

17 Mi Ma A b MiZ j a i, j A b j i j a i, j cosider removig he ma Z, i j i par, reformulaed problem is

18 . For shores pah we have, { f s C } Miimum s, s s 4. Eplai he differece bewee simulaed aealig ad abu search: abu search acceps o-improvig moves, whereas i simulaed aealig o-improvig moves are acceped oly wih probabiliy calculaios. There may be cyclig i simulaed aealig bu o i abu search.