CO-351-1065-Midterm_solutions

CO-351-1065-Midterm_solutions - UNIVERSITY OF WATERLOO...

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Unformatted text preview: UNIVERSITY OF WATERLOO MIDTERM EXAMINATION SPRING TERM 2006 Surname: M First Name: Name:(Signature) ‘ Id.#: Course Number C&O 351 I Course Title Network Flows Instructor Professor Bertrand Guenin Date of Exam June 13th, 2006 Time Period 5:00 PM — 7:00 PM and 7:00 PM - 9:00 PM Number of Exam Pages (including cover page) 11+1 pages Exam Type Closed Book Additional Materials Allowed Value Mark Awarded 2 14 3 l4 |_____l_ 4 20 5 16 6 20 Total 1 00 1 Exercise 1. (16pts) You just moved to Waterloo and bought a new car. You will need to have a car during each of the next 5 years. The cost of maintaining a car during a year depends on its age, as given in the following table: Age (in years) Annual Maintenance Cost (3) 0—1 2,000 1—2 4,000 2-3 5,000 3—4 9,000 4-5 12,000 To avoid the high maintenance cost, you can trade in your car and purchase a new one at the start of each year. The price you receive for trading in your old car depends on its age at the time you trade it in: To simplify the computations, we assume that at any time, it costs $12,000 to purchase a new car. We are interested in minimizing our net—cost (purchasing cost + maintenance cost - money received in trade-ins) during years 1, 2, 3, 4 and 5. As an example you could keep your new car for 3 years with maintenance costs of $2,000+ $4,000+ $5,000, and trade it in for $3,000. Then you could buy a new one (at the beginning of year 4) for $12,000, keep it for two year with maintenance costs of $2,000+$4,000 and trade it in for $6,000 after two years (at the end of the 5 year period). This would give you a total cost of: $20,000. (1) Model the problem of finding the minimum net-cost as a shortest path problem. (2) What is the best algorithm to solve the corresponding shortest path problem? 44 3 l El La ‘ t \' : (3‘13 Cour at“ bcgi V! V03 0‘, «a v‘ Auk‘ OWL Nb I K¢¢P w»er (M 0} gearg { ~ \ \ K . *“ s n 0 PAGE LEFT INTENTIONALLY BLANK 3 a A ev- i 191$ Dr‘” ea TBS Cd, 0;. Q V‘ r 4: Cos‘r o.\w w1h4'znnugz. [rah/Clo ‘m tvUO’s) 3 Exercise 2. (14pts) Use Dijkstra’s algorithm (the algorithm for non-negative weights) to find a tree of shortest dipaths rooted at s. The number next to each arc is its length. INDICATE EVERY STEP OF THE ALGORITHM. 9 m Q’qv A are; 4 Exercise 3. (14pts) Using the Ford-Fulkerson algorithm, find a maximum st-flow and a minimum st—cut, starting from the initial flow given in the figure. The first value next to each arc 1w is its capacity cm, the second number is the flow new. INDICATE EVERY STEP OF THE ALGORITHM. Exercise 4. (20pts) Consider a digraph D = (N , A) with nodes 3, t and capacities c 6 RA. (1) Let a: be an st-flow prove that for any st-cut 6(3) we have Mt) = a115(5)) - 9605(3))- Note your proof should be self contained. (2) Let x be a maximum st-flow and consider any S g N such that s E S and t ¢ S. Prove that 6(8) is a minimum st-cut if and only if 6;; (S) = (0 (i.e. in the residual digraph there are no arcs leaving 5). Hint: Use part (1). [42) (Km; 2 350’) m u e s Cmsldu earl. am Mr owl» $42.6, Low XW appears \\n at): Case, \t on? as XV“, lieu) wi‘ x \r E (flawl- U’ & Kvw “Wham. wiH’) 4’1 "‘4' gang“ 0!, 4K”) va— n u -l “ “ “ ix”) 3) («Xv-“s Cm c@\» Ca“, ‘5'. V63, we? K (P «ppm: “Alia kl Cur—i. iucw/Se 9* 4x W’) Ca“, 4‘. V6 Eur-65 Xuxr' appear “NH” " M4. Lacwu’ al— L/ W) 6 07:05 DI THIS PAGE LEFT INTENTIONALLY BLANK 6 TLA SD’CS):¢’ mm» ><( scs/H : L (5cm W Mg m ‘ ' ~ v“. MA erm : 45(9)) 4! 5“) K a WWW. 5L c (015 X :S a Mbk"¥luw MA “has kl"; “FAQ Wartmy‘ Exercise 5. (16pts) (1) Find a topological ordering of the nodes in the digraph given below using the algorithm described in class (indicate all steps). (2) The digraph is a PERT digraph. The dashed arcs do not represent tasks but are only there to indicate some precedence constraints. Using the algorithm for shortest dipath in acyclic digraphs, find for every node ’U the earliest time 615(1)) at which all tasks with head 11 can be completed. Deduce the makespan. (3) What is the latest time task E can be completed without increasing the makespan of the project. Justify your answer. “A ‘91 :0 9,1: Max 20 *4} = 4 t4} :May'1047—(4403:4 1pm,): 9 . 03* aMAY £9+‘/4+t3=5 V L95 1 M,“ £4-f2 ’47.;‘319rt0317 2.) MAkLSan=y (0‘ 2 MAY is“; 7‘13: 9 C37 oli‘l,w= 1/ waificcpma 9 2:) ‘k‘h‘ll. G meet/ks *0 b; cowpbqu £0 9' 2 a THIS PAGE LEFT INTENTIONALLY BLANK 9 Exercise 6. (20pts) Consider a digraph D = (N, A) with weights w E 3?". Consider the linear program (P): min such that f$(u)=0 uEN L120 (LEA a A wax“ (1) Write the dual (D) of (P). Explain how it is obtained (simply writing the dual is not sufficient). (2) Write the complementary slackness conditions for (P) and (D). (3) Show that if there is a negative dicycle then (P) is unbounded. (4) Deduce from (3) that if there is a negative dicycle then there are no feasible potentials. (5) Show that if there is no negative dicycle then ma = O for all a E A is an optimal solution of (P). (i) Max 040 (obv>/ w V (D\ K210 :3.) } (gr (D) (I v (I) wa >0 =J'9¢’°‘f)\r >/ ‘41.“;— m Let c, be “Tim din (, be that) V Thu +x¢(\l\ ; 0 VV 6 N as) LAXQL >\ ml V/\7/0 ,,\><“”7/0 hul‘ wkxaz>z~£ 1) vnbOVVWl‘A- <0 (5‘) BO (33 Aka-Oak Winn/H.144 a) inimgjiaip 2:) Vto lrekSA’h’ ()A'ewhinlfi Mo W¢§A¥;VL A‘LOOk “‘3 3 vol-tmkais (proM :4 45“» 1’) 34¢“?le $J» ai— Cb) leaL lML Nut 0 1"“) O is lchr hwwA on (F) a) X; o :s u‘plhmgl W (P) ...
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CO-351-1065-Midterm_solutions - UNIVERSITY OF WATERLOO...

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