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Unformatted text preview: UNIVERSITY OF WATERLOO
MIDTERM EXAMINATION
FALL TERM 2004 Surname: “
First Name: L Signature: M— Id..# L Course Number 0&0 351 Course Title Network Flow Theory Instructor Professor J. Cheriyan Date of Exam October 27, 2004 Time Period 5:00 — 7:00 OR 7:00 — 9:00 Number of Exam Pages 9 pages
(including this cover sheet) Exam Type Closed Book Additional Materials Allowed None (No Calculators) 1. Write your name, id number, and signature in the space provided above. 2. You may use without proof any result from the course, unless you are asked to prove that speciﬁc
result. You must quote in full any result that you use without proof. 3. If you need extra space to answer a question, then use the back of the previous page, or ask the
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#1. [18 marks] (a) Use Dijkstra’s algorithm from Chapter 2.3 (which is based on the primal—dual method) to find
[15] a tree of shortest dipaths rooted from the node 5 for the digraph D = (N, A) and arc costs w E IRA given below. (*) Start with the given feasible potentials y 6 RN. Show all of your work. (For every iteration, show the potentials, the equality arcs, and the set of nodes reachable from s using equality arcs. At the end, clearly indicate the tree of shortest dipaths.)
[3] (b) For each node 1;, write down a shortest svdipath P3,, and write down its length w(Pv).
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(different) nodes of D. [2] (a) (i) Deﬁne what is meant by a tree of shortest dipaths rooted from s. A brew? 0mm Months to a. amrvvwm m 6]) b 9 My, my
smupm mam/Md tan “no, m m mucttm Mama]; are [2] (ii) Deﬁne what is meant by feasible potentials. 0 C2) [2] (iii) Deﬁne What is meant by the reduced cost of an arc. “TN. iKCCU/iﬂﬁoi Casi: of) am We UN M mew = [d u+UUuV"Uj\J
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{lg Jﬁtm’ﬂg 12h . [2] 6‘!) Give a formula for the reduced cost of an st—diwalk Q in terms of w(Q)_
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ECG» : NLQ) +35“ (it [4] (b) Suppose that some arcs i j have ng < 0, but there is no negative dicycle, and suppose that
feasible potentials are available. How would you ﬁnd a shortest stdipath by running Dijkstra’s algorithm? Explain in brief. _[
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(#2, continued) b (N/m [Nam §Itél> (c) Suppose that there are no negative dicycles. Then prove that feasible potentials exist. Note
that there may be no node i E N such that all nodes are reachable from i. (Give a complete proof, except that you may quote results from Chapter 1 without proof. If you
use any other results, then give the proofs.) $61M WPQL QWWXJDWQWQ am WW vog=0 Hwn CUM
mm_ mm cam We be, ‘0. {peach Nbdm. H3 waxgmvxbt mommwmm .bza/iubupotemaﬁo. #3. [25 marks] \3/ +’\ +— 1 O: )3 @113; 5)
Let D: (N, A) be a digraph, let .5 and t be two (different) nodes of D, and let w E IRA assign a
real—valued cost to each arc. An arc ij may have 1051' < 0 or 10;) > 0. ° [5] (a) (i) Write down a linear programming formulation (P) for the problem of ﬁnding a slim
st —dipath. (Explain all of your notation.) M
(ii) Write down the linear programming dual (D) of ( 15) scalar notation.
(iii) Write down the complementary slackness (CS) conditions for (P) and (D). ($3 mm X/(IUWK Su ‘ec’rto 411—1) 0 1:3
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g bee C 3 99161111 owns [3] (b) (i) Give an interpretation of the constraints of the primal (P) as an st—ﬂow problem on the
®/7 digraph D = (N, A) (ignore the objective function). What are the arc capacities in your interpretation? i [2] (ii) Suppose D has an st—dipath. Does this imply that the stﬂow problem (in part (b)(i)) has a
fe ' :. solution? Explain in brief. [4] (iii) If D has no st—dipath, then prove that the stﬂow problem (in part (b)(i)) has no feasible
solution.
(H ‘T: Deduce the minimum capacity of an stcut, then apply the maxﬂow mincut theorem.) . (#3, continued)
[1] (b)(iv) If D as no stdipath, then what can you conclude for the primal (15)? (c) [10 marks: 3—6 marks per part]
[10] Are the following statements TRUE or FALSE? If true, give a complete (and brief) explanation.
If false, give one counterexample or a complete explanation. (You may use results from the course without proof, provided you quote them in full.) A A  rimal (P) has a feasible solution, or the dual (D) has a feasible solution.  (ml m mum/Mic! mmwm MLMWW
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* "t MW meJrW—mam m \2 = 0? WWW i) Suppose that w 2 0 (all arc costs are 0), and that T is a tree of shortest dipaths rooted from 3. Let ij be an arc in T. If we ecrease the cost wij of ij to a new value wﬁj 2 0 (while keeping all other .  ;  . e sam , then T stays a tree of shortest dipaths rooted from s for am mmmw ago summmmwwb‘a Wﬂm. AW mWw\S must“; 7/0 we, lunow
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#4. [18 marks] (a) Apply the FordFulkerson algorithm (from Chapter 3.3) to ﬁnd a maximum st—ﬂow and a
minimum stcut for the digraph D = (N, A) and arc capacities c E IRﬁ given below. (a:) Start with the given stﬂow ac. Clearly show a maximum stﬂow and its value, and a minimum st—cut 6(5) and its capacity (indicate
the node set S’ and the arc—set 6(5)). Show all of your work. (You may run the algorithm with or without residual digraphs. For each
iteration, show the incrementing path, the updated stﬂow, and other essential information.) (If you need extra sheets showing the digraph, you may ask a proctor.) [3] (b) Give a list of all the arcs ij such that increasing the capacity cij by a small positive number 6 (while keeping all other arc capacities the same), increases the maximum value of an st—ﬂow. C x @ qu® @ UVE® W300 incrimwlw‘tg with w D’ tarot @s t at St damn) what lbw) cam WWuw
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Let D = (N, A) be a digraph, let 3 and t be two (different) nodes of D, and let c E Z”: assign a nonnegative integer value to each arc. [3] (a) (i) Write down a linear programming formulation for the problem of ﬁnding a maximum st—ﬂow.
(Explain all of your notation.) m0“ >\ 0 n
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2V [4] (ii) Deﬁne What is meant by
(I) 6(Q), Where Q is a set of nodes, and
(II) an st— cut. 0 “mm “6;? (u) imidﬁgmomwﬁmw [8] (b) Let 73 = {P1, P2, . . . , Pk} be a collection of st—dipaths such that each arc ij 6 A is in 3 CU of the stdipaths in ’P.
Prove: The maximum value of an st—ﬂow is 2 I’PI. (Here, ’Pl denotes the the number of stdipaths in the multiset ’P, counting repetitions; e.g., if 73
consists of 5 copies of P1 and 3 copies of P2, then I’PI = 8.) ~~~~~~ THIS PAGE LEFT INTENTIONALLY BLANK ...
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