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Problem 1 (28%)
Consider the max flow instance given by the following flow network G with
capacities c(u, v) given as labels on the arcs (u, v).
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We define a flow f in G by the following diagram indicating all strictly positive
values f (u, v) as labels on the arcs (u, v).
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Question 1: What is the value of the flow f ? Justify your answer.
Question 2: What is c({s, 5, 6}, {1, 2, 3, 4, t}) and f ({s, 5, 6}, {1, 2, 3, 4, t})?
Justify your answer.
Question 3: Draw the residual network Gf with capacities as arc labels. No
justification is needed.
Question 4: Does the residual network Gf have an augmenting path? If so,
indicate it. No justification is needed.
Question 5: Draw a maximum flow in G (in the same way as the flow f was
drawn above) and prove that it is indeed maximum.
Question 6: State, as a pair of sets, a minimum-capacity cut in G and prove
that it is indeed minimum.
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Problem 2 (27%)
A company earns money by producing and selling dolls. The price of dolls as
well as the cost of producing them fluctuates. We want to plan a production
schedule for the year 2006. The following data are known in advance.
• At day i, for i = 1, 2, . . . , 365, the company is able to produce up to pi
dolls, but not more.
• At day i, for i = 1, 2, . . . , 365, the cost of producing each doll is vi Danish
kroner.
• At the end of day i, for i = 1, 2, . . . , 365, the company has contractual
rights allowing it to sell up to si dolls, but not any more than that. Also,
the company has contractual obligations implying that it must deliver and
sell at least ti dolls at the end of day i.
• At day i, for i = 1, 2, . . . , 365, the company will earn di Danish kroner for
each doll it sells.
• The company may produce dolls one day and sell them at a later date
but there is a cost associated with storing dolls. The cost of storing each
doll overnight between day i and day i + 1, for i = 1, . . . , 364, is ri Danish
kroner. Also, at most ui dolls can be stored overnight between day i and
day i + 1.
• The company is not allowed to have any dolls in stock at the end of day
365.
Question 7: Given the data (pi , vi , si , ti , di )i=1,...,365 and (ri , ui )i=1,...,364 , we
want to find a schedule of producing, storing and selling dolls that maximizes
the total profit of the company. Show how to formulate this as a min cost flow
problem, as this problem is defined in the note “The min cost flow problems”
by Miltersen. No formal proof of correctness is needed, but please explain how
the flow along each arc in a feasible flow in the network you construct is to be
interpreted in terms of the original problem.
Question 8: Show how to formulate the problem of Question 7 as a linear
program (no justification needed).
Question 9: Imagine that you next solve the min cost flow instance or the
linear program you have devised by giving either one of them to software that
solves such instances to optimality. Do you need any additional properties of
such software (besides that the software produces optimal solutions), in order
to be sure that you obtain a meaningful solution? If so, which algorithms for
min cost flow and linear programming do you know to possess these properties?
Please argue separately for the case of min cost flow and the case of linear
programming.
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Problem 3 (45%)
The standard game of Rock-Scissors-Papers is played as follows: Two players
each simultaneously choose a move from the set {rock, scissors, paper}. If the
two players choose the same move, no money is exchanged. If one player chooses
“scissors” and the other “rock”, the player choosing “scissors” pays the other
player 1 coin. If one player chooses “paper” and the other “scissors”, the player
choosing “paper” pays the other player 1 coin. If one player chooses “rock” and
the other “paper”, the player choosing “rock” pays the other player 1 coin.
Alice and Bob think it is very cool when “paper” beats “rock”, so they decide to
play the following variation of the game, called Paper-Rules: The rules are the
same as in Rock-Scissors-Paper, except that when one player chooses “paper”
and the other “rock”, the player choosing “rock” pays the other player 2 coins,
rather than just 1 coin.
We may model the game of Paper-Rules as a matrix game with Alice being the
row player, trying to maximize the payoff and Bob being the column player,
trying to minimize the payoff, and the moves “rock”, “scissors” and “paper”
being the pure strategies. The payoff matrix is then the following matrix
rock
scissors
paper
rock scissors paper
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Question 9: Write down a linear program expressing Alice’s optimal mixed
strategy in the game of Paper-Rules (no justification needed).
Question 10: Prove that the value of Paper-Rules is 0 and that an optimal
mixed strategy for Alice is to play “scissors” with probability 1/2, “rock” with
probability 1/4 and “paper” with probability 1/4.
Becoming tired of playing Paper-Rules, Alice and Bob invents a variation, Playwith-fire. In addition to the three original moves, a fourth move, “fire” is introduced. If neither player chooses “fire”, the outcome is the same as in the
original game of Paper-Rules. If both players choose “fire”, no money is exchanged. If one player chooses “fire” and the other “paper” or “scissors”, the
player choosing “fire” receives 1 coin from the other player (fire burns paper
and melts scissors). If one player chooses fire and the other “rock”, the player
choosing “fire” pays 1 coin to the player choosing “rock” (rock extinguishes fire).
Question 11: Write down the payoff matrix for the game of Play-With-Fire
(no justification needed).
We introduce a definition, applicable to any matrix game.
Useless pure strategies: For a matrix game, we say that a pure strategy j
of the column player is useless, if yj = 0 in every optimal mixed strategy y for
the column player, where yj denotes the j’th entry in the column vector y.
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Question 12: Identify the useless pure strategies in the game of Play-With-Fire
(no justification needed).
We introduce a second definition, applicable to any matrix game.
Exploitable pure strategies: For a matrix game given by a matrix A, we
say that a pure strategy j of the column player is exploitable, if there exists an
optimal mixed strategy x for the row player, so that (xA)j > v, where v is the
value of the game and (xA)j denotes the j’th entry in the row vector xA.
Question 13: Identify the exploitable pure strategies in the game of PlayWith-Fire (no justification needed).
The next questions are about arbitrary matrix games and not just about the
specific cases of Paper-rules and Play-with-fire.
Question 14: For a matrix game given by a matrix A with known value v and a
given pure strategy j for the column player, write down a linear program so that
the value of its optimal solution is 0 if and only if j is useless (no justification
needed).
Question 15: For a matrix game given by a matrix A of known value v and
a given pure strategy j of the column player, write down a linear program so
that the value of its optimal solution is 0 if and only if j is exploitable (no
justification needed).
Question 16: Prove that for all matrix games, each exploitable pure strategy
for the column player is also useless.
Question 171 : Is it true that in all matrix games, each useless pure strategy
for the column player is also exploitable? Justify your answer.
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may find that this question is more difficult than the other questions on this exam.
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