Introduction to Knowledge Engineering

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Introduction to Knowledge Engineering

• Operations Research
• Session 3 part A
• Joris van de Klundert
• j.vandeklundert_at_math.unimaas.nl

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Last weeks exercises
Solve the knapsack problem (that is check the
hackers solution and the optimal solution in the
graph model) Solve the 2 Thieves problem (that
is find an appropriate model and solution
algorithm).
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Solutions to the Knapsack problem
Magic bread Cookies Fish Lollies Gums Nuggets
Glow worms Black worms Red worms
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Knapsack a Hackers solution
Magic bread (11) Cookies (11) Fish Lollies
(7) Gums (7) Nuggets (6) Glow worms (1) Black
worms (1) Red worms (1)
cl
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Knapsack an optimal solution
Magic bread (11) Cookies (11) Fish Lollies
(7) Gums (7) Nuggets (6) Glow worms (1) Black
worms (1) Red worms (1)
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2 Thieves (problem statement)
Two thieves are sitting with there stolen
property, which consists of a set items 1,2,3.m,
all watches and jewelry. The thieves know the
value v(i) of each of the items i1,2,m. The
thieves strive to find a division of the set of
items into 2 sets, one for each thief, such that
the sets are of the same value. (the value of a
set is the sum of the values of the items in the
set.) Can you handle (model) this problem for
them?)
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The two thieves problem..

• 1
• 2
• 3
• .
• .
• .
• n

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Interpretation of the model
Define V ?ni1 v(i) V.
V/2
Item i
Thief X gets item i
Thief Y gets item i
v(i)
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Modelling the 2 thieves problem
1 2 3 4 5 . . . .
V/2
1 2 3 . . . n
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Repeat the following step until the yellow vertex
is red
Take any orange node v Consider all blue nodes u,
such that there is an arc from v to u. Eliminate
arc (v,u). If u has no incoming arcs anymore,
change the color of u from blue to orange. Change
the color of v from orange to red.
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Interpretation

• Any path from the red node to the yellow node
  correspond to a set of items of total value V/2,
  and therefore partitions the set of all items in
  2 sets of the same total value!
• This problem is called the 2 partitioning
  problem.
• How many steps does the algorithm take?
• Not more than n V.