Reducing%20Knapsack%20to%20TSP

1
Reducing Knapsack to TSP

• Pasi Fränti

2
Knapsack problem

• Input knapsack instance 2,3,5,7,11
• Size of the knapsack S15.

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Step 1 Create one node for every item

• Input knapsack instance 2,3,5,7,11
• Create a node for every knapsack element.

2
7
5
3
11
4
Step 2 Add start and end points

• Input knapsack instance 2,3,5,7,11
• Add node 0 as the home.
• Add node N1 as the turning point.

2
7
5
0
n1
3
11
N2 nodes needed to represent the knapsack
instance
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Step 3 Create forward links

• Input knapsack instance 2,3,5,7,11
• Draw links from smaller nodes to bigger ones.
• Set weights according to the bigger node
  w(i,j)j.

7
2
7
2
7
7
7
5
5
5
0
n1
5
11
11
3
11
3
3
11
11
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Step 4 Create forward links for node N1

• Input knapsack instance 2,3,5,7,11
• Draw links to N1 with weights w(i,N1)0.

7
2
7
2
7
7
7
0
5
0
5
5
0
0
n1
0
5
0
11
11
3
11
3
3
11
11
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Step 5 Create backward links

• Input knapsack instance 2,3,5,7,11
• Draw backward links from bigger to smaller nodes.
• Set weight of the link as w(j,i)0.

0
2
7
0
0
0
0
0
0
0
0
0
5
0
n1
0
0
0
0
0
0
0
3
11
0
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Solution for KP ? Solution for TSP

• KP 3,5,7

• ?
• TSP 0-3-5-7-(N1)-11-2-0 (S15)

Visit the nodes in an increasing order !
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Solution for TSP ? Solution for KP

• TSP 0-3-5-7-(N1)-11-2-0

?

• KP 3,5,7 (all nodes which arrival cost gt
  0)

Select nodes with entrance wgt0 !