Reducing Knapsack to TSP

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Cooks theorem and NP-reductions
6.11.2016
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Cooks theorem
Theorem Satisfiability problem (SAT) is
NP-complete Proof
TM p SAT
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Satisfiability problem
f1
f2
f3
F(X1? X2 ? ?X3) ? (?X1? X2) ? (X1? X3)
x1 x2 x3 f1 f2 f3 F
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1
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Non-deterministic algorithm
Satisfiability(F) FOR i?1 TO n DO xi ?
Choose0,1 IF F(x)1 THEN SUCCESS ELSE FAIL
T(n)O(n) ? SAT ? NP
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Notations needed
Implication
Equivalence
All are true
Some is true
Exactly one is true
At least oneis true
At most one is true
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Proof of Cooks theorem
Only valid Turing machine configurations can have
value TRUE
ai tape position s has symbol ai
qj machine is at state qj and R/W head is at
position s
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Cooks theorem
Machine is at state q0 Tape content is ai1,
ai2, , ain
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Cooks theorem
If not at position s, then content does not change
If at position s and content is ai, then change
state and content
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Cooks theorem
Can be in any position s but machine must be in
final state qk
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NP hard problems

1. Satisfiability problem (SAT)
2. Coloring problem (Color)
3. Exact cover problem (EC)
4. Knapsack problem (KP)
5. Traveling salesman problem (TSP)

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Satisfiability to Coloring
Complete k-clique
False color
c0
Connect to all but ci
Literals
Color for fi
xj
fi
fi
Connect to all but those literals in fi
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SP to Coloring example
(X1? X2) ? (?X1? X3)
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Additional example
f1
f2
f3
(X2 ? ?X3) ? (?X1? X3 ) ? (X1 ? ?X2)
x1 x2 x3 f1 f2 f3 F
0 0 0 1 1 1 1
0 0 1 0 1 1 0
0 1 0 1 1 0 0
0 1 1 1 1 0 0
1 0 0 1 0 1 0
1 0 1 0 1 1 0
1 1 0 1 0 1 0
1 1 1 1 1 1 1
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Knapsack to TSP
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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
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Step 2 Add start and end points

• 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

• Draw links from smaller to bigger with weights
• w(i,j) j
• 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 4 Create backward links

• 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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KP ? TSP

• All nodes have two incoming links with weights
• w(i,j) if item j is taken into knapsack (xj1)
• 0 if item j not is taken (xj0)
• Visit nodes selected in KP using wgt0 link

0
select
7
0
0
n1
0
5
select
0
select
3
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KP ? TSP

• TSP 0-3-5-7-(N1)-11-2-0

??
KP 3,5,7 (all nodes which arrival cost gt 0)
0
select
7
0
0
n1
0
5
select
0
select
3
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Empty space for notes