Prof. Busch - LSU1 More NP-complete Problems. Prof. Busch - LSU2 Theorem: If: Language is...
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Transcript of Prof. Busch - LSU1 More NP-complete Problems. Prof. Busch - LSU2 Theorem: If: Language is...
![Page 1: Prof. Busch - LSU1 More NP-complete Problems. Prof. Busch - LSU2 Theorem: If: Language is NP-complete Language is in NP is polynomial time reducible to.](https://reader035.fdocuments.net/reader035/viewer/2022070403/56649f285503460f94c4148e/html5/thumbnails/1.jpg)
Prof. Busch - LSU 1
More NP-complete Problems
![Page 2: Prof. Busch - LSU1 More NP-complete Problems. Prof. Busch - LSU2 Theorem: If: Language is NP-complete Language is in NP is polynomial time reducible to.](https://reader035.fdocuments.net/reader035/viewer/2022070403/56649f285503460f94c4148e/html5/thumbnails/2.jpg)
Prof. Busch - LSU 2
Theorem:
If: Language is NP-complete Language is in NP is polynomial time reducible to
A
A BB
Then: is NP-completeB
(proven in previous class)
![Page 3: Prof. Busch - LSU1 More NP-complete Problems. Prof. Busch - LSU2 Theorem: If: Language is NP-complete Language is in NP is polynomial time reducible to.](https://reader035.fdocuments.net/reader035/viewer/2022070403/56649f285503460f94c4148e/html5/thumbnails/3.jpg)
Prof. Busch - LSU 3
Using the previous theorem,we will prove that 2 problems are NP-complete:
Vertex-Cover
Hamiltonian-Path
![Page 4: Prof. Busch - LSU1 More NP-complete Problems. Prof. Busch - LSU2 Theorem: If: Language is NP-complete Language is in NP is polynomial time reducible to.](https://reader035.fdocuments.net/reader035/viewer/2022070403/56649f285503460f94c4148e/html5/thumbnails/4.jpg)
Prof. Busch - LSU 4
Vertex cover of a graph is a subset of nodes such that every edge in the graph touches one node in
Vertex Cover
SS
S = red nodes
Example:
![Page 5: Prof. Busch - LSU1 More NP-complete Problems. Prof. Busch - LSU2 Theorem: If: Language is NP-complete Language is in NP is polynomial time reducible to.](https://reader035.fdocuments.net/reader035/viewer/2022070403/56649f285503460f94c4148e/html5/thumbnails/5.jpg)
Prof. Busch - LSU 5
|S|=4Example:
Size of vertex-cover is the number of nodes in the cover
![Page 6: Prof. Busch - LSU1 More NP-complete Problems. Prof. Busch - LSU2 Theorem: If: Language is NP-complete Language is in NP is polynomial time reducible to.](https://reader035.fdocuments.net/reader035/viewer/2022070403/56649f285503460f94c4148e/html5/thumbnails/6.jpg)
Prof. Busch - LSU 6
graph contains a vertex cover of size }
VERTEX-COVER = { : kG,G
k
Corresponding language:
Example:G
COVER-VERTEX4, G
![Page 7: Prof. Busch - LSU1 More NP-complete Problems. Prof. Busch - LSU2 Theorem: If: Language is NP-complete Language is in NP is polynomial time reducible to.](https://reader035.fdocuments.net/reader035/viewer/2022070403/56649f285503460f94c4148e/html5/thumbnails/7.jpg)
Prof. Busch - LSU 7
Theorem:
1. VERTEX-COVER is in NP
2. We will reduce in polynomial time 3CNF-SAT to VERTEX-COVER
Can be easily proven
VERTEX-COVER is NP-complete
Proof:
(NP-complete)
![Page 8: Prof. Busch - LSU1 More NP-complete Problems. Prof. Busch - LSU2 Theorem: If: Language is NP-complete Language is in NP is polynomial time reducible to.](https://reader035.fdocuments.net/reader035/viewer/2022070403/56649f285503460f94c4148e/html5/thumbnails/8.jpg)
Prof. Busch - LSU 8
Let be a 3CNF formulawith variables and clauses
m
l
Example:
)()()( 431421321 xxxxxxxxx
4m
3l
Clause 1 Clause 2 Clause 3
![Page 9: Prof. Busch - LSU1 More NP-complete Problems. Prof. Busch - LSU2 Theorem: If: Language is NP-complete Language is in NP is polynomial time reducible to.](https://reader035.fdocuments.net/reader035/viewer/2022070403/56649f285503460f94c4148e/html5/thumbnails/9.jpg)
Prof. Busch - LSU 9
Formula can be converted to a graph such that:
G
is satisfied
if and only if
GContains a vertex coverof size lmk 2
![Page 10: Prof. Busch - LSU1 More NP-complete Problems. Prof. Busch - LSU2 Theorem: If: Language is NP-complete Language is in NP is polynomial time reducible to.](https://reader035.fdocuments.net/reader035/viewer/2022070403/56649f285503460f94c4148e/html5/thumbnails/10.jpg)
Prof. Busch - LSU 10
)()()( 431421321 xxxxxxxxx
1x1x 2x
2x 3x3x 4x
4x
1x
2x 3x
1x
2x 4x
1x
3x 4x
Clause 1 Clause 2 Clause 3
Clause 1 Clause 2 Clause 3
Variable Gadgets
Clause Gadgets
m2 nodes
l3 nodes
![Page 11: Prof. Busch - LSU1 More NP-complete Problems. Prof. Busch - LSU2 Theorem: If: Language is NP-complete Language is in NP is polynomial time reducible to.](https://reader035.fdocuments.net/reader035/viewer/2022070403/56649f285503460f94c4148e/html5/thumbnails/11.jpg)
Prof. Busch - LSU 11
)()()( 431421321 xxxxxxxxx
1x1x 2x
2x 3x3x 4x
4x
1x
2x 3x
1x
2x 4x
1x
3x 4x
Clause 1 Clause 2 Clause 3
Clause 1 Clause 2 Clause 3
![Page 12: Prof. Busch - LSU1 More NP-complete Problems. Prof. Busch - LSU2 Theorem: If: Language is NP-complete Language is in NP is polynomial time reducible to.](https://reader035.fdocuments.net/reader035/viewer/2022070403/56649f285503460f94c4148e/html5/thumbnails/12.jpg)
Prof. Busch - LSU 12
If is satisfied, then contains a vertex cover of size
G
lmk 2
First direction in proof:
![Page 13: Prof. Busch - LSU1 More NP-complete Problems. Prof. Busch - LSU2 Theorem: If: Language is NP-complete Language is in NP is polynomial time reducible to.](https://reader035.fdocuments.net/reader035/viewer/2022070403/56649f285503460f94c4148e/html5/thumbnails/13.jpg)
Prof. Busch - LSU 13
)()()( 431421321 xxxxxxxxx
Satisfying assignment1 0 0 1 4321 xxxx
Example:
We will show that contains a vertex cover of size
103242 lmk
G
![Page 14: Prof. Busch - LSU1 More NP-complete Problems. Prof. Busch - LSU2 Theorem: If: Language is NP-complete Language is in NP is polynomial time reducible to.](https://reader035.fdocuments.net/reader035/viewer/2022070403/56649f285503460f94c4148e/html5/thumbnails/14.jpg)
Prof. Busch - LSU 14
)()()( 431421321 xxxxxxxxx
1x1x 2x
2x 3x3x 4x
4x
1x
2x 3x
1x
2x 4x
1x
3x 4x
1 0 0 1 4321 xxxx
Put every satisfying literal in the cover
![Page 15: Prof. Busch - LSU1 More NP-complete Problems. Prof. Busch - LSU2 Theorem: If: Language is NP-complete Language is in NP is polynomial time reducible to.](https://reader035.fdocuments.net/reader035/viewer/2022070403/56649f285503460f94c4148e/html5/thumbnails/15.jpg)
Prof. Busch - LSU 15
)()()( 431421321 xxxxxxxxx
1x1x 2x
2x 3x3x 4x
4x
1x
2x 3x
1x
2x 4x
1x
3x 4x
1 0 0 1 4321 xxxx
Select one satisfying literal in each clause gadgetand include the remaining literals in the cover
![Page 16: Prof. Busch - LSU1 More NP-complete Problems. Prof. Busch - LSU2 Theorem: If: Language is NP-complete Language is in NP is polynomial time reducible to.](https://reader035.fdocuments.net/reader035/viewer/2022070403/56649f285503460f94c4148e/html5/thumbnails/16.jpg)
Prof. Busch - LSU 16
1x1x 2x
2x 3x3x 4x
4x
1x
2x 3x
1x
2x 4x
1x
3x 4x
This is a vertex cover since every edge isadjacent to a chosen node
![Page 17: Prof. Busch - LSU1 More NP-complete Problems. Prof. Busch - LSU2 Theorem: If: Language is NP-complete Language is in NP is polynomial time reducible to.](https://reader035.fdocuments.net/reader035/viewer/2022070403/56649f285503460f94c4148e/html5/thumbnails/17.jpg)
Prof. Busch - LSU 17
Explanation for general case:
1x1x 2x
2x 3x3x 4x
4x
Edges in variable gadgets are incident to at least one node in cover
![Page 18: Prof. Busch - LSU1 More NP-complete Problems. Prof. Busch - LSU2 Theorem: If: Language is NP-complete Language is in NP is polynomial time reducible to.](https://reader035.fdocuments.net/reader035/viewer/2022070403/56649f285503460f94c4148e/html5/thumbnails/18.jpg)
Prof. Busch - LSU 18
1x
2x 3x
1x
2x 4x
1x
3x 4x
Edges in clause gadgets are incident to at least one node in cover,since two nodes are chosen in a clause gadget
![Page 19: Prof. Busch - LSU1 More NP-complete Problems. Prof. Busch - LSU2 Theorem: If: Language is NP-complete Language is in NP is polynomial time reducible to.](https://reader035.fdocuments.net/reader035/viewer/2022070403/56649f285503460f94c4148e/html5/thumbnails/19.jpg)
Prof. Busch - LSU 19
1x1x 2x
2x 3x3x 4x
4x
1x
2x 3x
1x
2x 4x
1x
3x 4x
Every edge connecting variable gadgetsand clause gadgets is one of three types:
Type 1 Type 2 Type 3
All adjacent to nodes in cover
![Page 20: Prof. Busch - LSU1 More NP-complete Problems. Prof. Busch - LSU2 Theorem: If: Language is NP-complete Language is in NP is polynomial time reducible to.](https://reader035.fdocuments.net/reader035/viewer/2022070403/56649f285503460f94c4148e/html5/thumbnails/20.jpg)
Prof. Busch - LSU 20
If graph contains a vertex-coverof size then formula is satisfiable
G
lmk 2
Second direction of proof:
![Page 21: Prof. Busch - LSU1 More NP-complete Problems. Prof. Busch - LSU2 Theorem: If: Language is NP-complete Language is in NP is polynomial time reducible to.](https://reader035.fdocuments.net/reader035/viewer/2022070403/56649f285503460f94c4148e/html5/thumbnails/21.jpg)
Prof. Busch - LSU 21
Example:
1x1x 2x
2x 3x3x 4x
4x
1x
2x 3x
1x
2x 4x
1x
3x 4x
![Page 22: Prof. Busch - LSU1 More NP-complete Problems. Prof. Busch - LSU2 Theorem: If: Language is NP-complete Language is in NP is polynomial time reducible to.](https://reader035.fdocuments.net/reader035/viewer/2022070403/56649f285503460f94c4148e/html5/thumbnails/22.jpg)
Prof. Busch - LSU 22
1x1x 2x
2x 3x3x 4x
4x
1x
2x 3x
1x
2x 4x
1x
3x 4x
exactly one literal in each variable gadget is chosen
exactly two nodes in each clause gadget is chosen
To include “internal’’ edges to gadgets,and satisfy lmk 2
mchosen out of m2
l2 l3chosen out of
![Page 23: Prof. Busch - LSU1 More NP-complete Problems. Prof. Busch - LSU2 Theorem: If: Language is NP-complete Language is in NP is polynomial time reducible to.](https://reader035.fdocuments.net/reader035/viewer/2022070403/56649f285503460f94c4148e/html5/thumbnails/23.jpg)
Prof. Busch - LSU 23
For the variable assignment choose the literals in the cover from variable gadgets
1x1x 2x
2x 3x3x 4x
4x
1 0 0 1 4321 xxxx
)()()( 431421321 xxxxxxxxx
![Page 24: Prof. Busch - LSU1 More NP-complete Problems. Prof. Busch - LSU2 Theorem: If: Language is NP-complete Language is in NP is polynomial time reducible to.](https://reader035.fdocuments.net/reader035/viewer/2022070403/56649f285503460f94c4148e/html5/thumbnails/24.jpg)
Prof. Busch - LSU 24
1x1x 2x
2x 3x3x 4x
4x
1x
2x 3x
1x
2x 4x
1x
3x 4x
1 0 0 1 4321 xxxx
)()()( 431421321 xxxxxxxxx
since the respective literals satisfy the clauses
is satisfied with
![Page 25: Prof. Busch - LSU1 More NP-complete Problems. Prof. Busch - LSU2 Theorem: If: Language is NP-complete Language is in NP is polynomial time reducible to.](https://reader035.fdocuments.net/reader035/viewer/2022070403/56649f285503460f94c4148e/html5/thumbnails/25.jpg)
Prof. Busch - LSU 25
Theorem:
1. HAMILTONIAN-PATH is in NP
2. We will reduce in polynomial time 3CNF-SAT to HAMILTONIAN-PATH
Can be easily proven
HAMILTONIAN-PATH is NP-complete
Proof:
(NP-complete)
![Page 26: Prof. Busch - LSU1 More NP-complete Problems. Prof. Busch - LSU2 Theorem: If: Language is NP-complete Language is in NP is polynomial time reducible to.](https://reader035.fdocuments.net/reader035/viewer/2022070403/56649f285503460f94c4148e/html5/thumbnails/26.jpg)
Prof. Busch - LSU 26
)()( 2121 xxxx
)( 21 xx
)( 21 xx
Gadget for variable 1x
the directionschange
![Page 27: Prof. Busch - LSU1 More NP-complete Problems. Prof. Busch - LSU2 Theorem: If: Language is NP-complete Language is in NP is polynomial time reducible to.](https://reader035.fdocuments.net/reader035/viewer/2022070403/56649f285503460f94c4148e/html5/thumbnails/27.jpg)
Prof. Busch - LSU 27
)()( 2121 xxxx
)( 21 xx
)( 21 xx
Gadget for variable 2x
![Page 28: Prof. Busch - LSU1 More NP-complete Problems. Prof. Busch - LSU2 Theorem: If: Language is NP-complete Language is in NP is polynomial time reducible to.](https://reader035.fdocuments.net/reader035/viewer/2022070403/56649f285503460f94c4148e/html5/thumbnails/28.jpg)
Prof. Busch - LSU 28
)( 21 xx
)( 21 xx
1x
2x
)()( 2121 xxxx
![Page 29: Prof. Busch - LSU1 More NP-complete Problems. Prof. Busch - LSU2 Theorem: If: Language is NP-complete Language is in NP is polynomial time reducible to.](https://reader035.fdocuments.net/reader035/viewer/2022070403/56649f285503460f94c4148e/html5/thumbnails/29.jpg)
Prof. Busch - LSU 29
)( 21 xx
)( 21 xx
11 x
12 x
1)()( 2121 xxxx
![Page 30: Prof. Busch - LSU1 More NP-complete Problems. Prof. Busch - LSU2 Theorem: If: Language is NP-complete Language is in NP is polynomial time reducible to.](https://reader035.fdocuments.net/reader035/viewer/2022070403/56649f285503460f94c4148e/html5/thumbnails/30.jpg)
Prof. Busch - LSU 30
)( 21 xx
)( 21 xx
01 x
12 x
1)()( 2121 xxxx