?????????????? Traveling Salesman Problem (TSP)

1
??????????? ????????????????????????
? ??

• ????????????????????
• ????????????????????
• ???????????????????
• ??????????????????,???????

2
?????????????? Traveling Salesman Problem (TSP)
3
Definition
otherwise Asymmetric TSP
4
???????????????????

• ?????(??????)???????
• ??????? ??????????
• ??????????????????
• ??????????????????????????

5
TSP??????? naïve ???(????)

• ?????30????????????????????????????????3????!
  (???)
• (n-1)(n-2)321(n-1)!
• 1 PIPS (Peta Instruction Per Second) 1015
• 29!/1015 (?) ? p1014 (?)
• ? 105
  (??)
• James Stiring??? n! ?2vpn(n/e)n
• Tom Duff??? p??1????

6
World Records of Exact Algorithms for Euclidean
Benchmarks (TSPLIB)

• Dantzig, Fulkerson Johnson 49
  cities (1954)
• Held Karp
  65 cities (1971)
• Grotschel
  120 cities (1977)
• Crowder Padberg
  318 cities (1980)
• Padberg Rinaldi
  532 cities (1987)
• Grotschel Holland
  666 cities (1991)
• Padberg Rinaldi
  2392 cities (1991)
• Applegate, Bixby, Chvatal Cook 7392 cities
  (1996)
• Applegate, Bixby, Chvatal Cook 13509 cities
  (1998)
• Applegate, Bixby, Chvatal Cook 15112 cities
  (2001)
• Applegate, Bixby, Chvatal Cook 1904771 cities
  (????)

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DANTZIG42 solved byGeorge Dantzig, Ray Fulkerson
and Selmer Johnson (1954)
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GR120 solved by Groetschel (1977)
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The optimal tour of ATT532 (532 ATT switch
locations in the USA) found by Padberg and
Rinaldi (1987)
10
The optimal tour of GR666 (666 cities in the
world)found by Groetschel and Holland (1987)
11
The optimal tour through a layout of 2,392
(PR2392) found byPadberg and Rinaldi (1991)
12
PLA7397 (a programmed logic array application
containing 7,397 cities)
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The optimal tour of PLA7397 found by David
Applegate, Robert Bixby, Vasek Chvátal and
William Cook (1994)
14
USA13509 (13,509 cities in the United States
having a population of at least 500)
15
The optimal tour of USA13509 found by Applegate,
Bixby, Chvátal, and Cook (1998)
16
D15112 (15,112 cities in Germany)
17
The optimal tour of D15112 found by Applegate,
Bixby, Chvátal, and Cook (2001)
CPU time used is 22.6 years of computer time,
adjusted to a 500 MHz, EV6 Alpha processor
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1,904,711-city instance
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????TSP(1)Knight Tour
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A Knight Tour (by Leonhard Euler1707-1783)
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????TSP(2) Icosian Game (Origin of Hamiltonian
Circuit) Invented by W. R. Hamilton (1805-1865)
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Icosian Game (1)
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Icosian Game (2)
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Icosian Game (3)
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Icosian Game (4)
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Icosian Game (5)
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Applications of TSP
28
Applications
10 P
2
p
clustering a data array
circuit board assembly
computer wiring
3
circuit board drilling
vehicle routing
4
protein conformations
x-ray crystallography
5
VLSI Scan Chain Optimization
6
VLSI fabrication
7
29
??????????????????????

• ??
• ??(CPU??,??????,?????????),??????,????(????,????)
• ???
• ???,????????????,???
• ???
• ???,??????,??????,??????????
• ?????
• ???,???

30
??????????????

• ?????
• ??????????????(Christofides algorithm)
• ?????
• ????????????????(Karps algorithm)
• ?????
• ??????????????

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Theoretical Results(?????)
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Approximate Algorithms

• Construction Algorithms
• Nearest Neighbour
• Greedy
• Christiofides
• Insertion
• Karps
• Bucket
• Improvement Algorithm
• 2,3,..k-opt
• Lin-Kernighan opt

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Nearest Neighbor
?????????????????????????????
?????????????????
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Nearest Neighbor (Worst Case Results)
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Greedy (Multiple Fragment)
???????????2????????????????????
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Greedy (Worst Case Results)
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Christofides (1)
??????
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Christofides (2)
????????????????????????????
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Christofides (Worst Case Results)
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Karps Partitioning Method (1)
?????p?????????????
41
Karps Partitioning Method (2)
?????????????????????????
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Bucket
??????????????????????????????
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Karps Partitioning Method
Probabilisitcs Analysis(?????)
BHH Theorem (Beardwood, Halton and Hammersley,
1959) ??A????????????????? n??????? lim OPT(I)
(ß?0.7124)
ß0.749 BHH (1959)
ß0.765 Stein (1977)
44
?????

• ?????(??)???????????????? (application paper)
• (??????????)?????????????????? (salespitch paper)
• ????????????????????????????????? (experimental
  paper)

45
?????????????

• Application paper(????)
• ?????????????????
• Salespitch paper(?????)
• ????????(??????????????????????????????????????!)
  
• Experimental paper (?????)
• ?????????????(E.g., U(0,1 for bin packing)
  -gt?????
• ???????????????????????????????(E.g., U(a,b for
  bin packing)
• ????????????(?????????)

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Implementaion

• Geometric data structure
• Bucket
• Delaunay Triangulation
• K-d tree
• Tour data structure
• Array
• Two-level Tree
• Segment Tree
• Splay Tree

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????????
Percent Excess over OPT(I)
30
25
15
5
3
2
1
0
n
nlogn
Running Time
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???????????????

• ???????????(?????????????????)?????
• ????????????
• ???????????????????????(????????)
• ????????????????
• ???????????
• Sexy?????????(?????)
• ???(local search)???????????????(??????????)??????
  ??

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????????(???)
??????
???? f(x)
???????? F
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????????????
x
?? N(x)
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2-opt,3-opt neighborhood
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Local Search
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??????(??????)
?????
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Lin-Kernigan opt (3-opt Depth-first Tabu
Search using 2-opt
neighborhood)
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LK Search (Depth-first Tabu Search using
2-opt neighborhood)
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LK Search (Depth-first Tabu Search using
2-opt neighborhood)
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2-opt,3-opt,k-opt (Worst Case Results)
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Simulated Annealing ? (????????)

• ?????(annealing)???????????.

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Simulated Annealing?
60
????????????
????
1
??
??
0
?f(y)-f(x)
?????????
61
Genetic Algorithm(?????????)

• ??(?????????)?????????.
• ????????(??population)?????????.
• ?????????????,??(crossover)???????(mutation)??????
  ?????.

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???????
??(crossover)
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Tabu Search (?????)

• ????????????????.
• ??
• ????????? (steepest ascent mildest descent
  method)
• ???????(adaptive memory programming)
• ?????????????,??????,????????(?????????????????)??
  ??.

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Tabu Search(???)
TL2
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Tabu Search 1
????????
????????????!
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Tabu Search 2
TL2
67
Tabu Search 3
TL2
FIFO
68
Tabu Search 4
69
Tabu Search 5
70
Tabu Search 6
71
Tabu Search 7
72
Tabu Search 8
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Tabu Search 9
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Tabu Search 10
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???????????

• ?????????????
• ????,??,????????????
• ????????????????????
• ??,??,?????????