1. NP-困難な組合せ最適化問題に対する近似解法 -巡回セールスマン問題を例として-
Traveling Salesman Problem
Approximate Algorithm
Implementation
Experimental Analysis
Metaheuristics for
Combinatorial Optimization Problems

2. Traveling Salesman Problem (TSP)

3. Definition Given n points (cities) and a distance function
between two points, find a minimum length Hamiltonian circuit
|ab|=|ba| Symmetric TSP a b
|ab|=|ba|
otherwise Asymmetric TSP
ΔTSP a b c
|ac|≦|ab|+|bc|
points are in d-dim. Euclidean space
Euclidean TSP

4. Icosian Game (Origin of Hamiltonian Circuit) Invented by W. R. Hamilton

5. Icosian Game (1) １ ２ ４ ３ ２０

6. Icosian Game (2) １ ２ ７ ６ ５ ４ ３ ２０

7. Icosian Game (3) １ ２ ７ ６ ５ ４ ３ ９ ８ ２０

8. Icosian Game (4) １ ２ ７ ６ ５ ４ ３ ９ ８ １４ １３ １２ １１ １０ ２０

9. Icosian Game (5) １ ２ ７ ６ ５ ４ ３ １９ ９ ８ １８ １７ １６ １５ １４ １３ １２ １１ １０ ２０

10. Knight Tour

11. Knight Tour (by Leonhard Euler)

12. Applications of TSP
基盤配線
配送計画
タンパク質構造解析

13. 10 P Applications clustering a data array circuit board assembly2
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

14. World Record of Exact Algorithm for Euclidean Benchmarks (TSPLIB95)
Dantzig, Fulkerson & Johnson cities (1954)
Held & Karp cities (1971)
Crowder & Padberg cities (1980)
Padberg & Rinaldi cities (1987)
Grotschel & Holland cities (1991)
Padberg & Rinaldi cities (1991)
Applegate, Bixby, Chvatal, & Cook cities (1996)

15. Theoretical Results A(I)/OPT(I) ≦ 2 p(n) Symmetric TSP △TSP
Assuming P ≠NP
no polynomial-time algorithm
can guarantee
A(I)/OPT(I) ≦ 2 p(n)
for any fixed polynomial p
Symmetric TSP
△TSP
A(I)/OPT(I) ≦ 1+ε
for an ε>0
A(I)/OPT(I) ≦1.5
(Christofides’ O(n3)
algorithm)
d-dimensional
Euclidean TSP
A(I)/OPT(I) ≦ 1+ε
for an ε>0
if d is Θ(n)
for any ε>0 if d is O(1)
(Arora’s
O(n log n/ε)
algorithm)

16. Approximate Algorithms
Construction Algorithms
Nearest Neighbor
Greedy
Christiofides’
Insertion
Karp’s
Bucket
Improvement Algorithm
2,3,..k-opt
Lin-Kernighan opt

17. 適当な点から出発し、まだ訪問していない最も近い点へ移動する
Nearest Neighbor
全ての点を訪問したら出発点へもどる
適当な点から出発し、まだ訪問していない最も近い点へ移動する

18. Nearest Neighbor (Worst Case Results)
Running Time :
NN(I)
OPT(I) ∀I
NN(I)
OPT(I) ∃I

19. Greedy （Multiple Fragment)
閉路ができたり、次数が２を超えないように、枝を短い順に加えていく

20. Greedy (Worst Case Results)
Running Time :
Greedy(I)
OPT(I) ∀I ∃I
Greedy(I)
OPT(I)

21. 奇数次数を持つ頂点に対して最小マッチングを求める
Christofides’ 　（１）
奇数次数を持つ頂点を赤く塗りつぶす
奇数次数を持つ頂点に対して最小マッチングを求める
最小木を作る

22. Christofides’ （２） 一度通過した点をスキップすることにより、順回路をえる
まだ通過していない枝をグラフ非連結にならないようにたどる

23. Christofides’ (Worst Case Results)
Running Time :
Christofides(I)
OPT(I) ∀I ∃I
Christofides(I)
OPT(I)

24. Convex Hull +Insertion
Running Time :
凸包で点を囲むように巡回路をつくる
巡回路に入っていない，最も遠い点へ移動する

25. Karp’s Partitioning Method (1)
各小領域に対する最適巡回路を求める
長方形で、p個の点が入るように分割する

26. Karp’s Partitioning Method (2)
長方体と交わる点の枝を非連結にならないようにたどる

27. Karp’s Partitioning Method Probabilisitcs Analysis
BHH Theorem (Beardwood, Halton and Hammersley, 1959)
面積Aの正方領域にランダムにばらまかれた
n個の点に対する
lim OPT(I) ＝　　　　　　　 (β≒0.7124)
β= BHH (1959)
too optimistic!
β= Stein (1977)

28. Bucket Running Time : 決められた順序（小領域内では任意）で点を訪問する
全ての点を含む単位正方形で小領域に分割し、適当な順序をつける

29. 組合せ最適化問題(概念図)
大域的最適解
目的関数 f(x)
実行可能解の集合 F

30. 山頂を目指す闇夜の登山者 ｘ
近傍 N(x)

31. 2-opt,3-opt neighborhood

32. Local Search

33. 闇夜の登山者(ここが山頂?)
局所最適解

34. 2-opt,3-opt,k-opt (Worst Case Results)
Running Time : ∃I
PLS-complete ∃I
A k-opt(I)
OPT(I)
A 2-opt(I)
A 3-opt(I)
A 2,3,k-opt(I)
OPT(I) ∀I

35. Implementation for solving 108 TSP within 1% of optimum
Geometric data structure
Bucket
Delaunay Triangulation
K-d tree
Tour data structure
Array
Two-level Tree
Segment Tree
Splay Tree
for solving
108 TSP
within 1%
of optimum

36. Semidynamic K-d tree kdtree Build K-d tree (1)
A B C D E P O N F G H I J K M L
cutdim=X
cutval=x(N) K A G J F H D E I B M C L P O N

37. Build K-d tree (2) K A J G F H D E I B M C L P O N cutdim=X
cutval= X(N) K A G J F H D E
A B D E C P O N
cutdim=Y
cutval=Y(B) I B M C L P O N
F G H I J K M L

38. Build K-d tree (3) : O(Kn + n log n) time
cutdim=X
cutval= X(N) K A G J F H D E
cutdim=Y
cutval=Y(F)
F G J K H I M L I
cutdim=Y
cutval=Y(B) B M C L P O N
A B D E C P O N

39. Delete point Delete(D) (1)
φ=0
cutdim=X
cutval= X(N) K A G J F D H E
φ=0
φ=0 I
cutdim=Y
cutval=Y(B)
cutdim=Y
cutval=Y(F) B M C L P
φ=0 O N
A B D E C P O N F G J K H I M L
φ=0
φ=0
φ=0
LOPT
HIPT

40. Delete point Delete(D) (2)
φ=0
cutdim=X
cutval= X(N) K A G J F H E
φ=0
φ=0 I
cutdim=Y
cutval=Y(B)
cutdim=Y
cutval=Y(F) B M C L P
φ=0 O N
A B E D C P O N F G J K H I M L
φ=0
φ=0
φ=0
LOPT
HIPT

41. Delete (Undelete) point: O(1) amortized time
φ=0
cutdim=X
cutval= X(N)
φ=1
φ=0
cutdim=Y
cutval=Y(B)
cutdim=Y
cutval=Y(F)
φ=1
φ=1
A B D E C P O N F G J K H I M L
φ=0
φ=0
HIPT<
LOPT

42. Nearest Neighbor Search Find the nearest point of point nn(E) (1)
cutdim=X
cutval= X(N) A D E
cutdim=Y
cutval=Y(B) B
Find the nearest
point in the same
bucket!
A B D E

43. Nearest Neighbor Search Find the nearest point of point nn(E) (2)
cutdim=X
cutval= X(N) A D E
cutdim=Y
cutval=Y(B) B
Check the points
in the bucket, too! C P O N
A B D E C P O N
Conjecture (via experimental analysis)
The expected running time is O(1).

44. Nearest Neighbor Method nntour for TSP using K-d tree
Void nntour(point_set *N, int start_point, Int *tour)
{ tree = new kdtree(N); // O(n log n)
tour[0] = start_point;
tree -> Delete( tour[0] );
for ( i=1; i < n; i++){
tour[i] =tree->nn(tour[i-1]); // O(1) (average)
tree-> Delete(tour[i]); // O(1) (amortized) }
Average runing time: O(n log n)

45. Fixed-Radius Near Neighbor frnn (E, r, f) Find the point(s) f within radius r of point E (1)
cutdim=X
cutval= X(N) A D E r
cutdim=Y
cutval=Y(B) B
Check the point
in the same bucket!
A B D E

46. Fixed-Radius Near Neighbor frnn (E, r, f) Find the point(s) f within radius r of point E (2)
cutdim=X
cutval= X(N) A D E
cutdim=Y
cutval=Y(B) B
Check the points
in the bucket, too! C P O N
A B D E C P O N

47. |ab|+|cd| > |ac| + |bd|
2-opt (1)
frnn(a,|ab|)
b=next(a) a c
d=next(c)
|ab|+|cd| > |ac| + |bd|

48. 2-opt (2)
b=next(a) a c
flip(a,b,c,d)
d=next(c)

49. 2-opt procedure two_opt for TSP using K-d tree
void two_opt (point_set *N, int *tour)
{ restart:
for all a ∈unserched_list) {
b = next(a);
for all c ∈ frnn(a,|ab|){
d =next (c);
if (|ab|+|cd|>|ac|+bd|){
flip(a,b,c,d);
add b,c,d to unserched_list;
goto restart; }
remove point a from unserched_list; } }
Tour Database Type
Array Splay Tree
(amortized)
O(1) O(log n)
O(n) O(log n)
Average runing time: O(n log n)

50. Ball Search ballsearch(E, f) Each point has an associated radius
Ball Search ballsearch(E, f) Each point has an associated radius. Query asks which balls contain a given point.
cutdim=X
cutval= X(N) K A G J F H D E I
cutdim=Y
cutval=Y(B)
cutdim=Y
cutval=Y(F) B M C L P O N
A B D E C P O N F G J K H I M L
Balls that
intersect the bucket
A,B,D,E,F,P

51. Insertion Method using K-d tree(1)
ballsearch(a) a
b=nn(a)

52. Insertion Method using K-d tree(2)a
Average running time: O(n log n)

53. Tabu Search　1
に戻らないように
一番高い地点へ移動しよう！

54. Tabu Search　2
|TL|=2

55. Tabu Search　3
|TL|=2
FIFO

56. Tabu Search　4

57. Tabu Search 5

58. Tabu Search 6

59. Tabu Search 7

60. Tabu Search 8

61. Tabu Search 9

62. Tabu Search 10

63. Lin-Kernighan opt (=3-opt + Depth-first Tabu Search using 2-opt neighborhood)

64. LK Search (Depth-first Tabu Search using 2-opt neighborhood)

65. LK Search (Depth-first Tabu Search using 2-opt neighborhood)

66. Experimental Analysis
Using random Euclidean instances with n=100 to cities.
Using TSPLIB (real) instances n=48 to cities.
Using random distance instances.

67. Quality for tour generation heuristics
Percent Excess over
the Held-Karp Lower Bound P
(# of Cities =10 ) P

68. Running times for tour generation heuristics
CPU Time in Seconds on 150Mhz SGI Challenge P
(# of Cities =10 ) P

69. Quality for improvement heuristics
Percent Excess over
the Held-Karp Lower Bound P
(# of Cities =10 ) P

70. Running times for improvement heuristics
CPU Time in Seconds on 150Mhz SGI Challenge P
(# of Cities =10 ) P

71. Undominated Algorithms
Percent　Excess　
over　OPT(I)
Christofides’
Karp’s SA
Insertion
30%
Bucket
25%
Nearest Neighbor
15%
Greedy
2-opt 5% 3%
3-opt 2%
Lin-Kernighan
Iterated-LK 1% 0% n
nlogn
Running　Time