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
Traveling Salesman Problem (TSP)
Definition
Given n points (cities) and a distance function
between two points, find a minimum length Hamiltonian circuit
a
b
|ab|=|ba|
Symmetric TSP
|ab|=|ba|
otherwise
b
a
c
Asymmetric TSP
|ac|≦|ab|+|bc| ΔTSP
points are in d-dim. Euclidean space
Euclidean TSP
Icosian Game (Origin of Hamiltonian Circuit)
Invented by W. R. Hamilton
Icosian Game (1)
２０
４
２
１
３
Icosian Game (2)
２０
５
４
２
１
３
６
７
Icosian Game (3)
２０
９
５
４
２
１
３
６
７
８
Icosian Game (4)
１１
２０
１３
１２
１４
５
４
２
１
１０
３
６
７
８
９
Icosian Game (5)
１９
１８
１７
２０
１６
１１
１３
１２
１４
１５
２
１
１０
５
４
３
６
７
８
９
Knight Tour
Knight Tour (by Leonhard Euler)
Applications of TSP
基盤配線
配送計画
タンパク質構造解析
Applications
clustering a data array
10 P
p= 2
circuit board assembly computer wiring
3
circuit board drilling
4
vehicle routing
protein conformations
x-ray crystallography
5
VLSI Scan Chain Optimization
6
VLSI fabrication
7
World Record of Exact Algorithm
for Euclidean Benchmarks (TSPLIB95)
• Dantzig, Fulkerson & Johnson
49 cities (1954)
• Held & Karp
65 cities (1971)
• 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)
Theoretical Results
Symmetric TSP
A(I)/OPT(I) ≦1.5
(Christofides’ O(n3)
algorithm)
A(I)/OPT(I) ≦ 1+ε
for any ε>0 if d is O(1)
(Arora’s
O(n 12 2 13 log n/ε)
algorithm)
△TSP
d-dimensional
Euclidean TSP
Assuming P ≠NP
no polynomial-time algorithm
can guarantee
A(I)/OPT(I) ≦ 2 p(n)
for any fixed polynomial p
A(I)/OPT(I) ≦ 1+ε
for an ε>0
A(I)/OPT(I) ≦ 1+ε
for an ε>0
if d is Θ(n)
Approximate Algorithms
 Construction
–
–
–
–
–
–
Algorithms
Nearest Neighbor
Greedy
Christiofides’
Insertion
Karp’s
Bucket
 Improvement
Algorithm
– 2,3,..k-opt
– Lin-Kernighan opt
Nearest Neighbor
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
全ての点を訪問したら出発点へもどる
Nearest Neighbor (Worst Case Results)
 
2
Running Time :  n
1
NN(I)
   log 2 n   1
2
OPT(I)
log n 
NN(I)
OPT(I)
∀I
∃I
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
Greedy (Worst Case Results)
2
Running Time :  n log n
Greedy(I) 1
   log 2 n   1
2
OPT(I)
log n
Greedy(I)

3 log log n
OPT(I)
∀I
∃I
Christofides’ （１）
奇数次数を持つ頂点に対して最小マッチングを求める
奇数次数を持つ頂点を赤く塗りつぶす
最小木を作る
Christofides’ （２）
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
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
Christofides’ (Worst Case Results)
3
Running Time :  n 
Christofides(I)
3

2
OPT(I)
3

2
Christofides(I)
OPT(I)
∀I
∃I
Convex Hull +Insertion
2
Running Time :  n 
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
Karp’s Partitioning Method (1)
長方形で、p個の点が入るように分割する
各小領域に対する最適巡回路を求める
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
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) ＝  An (β≒0.7124)
β=0.749
BHH (1959)
too optimistic!
β=0.765
Stein (1977)
Bucket
Running Time :  n
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
決められた順序（小領域内では任意）で点を訪問する
組合せ最適化問題(概念図)
目的関数 f(x)
大域的最適解
実行可能解の集合 F
山頂を目指す闇夜の登山者
ｘ
近傍 N(x)
2-opt,3-opt neighborhood
Local Search
闇夜の登山者(ここが山頂?)
局所最適解
2-opt,3-opt,k-opt (Worst Case Results)
 n2 
Running Time :  2 
 
∃I
k   6  opt

PLS-complete 
 Lin  Kernighan　opt
1
n
4
1
6
1
n 
4
1 21k
n 
4
A 2-opt(I)
OPT(I)
A 3-opt(I)
OPT(I)
A k-opt(I)
OPT(I)





∃I
A 2,3,k-opt(I)
OPT(I)
4 n
∀I
Implementation
 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
Semidynamic K-d tree kdtree
Build K-d tree (1)
K
A
D E
F GH J
cutdim=X
cutval=x(N)
I
B
C
M
P
L
O N
ABCDEPONFGHIJKML
Build K-d tree (2)
K
A
D E
F GH J
I
B
C
cutdim=X
cutval= X(N)
cutdim=Y
cutval=Y(B)
M
P
L
O N
ABDECPON FGHIJKML
Build K-d tree (3) : O(Kn + n log n) time
K
A
D E
F GH J
I
B
C
cutdim=X
cutval= X(N)
cutdim=Y
cutval=Y(B)
cutdim=Y
cutval=Y(F)
M
P
L
O N
ABDECPON FGJK HIML
Delete point Delete(D) (1)
D
φ=0
cutdim=X
cutval= X(N)
K
A
D
E
F GH J
I
B
C
φ=0
φ=0
cutdim=Y
cutval=Y(B)
cutdim=Y
cutval=Y(F)
M
P
L
O N
φ=0
ABDECPON FGJK HIML
φ=0 φ=0 φ=0
LOPT HIPT
Delete point Delete(D) (2)
D
φ=0
cutdim=X
cutval= X(N)
K
A
E
F GH J
I
B
C
M
P
L
O N
φ=0
φ=0
cutdim=Y
cutval=Y(B)
cutdim=Y
cutval=Y(F)
φ=0
ABEDCPON FGJK HIML
LOPT HIPT
φ=0 φ=0 φ=0
Delete (Undelete) point:
O(1) amortized time
φ=0
cutdim=X
cutval= X(N)
φ=0
φ=1
cutdim=Y
cutval=Y(F)
cutdim=Y
cutval=Y(B)
φ=1
φ=1
ABDECPON FGJK HIML
φ=0
HIPT< LOPT
φ=0
Nearest Neighbor Search
Find the nearest point of point nn(E) (1)
cutdim=X
cutval= X(N)
A
D
B
E
cutdim=Y
cutval=Y(B)
Find the nearest
point in the same
bucket!
ABDE
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
C
Check the points
in the bucket, too!
P
O N
ABDECPON
Conjecture (via experimental analysis)
The expected running time is O(1).
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)
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
B
E
r
cutdim=Y
cutval=Y(B)
Check the point
in the same bucket!
ABDE
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
C
Check the points
in the bucket, too!
P
O N
ABDECPON
2-opt (1)
frnn(a,|ab|)
b=next(a)
a
c
|ab|+|cd| > |ac| + |bd|
d=next(c)
2-opt (2)
b=next(a)
a
c
flip(a,b,c,d)
d=next(c)
2-opt procedure two_opt
for TSP using K-d tree
Tour Database Type
void two_opt (point_set *N, int *tour)
Array Splay Tree
{ restart:
(amortized)
for all a ∈unserched_list) {
O(1)
O(log n)
b = next(a);
for all c ∈ frnn(a,|ab|){
O(1)
O(log n)
d =next (c);
if (|ab|+|cd|>|ac|+bd|){
O(n)
O(log n)
flip(a,b,c,d);
add b,c,d to unserched_list;
goto restart;
}
Average runing time: O(n log n)
}
remove point a from unserched_list; } }
Ball Search ballsearch(E, f)
Each point has an associated radius.
Query asks which balls contain a given point.
K
A
D E
F GH J
I
B
C
cutdim=X
cutval= X(N)
cutdim=Y
cutval=Y(B)
cutdim=Y
cutval=Y(F)
M
P
L
O N
ABDECPON FGJK HIML
Balls that
A,B,D,E,F,P intersect the bucket
Insertion Method
using K-d tree(1)
ballsearch(a)
a
b=nn(a)
Insertion Method
using K-d tree(2)
a
Average running time: O(n log n)
Tabu Search 1
に戻らないように
一番高い地点へ移動しよう！
Tabu Search 2
|TL|=2
Tabu Search 3
|TL|=2
FIFO
Tabu Search 4
Tabu Search 5
Tabu Search 6
Tabu Search 7
Tabu Search 8
Tabu Search 9
Tabu Search 10
Lin-Kernighan opt
(=3-opt + Depth-first Tabu Search using
2-opt neighborhood)
c
f
d
e
b
a
LK Search
(Depth-first Tabu Search using 2-opt neighborhood)
c
f
d
e
b
a
d
a
e
b
Tabu
c
f
LK Search
(Depth-first Tabu Search using 2-opt neighborhood)
d
a
a
f
a
f
e
c
b
c
e
b
d
f
e
d
b
c
Experimental Analysis
 Using
random Euclidean instances with
n=100 to 1000000 cities.
 Using TSPLIB (real) instances
n=48 to 85900 cities.
 Using random distance instances.
Quality for tour generation
heuristics
30
Percent Excess over
the Held-Karp Lower
Bound
25
20
Christofides
Clarke-Wright
Greedy
Nearest Neighbor
15
10
5
0
2
2.5
3
3.5
4
4.5
5
5.5
6
P
P
(# of Cities =10 )
Running times for tour
generation heuristics
CPU Time in
Seconds on 150Mhz
SGI Challenge
100000
10000
1000
100
Christofides
Clarke-Wright
Greedy
Nearest Neighbor
10
1
2
0.1
2.5
3
3.5
4
4.5
5
5.5
6
P
0.01
0.001
P
(# of Cities =10 )
Quality for improvement
heuristics
Percent Excess over
the Held-Karp
Lower Bound
5
4.5
4
3.5
2-opt
3-opt
Lin-Kernighan
Iterated LK
SA2
3
2.5
2
1.5
1
0.5
0
2
2.5
3
3.5
4
4.5
5
5.5
6
P
P
(# of Cities =10 )
Running times for improvement
heuristics
100000
CPU Time in
Seconds on 150Mhz
SGI Challenge
10000
1000
2-opt
3-opt
Lin-Kernighan
Iterated LK
SA2
100
10
1
2
0.1
2.5
3
3.5
4
4.5
5
5.5
6
P
0.01
P
(# of Cities =10 )
Undominated Algorithms
Percent Excess
over OPT(I)
30%
Karp’s
Bucket
25%
Nearest Neighbor
Insertion
Greedy
Christofides’
15%
2-opt
5%
3%
2%
1%
0%
3-opt
SA
Lin-Kernighan
Iterated-LK
n
nlogn
Running Time