1. Traveling Salesman Problem Basic Data Structure
巡回セールスマン問題に対する Lin-Kernighan 近傍を用いた Guided Local Search -大規模問題に対する実装法を中心として -
Traveling Salesman Problem
Basic Data Structure
Lin-Kernighan Neighborhood
Guided Local Search
Implementation
Experimental Analysis

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. 10 P Applications clustering a data array circuit board assembly2
clustering a data array p= 3
circuit board assembly
computer wiring 4
vehicle routing
circuit board drilling
protein conformations 5 6
x-ray crystallography
VLSI Scan Chain Optimization 7
VLSI fabrication

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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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).

15. 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

16. 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

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

18. 2-opt (2)
b=next(a) a c
flip(a,b,c,d)
d=next(c)

19. 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)

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

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

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

23. Combinatorial Optimization
Ground Set U c: U→Z
(⊆2 U )
f(x)=Σu∈x c(u)
∀ x ∈2 U
Set of Feasible Solutions F
Objective Function f : F→ Z
Min. f(x) sub. to x ∈ F
Traveling Salesman Problem (TSP)
(Find a minimum length tour of a salesman who wants to visit n customers.)
U = set of edges, F = set of Hamilton circuitｓ （ＨＣs）,
c = length of edge, f= total length of edges in HC.

24. Neighborhood Search (NS)F
Neighborhood N: F →2
Move(x)= an appropriate element in N(x)
NS　（ｘ）
1 while STOP ≠ TRUE do
2 x := Move(x)
Local Search, (Reactive) Tabu Search (TS), Tabu Threshold,
Simulated Annealing (SA), Threshold Accepting, Guided Local Search
Great Deluge (Drought), Record-to-Record Travel , Neural Net, etc.

25. Local Search (LS) LS （ｘ,f）
any x’ ∈N(x) with f(x’)<f(x) if such an x’ exists
“no” otherwise
Improve(x)={
LS　（ｘ,f）
1 while Improve(x) ≠ “no” do
2 x := improve (x)
3 return x

26. Snapshot of Guided Local Search (1)

27. Snapshot of Guided Local Search (2)

28. Snapshot of Guided Local Search (3)

29. Guided Local Search (GLS)
Penalty (≒long term memory) p: U → R
Modified Cost Function g(x)=Σu∈x c(u) + λΣu∈x p(u)
GLS　（ｘ）
1 p(u) := 0 (∀ u ∈U)
2 while STOP ≠ TRUE do
x := LS (x, g)
u* := arg max u∈x c(u)/(p(u)+1)
p(u*) :=p(u*)+1

30. How to memorize “p” (1) Candidate Graph (Digraph)b c d a c b d
c(a,c)
c(a,b)
c(a,d)
p(a,c)
p(a,b)
p(a,d)
(a,b) := arg max c(i,j)/(1+p(i,j))
p(a,b) :=p(a,b)+1
Memorize p(g,c) within the candidate graph data structure
(used when edge (a,b) will be added).

31. How to memorize “p” (1) Candidate Graph (Digraph)b d
c(a,c)
c(a,b)
c(a,d)
p(a,c)
p(a,b)
p(a,d) a b c d a c d b
c(a,c)
c(a,d)
c(a,b)
p(a,c)
p(a,d)
p(a,b)
Sort edges adjacent to vertex “a” so that
c(a,c)+λp(a,c)≦ c(a,d)+λp(a,d) ≦ c(a,b)+λp(a,b)

32. How to memorize “p” (2) Tour Data Structureb e f g c d c b
(g,c) := arg max c(i,j)/(1+p(i,j))
p(g,c) :=p(g,c)+1 d
Memorize p(g,c) within the tour data structure
(used when edge (g,c) will be deleted). f g e

33. Tour Data Structure: Arrayb c d 1 2 a d c b 1 2
tour p
p(a,b)
p(b,1)
p(1,2)
p(2,c)
p(c,d)
Flip(a,b,c,d) a c a c b d 2 1
tour 2 1 p
p(2,c)
p(1,2)
p(b,1) d b

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

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

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