This slide has been presented in ISOLA 2015 by Norhazwani Md Yunos, Aleksandar Shurbevski and Hiroshi Nagamochi.

1. A Polynomial-Space Exact Algorithm
for TSP in Degree-5 Graphs
Norhazwani Md Yunos, Aleksandar Shurbevski, Hiroshi Nagamochi
Graduate School of Informatics
Kyoto University, Japan
The 12th
International Symposium on Operations Research and Its Applications
in engineering, technology and management (ISORA 2015)
Luoyang, China
21-24 August 2015
Norhazwani et al. A Polynomial-Space Algorithm for TSP5 ISORA 2015 1 / 17

2. 平成28年度 数理工学専攻説明会
第１回： 平成２８年５月７日(土)
第２回： 平成２８年５月３０日(月)
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http://www.amp.i.kyoto-u.ac.jp
研究室見学できます．在学生から，入試勉強のしかた，
過去問の勉強方法などを聞くチャンスです．
京都大学大学院 情報学研究科 数理工学専攻
修士課程，博士課程の学生募集

3. Traveling Salesman Problem
One of the most widely studied problems in combinatorial optimization.
A famous and important NP-hard optimization problem.
Input:
An undirected edge-weighted graph
G = (V, E).
Output:
The minimum cost/length of a tour in
G that passes all vertices of V exactly
once; or
A message for the infeasibility of G.
5 2
3
6
2
1
4 3
2 2
G
Norhazwani et al. A Polynomial-Space Algorithm for TSP5 ISORA 2015 2 / 17

4. Traveling Salesman Problem
One of the most widely studied problems in combinatorial optimization.
A famous and important NP-hard optimization problem.
Input:
An undirected edge-weighted graph
G = (V, E).
Output:
The minimum cost/length of a tour in
G that passes all vertices of V exactly
once; or
A message for the infeasibility of G.
5 2
3
6
2
1
4 3
2 2
G
11
Norhazwani et al. A Polynomial-Space Algorithm for TSP5 ISORA 2015 2 / 17

5. Traveling Salesman Problem
One of the most widely studied problems in combinatorial optimization.
A famous and important NP-hard optimization problem.
Input:
An undirected edge-weighted graph
G = (V, E).
Output:
The minimum cost/length of a tour in
G that passes all vertices of V exactly
once; or
A message for the infeasibility of G.
5 2
3
6
2
1
4 3
2 2
G
2
1
21
1
infeasible
Norhazwani et al. A Polynomial-Space Algorithm for TSP5 ISORA 2015 2 / 17

6. TSP in Degree-k Graphs
Input:
An undirected edge-weighted degree-k graph G = (V, E).
Degree-k graphs = graphs in which vertices have maximum degree at most k.
Output:
The minimum cost of a tour in G that passes all vertices of V exactly
once; or
A message for the infeasibility of G.
Norhazwani et al. A Polynomial-Space Algorithm for TSP5 ISORA 2015 3 / 17

7. Previous Result
Graphs Time Space Method Authors (Year)
General 2n 2n Dynamic
Programming
Bellman (1960)
General 4nnlog n Poly.
Divide and
Conquer
Gurevich & Shelah
(1987)
Degree-3 1.2312n Poly.
Branching
Algorithm
Xiao & Nagamochi
(2013)
Degree-4 1.692n Poly.
Branching
Algorithm
Xiao & Nagamochi
(2015)
Degree-5 2.4531n Poly.
Branching
Algorithm
This presentation
(2015)
Degree-k,
k ≥ 6
open Poly.
Norhazwani et al. A Polynomial-Space Algorithm for TSP5 ISORA 2015 4 / 17

8. Forced TSP
Input:
An undirected edge-weighted graph G = (V, E),
Set of forced edges F ⊆ E.
Output:
The minimum cost of a tour in (G, F) that passes all vertices of V exactly
once, and all forced edges of F; or
A message for the infeasibility of (G, F).
Design a polynomial-space branching algorithm
Reduction procedure.
Branching operations.
Norhazwani et al. A Polynomial-Space Algorithm for TSP5 ISORA 2015 5 / 17

9. A Variety Type of Vertices
Norhazwani et al. A Polynomial-Space Algorithm for TSP5 ISORA 2015 6 / 17

10. Type of Vertices and their Weight, w
Forced
vertices:
f3-vertex f4-vertex f5-vertex
w3’ = 0.1567 ≤ w4’ = 0.3134 ≤ w5’ = 0.4701
Unforced
vertices:
u3-vertex u4-vertex u5-vertex
w3 = 0.2769 ≤ w4 = 0.6075 ≤ w5 = 1
: unforced edges : forced edges
w(v) = 0
otherwise
Norhazwani et al. A Polynomial-Space Algorithm for TSP5 ISORA 2015 7 / 17

11. Measure-and-Conquer
Measure µ for a given instance I = (G, F) of forced TSP:
µ(I) =
v∈V(G)
(w(v))
u3=0.2769
0
f3=0.1567 u4=0.6075
u3=0.2769 u4=0.6075
f5=0.4701
µ(I) = w3 + w3 + w3 + w5 + w4 + w4 + 0
= 2.3956
Norhazwani et al. A Polynomial-Space Algorithm for TSP5 ISORA 2015 8 / 17

12. Reduction Procedure
Infeasibility conditions:
i)
ii)
or
Reduction Rules:
i)
ii)
: unforced edges : forced edges : deleted edges
Norhazwani et al. A Polynomial-Space Algorithm for TSP5 ISORA 2015 9 / 17

13. Branching Operation
G
e
force(e) delete(e)
Instance I with size µ
G
Instance I’
with size µ-a
G
Instance I’’
with size µ-b
e
G
µ(I) = 2.3956
Choose edge e
and branch on
force(e) delete(e)
e
G
µ(I’’) = 1.9620
e
G
µ(I’) = 1.8053
: unforced edges : forced edges : deleted edges
(a, b) is a branching vector of the branching rules.
This implies the linear recurrence: T(µ) ≤ T(µ − a) + T(µ − b)
T(µ) = O(cµ)
Norhazwani et al. A Polynomial-Space Algorithm for TSP5 ISORA 2015 10 / 17

14. Branching Operation
G
e
force(e) delete(e)
Instance I with size µ
G
Instance I’
with size µ-a
G
Instance I’’
with size µ-b
e
G
µ(I) = 2.3956
Choose edge e
and branch on
force(e) delete(e)
e
G
µ(I’’) = 0.6268
e
G
µ(I’) = 0
: unforced edges : forced edges : deleted edges
(a, b) is a branching vector of the branching rules.
This implies the linear recurrence: T(µ) ≤ T(µ − a) + T(µ − b)
T(µ) = O(cµ)
Norhazwani et al. A Polynomial-Space Algorithm for TSP5 ISORA 2015 10 / 17

15. How to Choose an Edge to Branch On
Branching rules applied to an edge e = vt:
v
t
e
While there is a vertex of degree 5,
For the choice of a vertex v of degree-5:
High Priority Less Priority
f5-vertex u5-vertex
v v
Norhazwani et al. A Polynomial-Space Algorithm for TSP5 ISORA 2015 11 / 17

16. How to Choose an Edge to Branch On
For the choice of a vertex t:
High Priority Less Priority
v
t1
t2 t3
t4
e
t5
v
t1
t2 t3
t4
e
f3-vertex
v
t
u3-vertex
v
t
f4-vertex
v
t
u4-vertex
v
t
f5-vertex
v
t
u5-vertex
v
t
There are 14 cases which make our branching rules.
Norhazwani et al. A Polynomial-Space Algorithm for TSP5 ISORA 2015 12 / 17

17. Switching to TSP in Degree-4 Graphs
When the graph has no degree-5 vertices, switch and use the
O∗(1.69193n)-time algorithm for TSP in degree-4 graphs
by Xiao & Nagamochi (2015).
Norhazwani et al. A Polynomial-Space Algorithm for TSP5 ISORA 2015 13 / 17

18. Analysis (Example for c-3)
force(vt1) delete(vt1)
: unforced edges
: forced edges
: newly deleted edges : newly forced edges
v
t1
t2 t3
t4
e
t5 t6
v
t1
t2 t3
t4
e
t5 t6
v
t1
t2 t3
t4
e
t5 t6
Branching vector:
(w5 + w3 − w3 + 3m2, w5 − w4 + w3 + 2m3)
where
m2 = min{w3, (w4 − w3 ), (w4 − w3), (w5 − w4 ), (w5 − w4)}.
m3 = min{w3 , (w3 − w3 ), w4 , (w4 − w4 ), w5 , (w5 − w5 )}.
Norhazwani et al. A Polynomial-Space Algorithm for TSP5 ISORA 2015 14 / 17

19. Analysis
When there exist degree-5 vertices:
Each of the 14 branching vectors has a branching factor ≤ 2.453051.
For switching to TSP in degree-4 graphs:
Measure µ is calculated based on the maximum ratio of vertex weights for
TSP in degree-4 graphs and TSP in degree-5 graphs.
The running bound for TSP in degree-4 graphs is:
T(µ) ≤ O(1.69193z
)
where z = max{0.21968
w3
, 0.45540
w3
, 0.59804
w4
, 1
w4
}
Norhazwani et al. A Polynomial-Space Algorithm for TSP5 ISORA 2015 15 / 17

20. Conclusion and Future Works
Result:
The TSP in an n-vertex graph G with maximum degree 5 can be solved
in O∗(2.4531n)-time and polynomial-space.
Future Work:
It is interesting to obtain a polynomial-space algorithm with a running
time of O∗(2n) or less.
Modified analysis technique.
Re-examination of the branching rules.
Work on TSP in higher degree graphs.
Norhazwani et al. A Polynomial-Space Algorithm for TSP5 ISORA 2015 16 / 17

21. Thank you
Norhazwani et al. A Polynomial-Space Algorithm for TSP5 ISORA 2015 17 / 17