a polynomial-space exact algorithm for tsp in degree-5 graphs

A Polynomial-Space Exact Algorithmfor TSP in Degree-5 Graphs

Norhazwani Md Yunos, Aleksandar Shurbevski, Hiroshi Nagamochi

Graduate School of InformaticsKyoto University, Japan

The 12th International Symposium on Operations Research and Its Applicationsin engineering, technology and management (ISORA 2015)

Luoyang, China21-24 August 2015

Norhazwani et al. A Polynomial-Space Algorithm for TSP5 ISORA 2015 1 / 17

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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 graphG = (V,E).

Output:The minimum cost/length of a tour inG that passes all vertices of V exactlyonce; or

A message for the infeasibility of G.

5 2

3

6

2

1

4 3

2 2

G








5 2

3

6

2

1

4 3

2 2

G

11








5 2

3

6

2

1

4 3

2 2

G

2

1

21

1

infeasible


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 exactlyonce; or



Previous Result

Graphs Time Space Method Authors (Year)

General 2n 2n DynamicProgramming

Bellman (1960)

General 4nnlog n Poly.Divide andConquer

Gurevich & Shelah(1987)

Degree-3 1.2312n Poly.BranchingAlgorithm

Xiao & Nagamochi(2013)


Xiao & Nagamochi(2015)


This presentation(2015)

Degree-k,k ≥ 6

open Poly.


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 exactlyonce, and all forced edges of F; or

A message for the infeasibility of (G,F).

Design a polynomial-space branching algorithmReduction procedure.

Branching operations.


A Variety Type of Vertices


Type of Vertices and their Weight, w

Forced

vertices:

f3-vertex f4-vertex f5-vertexw3’ = 0.1567 ≤ w4’ = 0.3134 ≤ w5’ = 0.4701

Unforced

vertices:

u3-vertex u4-vertex u5-vertexw3 = 0.2769 ≤ w4 = 0.6075 ≤ w5 = 1

: unforced edges : forced edges

w(v) = 0

otherwise


Measure-and-Conquer

Measure µ for a given instance I = (G,F) of forced TSP:

µ(I) =∑

v∈V(G)

(w(v))

u3=0.27690

f3=0.1567u4=0.6075

u3=0.2769 u4=0.6075

f5=0.4701

µ(I) = w3 + w3 + w3′ + w5′ + w4 + w4 + 0

= 2.3956


Reduction Procedure

Infeasibility conditions:

i)

ii)

or

Reduction Rules:

i)

ii)

: unforced edges : forced edges : deleted edges


Branching Operation

Ge

force(e) delete(e)

Instance I with size µ

G

Instance I’

with size µ-a

G

Instance I’’

with size µ-b

eG

µ(I) = 2.3956

Choose edge e

and branch on

force(e) delete(e)

eG

µ(I’’) = 1.9620

eG

µ(I’) = 1.8053


(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

Branching Operation

Ge

force(e) delete(e)

Instance I with size µ

G

Instance I’

with size µ-a

G

Instance I’’

with size µ-b

eG

µ(I) = 2.3956

Choose edge e

and branch on

force(e) delete(e)

eG

µ(I’’) = 0.6268

eG

µ(I’) = 0


(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

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


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

t1t2 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.


Switching to TSP in Degree-4 Graphs

When the graph has no degree-5 vertices, switch and use theO∗(1.69193n)-time algorithm for TSP in degree-4 graphsby Xiao & Nagamochi (2015).


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)

wherem2 =min{w3, (w4′ − w3′), (w4 − w3), (w5′ − w4′), (w5 − w4)}.m3 =min{w3′ , (w3 − w3′),w4′ , (w4 − w4′),w5′ , (w5 − w5′)}.


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 forTSP 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{0.21968w3′

, 0.45540w3

, 0.59804w4′

, 1w4}


Conclusion and Future Works

Result:

The TSP in an n-vertex graph G with maximum degree 5 can be solvedin O∗(2.4531n)-time and polynomial-space.

Future Work:It is interesting to obtain a polynomial-space algorithm with a runningtime of O∗(2n) or less.

Modified analysis technique.Re-examination of the branching rules.

Work on TSP in higher degree graphs.


Thank you


a polynomial-space exact algorithm for tsp in degree-5 graphs

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