The Shortest Path Tour Problem is an extension to the normal Shortest Path Problem and appeared in the scientific literature in Bertsekas's dynamic programming and optimal control book in 2005, for the first time. This paper gives a description of the problem, two algorithms to solve it. Results to the numeric experimentation are given in terms of graphs. Finally, conclusion and discussions are made.

1. Outline
Introduction
Applications
Problem Description
The state-of-the-art
A modified graph method
A dynamic programming based algorithm
Computational results
Conclusions
Solving The Shortest Path Tour Problem
P.Festa, F.Guerriero, D.Lagana, R.Musmanno
by
Shokirov Nozir
Sabanci University
Shokirov Nozir Solving The Shortest Path Tour Problem

2. Outline
Introduction
Applications
Problem Description
The state-of-the-art
A modified graph method
A dynamic programming based algorithm
Computational results
Conclusions
Outline
1 Introduction
2 Applications
Robot Motions Control
Warehouse Management
Scientific Contribution
3 Problem Description
4 The state-of-the-art
5 A modified graph method
On the reduction of the SPTP to the SPP
Soving the SPTP as SPP
6 A dynamic programming based algorithm
7 Computational results
Test Problems
8 Conclusions
Shokirov Nozir Solving The Shortest Path Tour Problem

3. Outline
Introduction
Applications
Problem Description
The state-of-the-art
A modified graph method
A dynamic programming based algorithm
Computational results
Conclusions
Introduction
The shortest path tour problem (SPTP) is a
variant of the shortest path problem (SPP) and
appeared for the first time in the scientific
literature in Bertsekas’s dynamic programming
and optimal control book in 2005.
Shokirov Nozir Solving The Shortest Path Tour Problem

4. Outline
Introduction
Applications
Problem Description
The state-of-the-art
A modified graph method
A dynamic programming based algorithm
Computational results
Conclusions
Robot Motions Control
Warehouse Management
Scientific Contribution
Robot Motions Control
Shokirov Nozir Solving The Shortest Path Tour Problem

5. Outline
Introduction
Applications
Problem Description
The state-of-the-art
A modified graph method
A dynamic programming based algorithm
Computational results
Conclusions
Robot Motions Control
Warehouse Management
Scientific Contribution
Applications of the SPTP arise for example in the
context of the manufacture workpieces,
where a robot has to perform at least one
operation selected from a set of S types of
operations. In such case, the problem may be
modeled as a SPTP in which operations are
associated with nodes of a directed graph
and the time needed for a tool change is
represented by the distance between two
nodes.
Shokirov Nozir Solving The Shortest Path Tour Problem

6. Outline
Introduction
Applications
Problem Description
The state-of-the-art
A modified graph method
A dynamic programming based algorithm
Computational results
Conclusions
Robot Motions Control
Warehouse Management
Scientific Contribution
Warehouse Management
Shokirov Nozir Solving The Shortest Path Tour Problem

7. Outline
Introduction
Applications
Problem Description
The state-of-the-art
A modified graph method
A dynamic programming based algorithm
Computational results
Conclusions
Robot Motions Control
Warehouse Management
Scientific Contribution
Main Scientific Contribution
1 Analysing some basic theoretical properties of the SPTP
2 Designing a dynamic programming-based algorithm
(DPA) for solving it
3 Efficiently reducing SPTP to a classical SPP through
modified graph algorithm (MGA)
Shokirov Nozir Solving The Shortest Path Tour Problem

8. Outline
Introduction
Applications
Problem Description
The state-of-the-art
A modified graph method
A dynamic programming based algorithm
Computational results
Conclusions
Problem Description
Consider a directed graph G = (N, A) defined by a set of
nodes N := 1, ..., n and a set of arcs A := (i, j) ∈ NxN : i = j,
where |A| = m. A non-negative length cij is assigned to each
arc (i, j) ∈ A. Moreover, lets S denote a certain number of
node subsets T1, ..., TS such that Th ∩ Tk = , h, k = 1, ..., S,
h = k.
Given two nodes i1, iπ ∈ N, i1 = iπ, the path Pi1,iπ from i1 to iπ is
defined as a sequence of nodes Pi1,iπ = i1, ..., iπ such that
(ij , ij+1) ∈ A, j = 1, ..., π − 1.
Length of path Pi1,iπ is l(Pi1,iπ ) = π−1
j=1 cj,j+1
Shokirov Nozir Solving The Shortest Path Tour Problem

9. Outline
Introduction
Applications
Problem Description
The state-of-the-art
A modified graph method
A dynamic programming based algorithm
Computational results
Conclusions
Problem Description
The SPTP aims at finding a shortest path Ps, d
from origin node s ∈ V to destination d ∈ V
in the directed graph G, such that it visits
successively and sequentially the following
subsets Tk, k = 0, ..., S + 1, such that T0 = s
and TS+1 = d. Note that sets Tk, k = 1, ..., S,
must be visited in exactly the same order in
which they are defined.
Shokirov Nozir Solving The Shortest Path Tour Problem

10. Outline
Introduction
Applications
Problem Description
The state-of-the-art
A modified graph method
A dynamic programming based algorithm
Computational results
Conclusions
Instance on a small directed graph G
N = {s = 1, 2, 3, 4, 5, 6, d = 7} , S = 2, T0 = {s = 1} , T1 = {3} , T2 = {2, 4} , T4 =
{d = 7} .
Shokirov Nozir Solving The Shortest Path Tour Problem

11. Outline
Introduction
Applications
Problem Description
The state-of-the-art
A modified graph method
A dynamic programming based algorithm
Computational results
Conclusions
The state-of-the-art
The state of art consits of the expanded graph
method (EGA) propesed by Festa.
The EGA relies on a polynomial-time reduction algorithm that
transforms any SPTP instance defined on a single-stage graph
G into a single-source single-destination SPP instance defined
on a multi-stage graph G’ = (V’, A’) with S+2 stages each
replicating G, and such that V’ = 1, ... , (S+2)n and |A’| =
(S+1)m.
The overal worst coplexity of EGA is O(Smlogn)
Shokirov Nozir Solving The Shortest Path Tour Problem

12. Outline
Introduction
Applications
Problem Description
The state-of-the-art
A modified graph method
A dynamic programming based algorithm
Computational results
Conclusions
SPTP reduction to SPP
Shokirov Nozir Solving The Shortest Path Tour Problem

13. Outline
Introduction
Applications
Problem Description
The state-of-the-art
A modified graph method
A dynamic programming based algorithm
Computational results
Conclusions
On the reduction of the SPTP to the SPP
Soving the SPTP as SPP
On the reduction of the SPTP to the SPP
Given an istance of the SPTP on a directed graph G = (N, A) the following
definition is applied.
Definition
Let G(a)
= (N(a)
, A(a)
, c(a)
) be a weighted directed graph obtained from G in
such a way that;
1 N(a)
=
S+1
k=0
Tk
2 A(a)
=
S
k=0
A
(a)
k where A
(a)
k
:= (i, j) ∈ Tk xTk+1 : i ∈ Tk andj ∈ Tk + 1 ;
3 c(a)
: A(a)
→ Z+
is a function that associates an integer non-negative
number c(a)
to each arc (i, j) ∈ A(a)
, where c
(a)
ij := l(Pij ) is the length of
a shortest path from node i ∈ Tk to node j ∈ Tk+1 on graph G
Shokirov Nozir Solving The Shortest Path Tour Problem

14. Outline
Introduction
Applications
Problem Description
The state-of-the-art
A modified graph method
A dynamic programming based algorithm
Computational results
Conclusions
On the reduction of the SPTP to the SPP
Soving the SPTP as SPP
Modefied Graph Algorithm (MGA)
Theorem
The worst case computational complexity of MGA is O(n3
)
Shokirov Nozir Solving The Shortest Path Tour Problem

15. Outline
Introduction
Applications
Problem Description
The state-of-the-art
A modified graph method
A dynamic programming based algorithm
Computational results
Conclusions
A dynamic programming based algorithm
We view the SPTP as an extension of the weight constrained
shortes path problem. To each path Ps,i from node s to node i
we associate a label yi , which stores information about the
length of the path and the resource consumption along the
path, that is, yi = (l(Ps,i , ri )).
Theorem
The computational complexity of DPA is O(S2
n3
cmax ).
Shokirov Nozir Solving The Shortest Path Tour Problem

16. Outline
Introduction
Applications
Problem Description
The state-of-the-art
A modified graph method
A dynamic programming based algorithm
Computational results
Conclusions
Test Problems
Test Problems
(a) Table1: Grid random networks
Shokirov Nozir Solving The Shortest Path Tour Problem

17. Outline
Introduction
Applications
Problem Description
The state-of-the-art
A modified graph method
A dynamic programming based algorithm
Computational results
Conclusions
Test Problems
Test Problems
(b) Table2: Fully random networks
Shokirov Nozir Solving The Shortest Path Tour Problem

18. Outline
Introduction
Applications
Problem Description
The state-of-the-art
A modified graph method
A dynamic programming based algorithm
Computational results
Conclusions
Test Problems
Results on grid random networks
Shokirov Nozir Solving The Shortest Path Tour Problem

19. Outline
Introduction
Applications
Problem Description
The state-of-the-art
A modified graph method
A dynamic programming based algorithm
Computational results
Conclusions
Test Problems
Results on grid random networks
Shokirov Nozir Solving The Shortest Path Tour Problem

20. Outline
Introduction
Applications
Problem Description
The state-of-the-art
A modified graph method
A dynamic programming based algorithm
Computational results
Conclusions
Test Problems
Results on grid random networks
Shokirov Nozir Solving The Shortest Path Tour Problem

21. Outline
Introduction
Applications
Problem Description
The state-of-the-art
A modified graph method
A dynamic programming based algorithm
Computational results
Conclusions
Test Problems
Results on fully random networks
Shokirov Nozir Solving The Shortest Path Tour Problem

22. Outline
Introduction
Applications
Problem Description
The state-of-the-art
A modified graph method
A dynamic programming based algorithm
Computational results
Conclusions
Test Problems
Results on fully random networks
Shokirov Nozir Solving The Shortest Path Tour Problem

23. Outline
Introduction
Applications
Problem Description
The state-of-the-art
A modified graph method
A dynamic programming based algorithm
Computational results
Conclusions
Test Problems
Results on dense network-based test problems
Shokirov Nozir Solving The Shortest Path Tour Problem

24. Outline
Introduction
Applications
Problem Description
The state-of-the-art
A modified graph method
A dynamic programming based algorithm
Computational results
Conclusions
Test Problems
Results on dense network-based test problems
Shokirov Nozir Solving The Shortest Path Tour Problem

25. Outline
Introduction
Applications
Problem Description
The state-of-the-art
A modified graph method
A dynamic programming based algorithm
Computational results
Conclusions
Conclusions
1.This paper exhibits the results of the study
concerning a variant of SPP, namely the SPPT
2. Two competitve solving algorithms are provided
1 Modefied graph-based algortihm (MGA)
2 Dynamic programming-based algorithm
(DPA)
Shokirov Nozir Solving The Shortest Path Tour Problem

26. Outline
Introduction
Applications
Problem Description
The state-of-the-art
A modified graph method
A dynamic programming based algorithm
Computational results
Conclusions
Conclusions
3. The DPA outperforms MGA on the SPTP instances that are
generated from grid random netwroks, while the latter is
more competitive than the former as long as fully random
and fully dense netwroks are involved.
4.The most efficient proposed algorithm outperforms clearly
the state-of-art solution strategy
5.Perfromance of proposed algorithms depend mainly on the
structure of the networks.
Shokirov Nozir Solving The Shortest Path Tour Problem

27. Outline
Introduction
Applications
Problem Description
The state-of-the-art
A modified graph method
A dynamic programming based algorithm
Computational results
Conclusions
Questions and Comments
Shokirov Nozir Solving The Shortest Path Tour Problem