Download Rishi B. Jethwa and Mayank Agarwal

Analysis of the Traveling
Salesman Problem and
current approaches for
solving it.
Rishi B. Jethwa and Mayank Agarwal.
CSE Department.
University of Texas at Arlington
.
Table of Contents.
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Introduction.
Naïve and simple approaches.
Approaches using Dynamic Programming.
Approaches using Neural Network and
Genetics.
Approaches using Parallel Branch and Bound
Technique.
Traveling Salesman Problem
Given N points, find the shortest circular path that
links all the points together. The constraint on
TSP is that the first and the last point the tour
should be same and the edges cannot be
repeated. Vertices can be repeated.
No polynomial solution exists for this type of
problems which has alas combinatory solutions.
So we are interested in mainly good solutions, not
exact.
A naïve approach..
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Suppose you are given with n points and you
start with first point for second possibility you
are left with n-1 options then n-2 and so on.
Hence there are altogether !(n-1)/2 different
paths i.e. 0.5 * !(n-1) in APL notations.
For n = 36, there are 10*38 different paths.
Other simple approaches.
Nearest Neighbour.
Start at an arbitrary point and successively visit the
nearest unvisited point. After all the points have
been visited, return to the start point.
Other simple approaches.
Minimum Spanning Tree.
Construct the minimum spanning tree of the
point set and duplicate all the links on the tree.
Sequence the points as the would appear in a
traversal of the doubled tree. Pass through the
sequence and remove all representations after
the first of each point.
Other simple approaches.
Strip.
Partition the square into sqrroot N/3 vertical
strips. Sequence the points in the each strip
by the vertical position , alternately top-tobottom and bottom-to-top, and visit the
strips from left to right. At last return to the
starting point.
Spacefilling Approach
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The approach goes like this. Have a square that
has all the point inside. bisect it into two
triangle, name it 0 and 1. Then again bisect each
into two triangle and name them for 0, 00 and
01 and for 1, 10 and 11. Continue this partition
and name each partition accordingly. At last
traverse the small triangle for points in them.
Comparisons
NN
MST
STRIP
Spacefilling
Ease of
coding
Good
Poor
Good
Good
Memory
O(N)
O(N)
O(N)
O(N)
poor
Good
Good
Parallelizabilit Poor
y
To solve
O(N2 ) O(N2 LogN ) O(N Log N) O(N Log N
To modify
Resolve Resolve
O(Log N)
O(Log N)
Good
Poor
Good
Performance on
non-uniform
data
Good
Diamond Method to solve TSP
problems.
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Consider a big-diamond APL symbol, divide it
into 4 parts.
Approximately 1/4 points are located at each of
the four quadrants. Apply the TSP to each of
the four quadrants. A catenation of these four
sub paths produces a path with no loop, which is
what we wanted.
Dynamic Programming treatment.
Apart from the vertices it also defines d(i,j) be the
distance between ith and jth vertices.
 Then the final answer would be:
f(i;j1, j2, ....,jk) = min as long as 1 <= m <= k {d(i,jm)
+ f(i;j1, j2, ..,jm-1,jm+1,..,jk}
 first we get f(i; j1, j2) then f(i; j1, j2, j3) so on until we
get f(i; j1, j2.....jn).
 The running time is reduced to n2 2n-1 and the space
requirement is 6 times that of Striling’s formula i.e.
22m / sqrt(3.14 *m).
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Dynamic Programming treatment.
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The GTSP is linked to the fact that it is a combination
of two problems.
Once a vertex is chosen, we must choose which cluster
to visit first and then which vertex to visit first. Once
the cluster path is determined, we are confronted to the
minimum cycle path problem.
This paper proposes the genetic algorithm for
choosing a path. Here sequence of chromosomes
represents a path.
The Self-Organizing Neural
Network (SONN)
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The SONN is arranged with M neurons and if
each neuron has N-1 dimensional weight vector.
SONN will exhibit non-convergence in case
when M=N. This paper proposed a new density
function that guarantees the convergence even
when M=N.
Also for a problem with N cities original SONN
requires 2N neurons but this approach requires
only N neurons.
Parallel Branch and Bound
Technique.
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Uses parallel computers to solve large randomly generated
ATSP.
The principle components of the algorithm are as follows:
i) Lower Bounding Technique:- To find the lower bounds for
the parallel ATSP algorithm by solving the assignment problem.
ii) Upper Bounding Heuristic:- Use the solution to the
assignment problem to construct a solution to the ATSP.
iii) Branching rules:- Create two or more new sub-problems
based on an assignment problem solution.
This algorithm is capable of using tens to hundreds of
processors depending on the problem size and difficulty.
Parallel Branch and Bound
Technique.
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Edward discusses the issues surrounding
implementation of a particular branch and
bound algorithm for the TSP on a hypercube
multi-computer.
This paper uses the Best-First Search technique
for branching implementation.
TSP is one of the very few fully asynchronous
applications that have been written on
hypercube thus far.
THANKS.