This document proposes a new paradigm called Interaction Theory for solving the Traveling Salesman Problem (TSP) in a more efficient manner. It discusses how traditional TSP solving methods have exponential time complexity as the number of cities increases. Interaction Theory uses a relative interaction coefficient approach rather than absolute values to prioritize city connections. This allows it to solve TSP problems deterministically with limited resources in short operating time. The document provides several examples of TSP problems solved using Interaction Theory and argues it establishes that P=NP, meaning TSP and other NP-complete problems can be solved in polynomial time.

1. INTERACTION THEORY
NEW PARADIGM FOR SOLVING THE
TRAVELING SALESMAN PROBLEM
(TSP)
Anang Z.Gani
Department of Industrial Engineering
Institut Teknologi Bandung
Bandung, Indonesia 2012
ganiaz@gmail.com
ganiaz@mail.ti.itb.ac.id

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3. 3

4. 4
 INTRODUCTION
 OBJECTIVE
 BACKGROUND
 INTERACTION THEORY
 COMPUTATIONAL
EXPERIENCES AND
EXAMPLE
 CONCLUSION
AZG
2012
(Keywords: Graph; P vs NP; Combinatorial Optimization;
Traveling Salesman Problem; Complexity Theory; Interaction
Theory; Linear Programming; Integer Programming ;
Network).

5. 5
The area of Applications :
Robot control
Road Trips
Mapping Genomes
Customized Computer Chip
Constructing Universal DNA Linkers
Aiming Telescopes, X-rays and lasers
Guiding Industrial Machines
Organizing Data
X-ray crytallography
Tests for Microprocessors Scheduling Jobs
Planning hiking path in a nature park
Gathering geophysical seismic data
Vehicle routing
Crystallography
Drilling of printed circuit boards
Chronological sequencing
AZG
2012

6. The problem of TSP is to find the shortest
possible route to visit N cities exactly once and
returns to the origin city.
The TSP very simple and easily stated but it is
very difficult to solve.
The TSP - combinatorial problem
the alternative routes exponentially increases
by the number of cities.
1/2 (N-1)!
4 cities = 3 possible routes
4 times to 16 cities = to 653,837,184,000.
10 times to 40 cities =1,009 x1046
IF 100,000 CITIES...... (possible routes?)
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7. SOAL 33 KOTA
ALTERNATIVE RUTE 32!/2 =
131.565.418.466.846.756.083.609.606.080.000.000
KOMPUTER PALING TOP $ 133.000.000 ROADRUNNER
CLUSTER DARI UNITED STATES DEPARTMENT OF
ENERGY DIMANA 129.6600 CORE MACHINE TOPPED
THE 2009 RANKING OF THE 500 WORLD’S FASTES
SUPER COMPUTERS, DELIVERING UP TO 1.547
TRILION ARITHMETIC OPERATIONS PER SECOND.
DIPERLUKAN WAKTUN 28 TRILIUN TAHUN
SEDANGKAN UMUR UNIVERS HANYA 14
MILIAR TAHUN
INI MEMANG GILA
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8. 7 (tujuh)
problem
matematika
pada
millenium ini
1. The Birch and Swinnerton-
Dyer Conjecture
2. The Poincare Conjecture
3. Navier-Stokes Equations
4. P versus NP Problem
5. Riemann Hypothesis
6. The Hodge Conjecture
7. Yang-Mills Theory and The
Mass Gap Hypothesis.
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9. 9
. "The P versus NP
Problem" is considered one
of the seven greatest
unsolved mathematical
problems
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10. 10
One important statement about the NP-complete
problem (Papadimitriou & Steiglitz) :
a. No NP-complete problem can be solved by
any known polynomial algorithm (and this
is the resistance despite efforts by many
brilliant researchers for many decades).
b. If there is a polynomial algorithm for any
NP-complete problem, then there are
polynomial algorithms for all NP-complete
problems.
THIS IS CHALLENGE TO PROVE
P= NP MUST BE PURSUED!
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11. TSP dealing with the resources :
1. Time (how many iteration it takes to
solve a problem)
2. space (how much memory it takes to
solve a problem).
THE MAIN PROBLEM :
1. THE NUMBER OF STEPS (TIME) INCREASES
EXPONENTIALLY ALONG WITH THE INCREASE IN
THE SIZE OF THE PROBLEM.
2. HUGE AMOUNT COMPUTER RESOURCES ARE
REQUIRED
NEW PARADIGM
(BREAKTHROUGH)
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12. PARADIGM
OLD NEW
1. LP & DERIVATIVES
2. HEURISTIC (PROBABILISTIC)
3. PROCEDURE IS
COMPLICATED
4. NEEDS RESOURCES OF TIME
AND MEMORY UNLIMITED
5. CHECKING ALL ELEMENTS
6. P = NP VS P ≠ NP ?
7. KNOWLEDGE IS HIGH
8. LONG OPERATING TIME
1. INTERACTION THEORY
2. DETERMINISTIC
3. PROCEDURE IS SO SIMPLE
4. RESOURCES NEED IS
LIMITED
5. CHECKING LIMITED
ELEMENTS (PRIORITY)
6. P=NP
7. SIMPLE ARITHMATIC
8. SHORT OPERATING TIME
(EFFICIENT AND EFFECTIVE)
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13. SUMMARIZES THE MILESTONES OF SOLVING
TRAVELING SALESMAN PROBLEM.
Year Research Team Size of Instance
1954 G. Dantzig, R. Fulkerson, and S.
Johnson
49 cities
1971 M. Held and R.M. Karp 64 cities
1975 P.M. Camerini, L. Fratta, and F.
Maffioli
67 cities
1977 M. Grötschel 120 cities
1980 H. Crowder and M.W. Padberg 318 cities
1987 M. Padberg and G. Rinaldi 532 cities
(109,5 secon)
1987 M. Grötschel and O. Holland 666 cities
1987 M. Padberg and G. Rinaldi 2,392 cities
1994 D. Applegate, R. Bixby, V.
Chvátal, and W. Cook
7,397 cities
1998 D. Applegate, R. Bixby, V.
Chvátal, and W. Cook
13,509 cities
(4 Years)
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14. SUMMARIZES THE MILESTONES OF SOLVING
TRAVELING SALESMAN PROBLEM.
Year Research Team Size of Instance
2001 D. Applegate, R. Bixby, V. Chvátal,
and W. Cook
15,112 cities
(ca. 22 Years)
2004 D. Applegate, R. Bixby, V. Chvátal,
W. Cook and K. Helsgaun
24,978 cities
2006 D. Applegate, R. Bixby, V. Chvátal,
and W. Cook
85,900 cities
2009 D. Applegate, R. Bixby, V. Chvátal,
and W. Cook
1,904,711 cities
2009 Yuichi Nagata 100.000
Mona Lisa
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15. 15
TECHNIQUE AND METHOD
FOR SOLVING TSP
HEURISTIC EXACT SOLUTION
• NEURAL NETWORK
• GENETIC ALGORITHM
• SIMULATED ANNEALING
• ARTIFICIAL INTELLEGENT
• EXPERT SYSTEM
• FRACTAL
• TABU SEARCH
• NEAREST NEIGBOR
• THRESHOLD ALGORITHM
• ANT COLONY OPTIMIZATION
• LINEAR PROGRAMMING
INTEGER PROGRAMMING
• CUTTING PLANE
• DYNAMIC PROGRAMMING
• THE MINIMUM SPANNING
TREE
• LAGRANGE RELAXATION
• ELLIPSOID ALGORITHM
• PROJECTIVE SCALING
ALGORITHM
• BRANCH AND BOUND
• ASAINMENT
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OBJECTIVE FUNCTION
n

n

z  x(i, j)d(i, j)
j 1
i1
• d(i,j) = (direct) distance between
city i and city j.
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17. Constraints
n
 x(i, j)
1 , i 1,2, ...,n
j 1
n
 x(i, j)
1 , j 1,2, ...,n
i 1
• Each city must be “exited” exactly once
• Each city must be “entered” exactly once
Subtour elimination constraint
 x(i, j)  S
 1,  S  {1, 2, ...,n}
i , jS
• S = subset of cities
• |S| = cardinality of S (# of elements in S)
• There are 2n such sets !!!!!!!
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NUMBER OF LINIER INEQUALITIES
AS CONSTRAINS IN TSP
• If n=15 the number of countraints is
1.993.711.339.620
• If n=50 the number of countraints 1060
• If n=120 the number of countraints 2 x 10179
or to be exact :
267925490760634893755546189948219873995788690377687
078048465194329577247030862734015632117088075939986
913459296483643418942533445648036828825541887362427
99920969079258554704177287
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Grotschel

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 INTERACTION THEORY

20. INTERACTION THEORY
In 1965 Anang Z. Gani [28] did research on the Facilities Planning
problem as a special project (Georgia Tach in 1965)
Supervision James Apple
Later, J. M. Devis and K. M. Klein further continued the original
work of Anang Z. Gani
Then M. P. Deisenroth “ PLANET” direction of James Apple
(Georgia Tech in1971)
Since 1966, Anang Z. Gani has been continuing his research and
further developed a new concept which is called “The Interaction
Theory” (INSTITUT TEKNOLOGI BANDUNG)
The model is the From - To chart the which provides quantitative
information of the movement between departments
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21. The model is the From - To chart the which
provides quantitative information of the
movement between departments (common
mileage chart on the road map).
The absolute value or the number of a
element as an individual of a matrix can not
be used in priority setting
the TSP matrix has two values,
1. the initial absolute value (interaction
value)
2. the relative value (interaction coefficient)
DIM = The Delta Interaction Matrix
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22. AZG
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Two parallel lines

23. Two parallel lines distorted
(Hering illusion)
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24. 1 2 3 4
1 0 700 10 20
2 2 0 800 15
3 4 3 0 10
4 10 2 30 0
RELATIVE VALUE
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25. The formula for the interaction
coefficient ( c) is:
i,j c= x2/(X.X).
i,j i,j
i. .jm
Xi. = 
j 1

xij (i = 1 ……. m )
n
X.j = 
i 1
xij (j = 1 ……. n )
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26. TSP
INTERACTION
THEORY
TSP P=NP
GENERAL
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27. APPLICATION OF THEORY INTERACTION
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2012
• Traveling Salesman Problem (Symmetric and
Asymmetric, minimum and maximum).
• Transportation Problem.
• Logistic.
• Assignment problem.
• Network problem
• Set Covering Problem.
• Minimum Spanning Tree (MST)
• Decision Making.
• Layout Problem.
• Location Problem
• Financial Analysis.
• Clustering.
• Data Mining

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Transportation
Problems
TSP
(Symmetric & Asymmetric
Graph
Network Problems
Scheduling
Decision
Making Clustering
Layout
Problems
Location
Problems
Financial
Analysis
Assignment
Problems
Routing Data Mining
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Computer Science
Transportasi
Militer
Ekonomi
Strategi
Finansial
Distribusi / Logistik
Psikologi
Kimia
Fisika
Biologi
Operations Research
Telekomunikasi
Industri Sosial
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30. 90
80
70
60
50
40
30
20
10
0
49
45
17
86
38
47
16
8
48
84
61
64
19
82
85
83
5
91
60
93
98
100
11
7
18
99
37
96
59
92
62
6
89
88 31
52
36
46
63
90
32
10
70
30
20
66
71
65
35
34
81 78
9
51
33
79
3
76 77
50
1
69
27
101
53
28
26
12
68
80
29
24
54
55
4 25
39
23 67
56
75
41
21
73 72
74
22
40
58
94 13
95
97
87
2
57
15
43
42
14
44
0 10 20 30 40 50 60 70 80
Route for 101 cities ( 8 Optimal solutions)
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31. Portrait of Mona Lisa with Solution of a Traveling
Salesman Problem. Courtesy of Robert Bosch ©2012
( 7 Optimal solutions)
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32. AZG CONCLUSION
2012
• The conclusion is that the
Interaction Theory creates a new
paradigm to the new efficient and
effective algorithm for solving the
TSP easily (N=NP).
• Overall, the Interaction Theory
shows a new concept which has
potential for development in
mathematics, computer science
and Operations Research and their
applications
32

33. 33
THANK
YOU
AZG
2012
SIMPLICITY IS
POWER