Automatski - NP-Complete - TSP - Travelling Salesman Problem Solved in O(N^4)

NP-COMPLETE TRAVELLING
SALESMAN PROBLEM SOLVED IN
O(N4)
Automatski Solutions
http://www.automatski.com
E: Aditya@automatski.com , Founder & CEO
M: (904)-410-4617
© Automatski Solutions 2017. All Rights Reserved.

AUTOMATSKI’S EFFORTS
(LAST 25 YEARS)
1. 25+ years of Fundamental Research
2. Solved 50-100 of the Toughest
Problems on the Planet considered
unsolvable in a 1000 years given the
current state of Human Capability
and Technology
3. Including 7 NP-Complete + 4 NP-
Hard Problems
4. Broke RSA 2048
5. … in 1990’s

AUTOMATSKI HISTORY &
TIMELINE

THE TRAVELLING SALESMAN
PROBLEM
 The TSP - considered one of the most important theoretical problems in theoretical mathematics &
computer science.
 It is one of the most intensely studied problems in computational mathematics and yet no effective solution
method is known for the general case. Indeed, the resolution of the TSP would settle the P versus NP
problem and fetch a $1,000,000 prize from the Clay Mathematics Institute.
 We have solved 7 NP-Complete & 4 NP-Hard problems since 1990's. This is one of them.
 Given N cities, the goal of a traveling salesman is to visit each of them exactly once (and arrive back home)
while keeping the total distance traveled as short as possible. Stated more abstractly, the goal is to find a
path connecting N points in a plane that passes through each point exactly once.
 The importance of the TSP does not arise from an overwhelming demand of salespeople to minimize their
travel distance, but rather from a wealth of other applications such as vehicle routing, circuit board drilling,
VLSI design, robot control, X-ray crystallography, machine scheduling, and computational biology.
 It turns out to be convenient to distinguish between two ways of thinking about the TSP situation.
1. In one perspective, what one has is a decision problem. Given a weighted complete graph with n vertices, locate a
tour which visits each vertex once and only once whose total weight is less than a fixed constant k.
2. However, there is also the optimization version of the problem, the one we have focused on here, where the goal is
finding the tour of the vertices once and only once with total weight the smallest possible.

COMPLEXITY THEORY
1. travelling salesman problem (the prototype problem)
2. location und routing
3. set-packing, partitioning, -covering
4. max-cut
5. linear ordering
6. scheduling (with a few exceptions)
7. node and edge coloring
8. …
 These problems are NP-hard
 Complexity theory came formally into being in the years 1965 – 1972 with the work of
Cobham (1965), Edmonds(1965), Cook (1971), Karp(1972) and many others

THE TECHNIQUES
 Branch and bound technique
 Cutting plane methodology
 Heuristics
 Dynamic Programming

HEURISTICS
 Many real-world instances of hard combinatorial optimization problems are (still) too
large for exact algorithms.
 Or the time limit stipulated by the customer for the solution is too small.
 Therefore, we need heuristics!
 Exact algorithms usually also employ heuristics.

HEURISTICS
 Greedy Algorithms
 Exchange & Insertion Algorithms
 Neighborhood/Local Search
 Variable Neighborhood Search, Iterated Local Search
 Random sampling
 Simulated Annealing
 Taboo search
 Great Deluge Algorithms
 Simulated Tunneling
 Neural Networks
 Scatter Search
 Greedy Randomized Adaptive Search Procedures

HEURISTICS
 Genetic, Evolutionary, and similar Methods
 DNA-Technology
 Ant and Swarm Systems
 (Multi-) Agents
 Population Heuristics
 Memetic Algorithms (Meme are the “missing links” gens and mind)
 Space Filling Curves
 Fuzzy Logic Based…
 Fuzzy Genetics-Based Machine Learning
 Fast and Frugal Method (Psychology)
 Ecologically rational heuristic (Sociology)
 Method of Devine Intuition (Psychologist Thorndike)

DYNAMIC PROGRAMMING
 Dynamic Programming is a method for solving a
complex problem by breaking it down into a
collection of simpler subproblems, solving each of
those subproblems just once, and storing their
solutions using a memory-based data structure
(array, map,etc).
 Dynamic programming is both a mathematical
optimization method and a computer programming
method. The method was developed by Richard
Bellman in the 1950s and has found applications in
numerous fields, from aerospace engineering to
economics.
 2n sub-problems
 n2 matrix
 Order of Best Known Solution O(2nn2)

BO
OKS

Automatski - NP-Complete - TSP - Travelling Salesman Problem Solved in O(N^4)

WHERE ARE WE
GOING WITH ALL
THIS?

ORDER OF
ALGORITHMS
 Problem Complexity (n-1)! / 2
 Best Known Solution so far n22n
 Our Solution n4

ADITYA’S FIRST NP-
COMPLETE PROBLEM
SOLVING CONJECTUREconjecture
/kənˈdʒɛktʃə/
noun
an opinion or conclusion formed on the basis of incomplete
information.
1. All NP Complete Problems can mostly be solved
by Classical Computing in O(N^3)<= Order <=
O(N^5)
2. It is near impossible to do so in O(N) and rarely
requires Order >= O(N^7)
3. Sometimes can be done in O(N^2) and needs
O(N^6)
*** Empirically Based on solving 7 NP-Complete & 4
NP-Hard problems in Polynomial Time

TSP IS
NP-HARD
 TSP is NP-Hard, if you solve it in P,.
 You crack RSA 2048

MACHINES USED
 Development of Solution
 Machine #1
 AMD FX-6300 (6-Core), 32GB RAM
 Eight Year Old Machine
 Physical
 Testing of Parallelization & Concurrency
 Machine #2
 AWS EC2 m4.16xlarge ($3.2/hr)
 64 Core, 256 GB RAM
 Virtual Machine
 Machine #3
 AWS EC2 x1.32xlarge ($13.4/hr)
 128 Core, 1952 GB RAM
 Virtual Machine

BEFORE THE
A Message from our CFO!!!

MESSAGE FROM THE CFO!!!
 We are looking for Research & Development Grants
 Between
 $1m to $100m
 $1m will help us with the Research
 $100m will help us develop Path Breaking Solutions for over 10+ Domains using the
Breakthrough with a Big Bang.

AUTOMATSKI FUNDAMENTAL
RESEARCH
 Fundamental Research at Automatski has been working for the last 20-25+ years on
solving the toughest problems on the Planet.
 We have solved 7 NP-Complete Problems and 4 NP-Hard Problems, including the N-
Queens Completion Millennium Problem.
 We are applying them towards breakthroughs in 50+ Technology Domains
 These problems are considered unsolvable in a 1000 years given the current state of
Human Technology and Capability

SOLVED PROBLEMS
 N-Queens Completion
(Millennium Puzzle) Clay
Math Institute
 3-Sat/k-Sat
 Knapsack***
 Longest Common
Subsequence
 Travelling Salesman
Problem***
 3DM/nDM
 Graph Coloring -
Chromatic Number
 Linear Programming
 Integer Programming
 Mixed Integer
Programming
 Quadratic Programming
 Universal Expression
Programming
 Global Optimum in Hyper
Dimensional Space
 K-Means Clustering
 Universal Clustering
Algorithm
 Universal Constraint
Programming/Scheduling
 Integer Factorization***
 Prime Number Test***
 Universal Regression
 Non-Linear Random
Number Generation
 Automatic Theorem Proving
 Universal Experience
 Universal Heuristics
 Consciousness, Mind, Brain
 Genomics
 Billion & Trillion Actor Nano
Second Framework
 Universal Multi-Scale Simulations
 Internet Scale Rule Engine
 Internet Scale Workflow Engine
 Perfect Finance/Markets
 Perfect Environment
 Compromised All Cryptography
(RSA-2048, Elliptic Curve etc.)
 Post Quantum Cryptography
 Logarithmic Gradient Descent
Convergence
 Blackbox Function
Cracking/Reversal
 Hash Reversal (Incl. SHA-
256/512, LanMan etc.)
 NP-Complete Machine Learning
Algorithms (Clustering,
Regression, Classification)
 NP-Complete Deep Learning
Algorithms (ALL)
 Artificial General Intelligence
 Robotics (Simulations + RAD)***
 Universal Emotions
 Universal IQ
 Universal Creativity

FUND OUR RESEARCH
Together we can build the foundations of a better world

THE DEMO!

WHAT JUST HAPPENED?
 We Deterministically “Solved” The Travelling Salesman Problem
 It seemed like a Heuristic because we hit the Global optima with 12.5% Probability
 Just like the Shor’s Algorithm
 But it is Deterministic and in O(N4) time
 NO Heuristics gets Numbers such as these in < O(n) Steps.
 Lets see the Explanation.
 This problem is extremely well studied in TSP
 Absolute Value of Distances and Relative Values of Distances

THE ALGORITHM O(N4)
 Constant Memory Usage
 Retains only the best solution at each step
 Constant Rate/Time Progress per Step
 Matrix Calculations in Higher Order Dimensions
 General Case
 Symmetric TSP (Asymmetric TSP possible)
 Euclidean Distance
 Guaranteed Convergence
 Deterministic
 No Concept of Best/Worst Case
 Polynomial Order
 Numerical Errors
 Any Random Starting Point
 The Phantom Confusion!!!

THE
PHANTOM
CONFUSI
ON!!!

HENCE…
 RSA 2048 is Cracked!
 Yet Again!
 For the Nth time using N different schemes. Recall…
 P = NP !!!
 7 NP-Complete & 4 NP-Hard Problems Solved in Polynomial time
 N-Queens Completion etc. Including K-Means
 Quantum Computing
 Other Schemes…
 Integer Factorization
 K-SAT
 …
 You know the best part?
 Polynomial Order Solutions are great from a theoretical point of view but are rarely practically
feasible.
 Logarithmic Order Solutions to NP-Complete Problems will define the Future of Mankind.

AVAILABLE
CONFIGURATIONS
 1k locations
 10k locations
 100k locations
 1m locations

FROM HERE…
 In the future video’s you will see
 A Deterministic Algorithm to Find the Global Optima in a Billion Dimensions/Variables.
 I’m sure you will find it very interesting!!!

JET.COM ACQUIRED BY
WALMART FOR $3.3BN
 Walmart acquired ecommerce company Jet.com in 2017 because this start-up’s
employees understood the urban millennial consumer mindset really well. Walmart
needed this skill to take on Amazon better.
 Until recently everyone said it was an overpriced acquisition or acquihire to gain the
services of Jet.com Founder Marc Lore
 1 Year Later, Wal-Mart's Jet.com Acquisition Is an Undeniable Success
*** https://www.fool.com/investing/2017/10/03/1-year-later-wal-marts-jetcom-acquisition-is-
an-un.aspx
 Value
 Associate Delivery (Attempt at solving The Last Mile Problem)
 Store + eCommerce Integration
 Virtual Reality
 Engineering Capability

THE ASTRONOMICAL MATH
BEHIND UPS' NEW TOOL TO
DELIVER PACKAGES
FASTER ORION, or On-Road Integrated Optimization and Navigation (Uses Heuristics… suboptimal)
 Here's a few more numbers that play into the math behind UPS' quest for efficiency
1. $30 million—The cost to UPS per year if each driver drives just one more mile each day than necessary. By that same logic,
the company saves $30 million if each driver finds a way to drive one mile less.
2. 15 trillion trillion—The number of possible routes a driver with just 25 packages to deliver can choose from. As illustrated by
the classic traveling salesman problem, the mathematical phenomenon that makes figuring out the best delivery routes so
difficult is called a combinatorial explosion.
3. 55,000—The number of "package cars" (the brown trucks) in UPS' U.S. fleet. If the figures involved in determining the most
efficient route for one driver are astronomical in scale, imagine how those numbers look for the entire fleet.
4. 85 million—The number of miles Levis says UPS' analytics tools are saving UPS drivers per year.
5. 16 million—The number of deliveries UPS makes daily.
6. 30—The maximum number of inches UPS specifies a driver should have to move to select the next package. This is
accomplished through a meticulous system for loading packages into the truck in the order in which they'll be delivered.
7. 200 million—The number of addresses mapped by UPS drivers on the ground.
8. 74—The number of pages in the manual for UPS drivers detailing the best practices for maximizing delivery efficiency.
9. 100 million—The reduction in the number of minutes UPS trucks spend idling thanks in part, the company says, to onboard
sensors that helped figure out when in the delivery process to turn the truck on and off.
10. 200—The number of data points monitored on each delivery truck to anticipate maintenance issues and determine the most
efficient ways to operate the vehicles.
*** https://www.wired.com/2013/06/ups-astronomical-math/

DO YOU HAVE ANY IDEA?
 What we can do with all the things
Automatski is demo’ing?
 Even though we have just shown
everyone 3-4 out of a 100+
Inventions and Innovations?
 Can you put a $ value on it? How
much will it be? Millions? Billions?
Trillions???
 Any guesses? Wanna try???

THIS MEANS TRILLIONS OF
DOLLARS!!!

NEXT STEPS FOR
PROSPECTIVE CUSTOMERS
 Please contact sales at info@automatski.com
 Send an email with the following information…
 What is the problem(s) you are trying to solve?
 We will take it from there…
*** No Free Trials/Pilots/POCs Offered

WARNING!!!
 Don’t contact us asking for The Source Code
 Don’t contact us asking us to
 File Patents
 Make Public Disclosures of our Algorithm(s)
 Publish Academic Papers

SAMPLE PROBLEMS &
RESULTS
 The .Zip File Contains
1. Sample Problems
2. And their Solutions
 Download .Zip File here
 http://bit.ly/2P6Rb5c
 Download this Presentation here
 http://bit.ly/2TQX89T

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