Neural Network to solve Traveling Salesman Problem

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Neural Network to solve Traveling Salesman Problem

• Amit Goyal 01005009
• Koustubh Vachhani 01005021
• Ankur Jain 01D05007

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Roadmap

• Hopfield Neural Network
• Solving TSP using Hopfield Network
• Modification of Hopfield Neural Network
• Solving TSP using Concurrent Neural Network
• Comparison between Neural Network and SOM for
  solving TSP

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Background

• Neural Networks
• Computing device composed of processing elements
  called neurons
• Processing power comes from interconnection
  between neurons
• Various models are Hopfield, Back propagation,
  Perceptron, Kohonen Net etc

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Associative memory

• Associative memory
• Produces for any input pattern a similar stored
  pattern
• Retrieval by part of data
• Noisy input can be also recognized

Original
Degraded
Reconstruction
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Hopfield Network

• Recurrent network
• Feedback from output to input
• Fully connected
• Every neuron connected to every other neuron

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Hopfield Network

• Symmetric connections
• Connection weights from unit i to unit j and
  from unit j to unit i are identical for all i
  and j
• No self connection, so weight matrix is
  0-diagonal and symmetric
• Logic levels are 1 and -1

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Computation

• For any neuron i, at an instant t input is
• Sj 1 to n, j?i wij sj(t)
• sj(t) is the activation of the jth neuron
• Threshold function ? 0
• Activation si(t1)sgn(Sj1 to n, j?i wijsj(t))
• where

Sgn(x) 1 xgt0
Sgn(x) -1 xlt0
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Modes of operation

• Synchronous
• All neurons are updated simultaneously
• Asynchronous
• Simple Only one unit is randomly selected at
  each step
• General Neurons update themselves independently
  and randomly based on probability distribution
  over time.

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Stability

• Issue of stability arises since there is a
  feedback in Hopfield network
• May lead to fixed point, limit cycle or chaos
• Fixed point unique point attractor
• Limit cycles state space repeats itself in
  periodic cycles
• Chaotic aperiodic strange attractor

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Procedure

• Store and stabilize the vector which has to be
  part of memory.
• Find the value of weight wij, for all i, j such
  that
• lts1, s2, s3 sNgt is stable in Hopfield Network
  of N neurons.

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Weight learning

• Weight learning is given by
• wij 1/(N-1) si sj
• 1/(N-1) is Normalizing factor
• si sj derives from Hebbs rule
• If two connected neurons are ON then weight of
  the connection is such that mutual excitation is
  sustained.
• Similarly, if two neurons inhibit each other then
  the connection should sustain the mutual
  inhibition.

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Multiple Vectors

• If multiple vectors need to be stored in memory
  like
• lts11, s21, s31 sN1gt
• lts12, s22, s32 sN2gt
• .
• lts1p, s2p, s3p sNpgt
• Then the weight are given by
• wij 1/(N-1) Sm1 to psim sjm

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Energy

• Energy is associated with the state of the
  system.
• Some patterns need to be made stable this
  corresponds to minimum energy state of the system.

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Energy function

• Energy at state s lts1, s2, s3 sNgt
• E(s) -½ Si Sj?i wij sisj
• Let the pth neuron change its state from
  spinitial to spfinal so
• Einitial -½ Sj?p wpj spinitial sj T
• Efinal -½ Sj?p wpj spfinal sj T
• ?E Efinal Einitial
• T is independent of sp

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Continued

• ?E - ½ (spfinal - spinitial ) Sj?p wpj sj
• i.e. ?E -½ ?sp Sj?p wpj sj
• Thus ?E -½ ?sp x (netinputp)
• If p changes from 1 to -1 then ?sp is negative
  and netinputp is negative and vice versa.
• So, ?E is always negative. Thus energy always
  decreases when neuron changes state.

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Applications of Hopfield Nets

• Hopfield nets are applied for Optimization
  problems.
• Optimization problems maximize or minimize a
  function.
• In Hopfield Network the energy gets minimized.

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Traveling Salesman Problem

• Given a set of cities and the distances between
  them, determine the shortest closed path passing
  through all the cities exactly once.

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Traveling Salesman Problem

• One of the classic and highly researched problem
  in the field of computer science.
• Decision problem Is there a tour with length
  less than k" is NP - Complete
• Optimization problem What is the shortest
  tour? is NP - Hard

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Hopfield Net for TSP

• N cities are represented by an N X N matrix of
  neurons
• Each row has exactly one 1
• Each column has exactly one 1
• Matrix has exactly N 1s

skj 1 if city k is in position j skj 0
otherwise
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Hopfield Net for TSP

• For each element of the matrix take a neuron and
  fully connect the assembly with symmetric weights
• Finding a suitable energy function E

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Determination of Energy Function

• E function for TSP has four components satisfying
  four constraints
• Each city can have no more than one
• position i.e. each row can have no more
• than one activated neuron
• E1 A/2 Sk Si Sj?i ski skj A -
  Constant

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Energy Function (Contd..)

• Each position contains no more than one city i.e.
  each column contains no more than one activated
  neuron
• E2 B/2 Sj Sk Sr?k skj srj B - constant

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Energy Function (Contd..)

• There are exactly N entries in the output matrix
  i.e. there are N 1s in the output matrix
• E3 C/2 (n - SkSi ski)2 C - constant

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Energy Function (cont..)

• Fourth term incorporates the requirement of the
  shortest path
• E4 D/2 SkSr?kSj dkr skj(sr(j1) sr(j-1))
• where dkr is the distance between city-k and
  city-r
• Etotal E1 E2 E3 E4

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Energy Function (cont..)

• Energy equation is also given by
• E -½SkiSrj w(ki)(rj) ski srj
• ski City k at position i
• srj City r at position j
• Output function ski
• ski ½ ( 1 tanh(uki/u0))
• u0 is a constant
• uki is the net input

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Weight Value

• Comparing above equations with the energy
  equation obtained previously
• W(ki)(rj) -A dkr(1 drj) - Bdij(1 dkr) C
  Ddkr(dj(i1) dj(i-1))
• Kronecker Symbol dkr
• dkr 1 when k r
• dkr 0 when k ? r

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Observation

• Choice of constants A,B,C and D that provide a
  good solution vary between
• Always obtain legitimate loops (D is small
  relative to A, B and C)
• Giving heavier weights to the distances (D is
  large relative to A, B and C)

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Observation (cont..)

• Local minima
• Energy function full of dips, valleys and local
  minima
• Speed
• Fast due to rapid computational capacity of
  network

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Concurrent Neural Network

• Proposed by N. Toomarian in 1988
• It requires N(log(N)) neurons to compute TSP of N
  cities.
• It also has a much higher probability to reach a
  valid tour.

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Objective Function

• Aim is to minimize the distance between city k
  at position i and city r at position i1
• Ei Sk?rSrSi dkidr(i1) dkr
• Where d is the Kronecers Symbol

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Cont

• Ei 1/N2 Sk?rSrSi dkr ?i 1 to ln(N) 1 (2?i
  1) ski 1 (2µi 1) sri
• Where (2µi 1) (2?i 1) 1 ?j 1 to i-1 ?i
  
• Also to ensure that 2 cities dont occupy same
  position
• Eerror Sk?rSr dkr

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Solution

• Eerror will have a value 0 for any valid tour.
• So we have a constrained optimization problem to
  solve.
• E Ei ? Eerror
• ? is the Lagrange multiplier to be calculated
  form the solution.

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Minimization of energy function

• Minimizing Energy function which is in terms of
  ski
• Algorithm is an iterative procedure which is
  usually used for minimization of quadratic
  functions
• The iteration steps are carried out in the
  direction of steepest decent with respect to the
  energy function E

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Minimization of energy function

• Differentiating the energy
• dUki/dt - dE/ dski - dEi/ dski -
  ?dEerror/ dski
• d?/dt dE/ d? Eerror
• ski tanh(aUki) , a const.

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Implementation

• Initial Input Matrix and the value of ? is
  randomly selected and specified
• At each iteration, new value of ski and ? is
  calculated in the direction of steepest descent
  of energy function
• Iterations will stop either when convergence is
  achieved or when the number of iterations exceeds
  a user specified number

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Comparison Hopfield vs Concurrent NN

• Converges faster than Hopfield Network
• Probability to achieve valid tour is higher than
  Hopfield Network
• Hopfield doesnt have systematic way to determine
  the constant terms.

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Comparison SOM and Concurrent NN

• Data set consists of 52 cities in Germany and its
  subset of 15 cities.
• Both algorithms were run for 80 times on 15 city
  data set.
• 52 city dataset could be analyzed only using SOM
  while Concurrent Neural Net failed to analyze
  this dataset.

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Result

• Concurrent neural network always converged and
  never missed any city, where as SOM is capable of
  missing cities.
• Concurrent Neural Network is very erratic in
  behavior , whereas SOM has higher reliability to
  detect every link in smallest path.
• Overall Concurrent Neural Network performed
  poorly as compared to SOM.

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Shortest path generated
Concurrent Neural Network (2127 km)
Self Organizing Maps (1311km)
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Behavior in terms of probability
Concurrent Neural Network
Self Organizing Maps
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Conclusion

• Hopfield Network can also be used for
  optimization problems.
• Concurrent Neural Network performs better than
  Hopfield network and uses less neurons.
• Concurrent and Hopfield Neural Network are less
  efficient than SOM for solving TSP.

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References

• N. K. Bose and P. Liang, Neural Network
  Fundamentals with Graphs, Algorithms and
  Applications, Tata McGraw Hill Publication,
  1996
• P. D. Wasserman, Neural computing theory and
  practice, Van Nostrand Reinhold Co., 1989
• N. Toomarian, A Concurrent Neural Network
  algorithm for the Traveling Salesman Problem,
  ACM Proceedings of the third conference on
  Hypercube concurrent computers and applications,
  pp. 1483-1490, 1988.

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References

• R. Reilly, Neural Network approach to solving
  the Traveling Salesman Problem, Journals of
  Computer Science in Colleges, pp. 41-61,October
  2003
• Wolfram Research inc., Tutorial on Neural
  Networks, http//documents.wolfram.com/applicatio
  ns/neuralnetworks/NeuralNetworkTheory/2.7.0.html,
  2004
• Prof. P. Bhattacharyya, Introduction to
  computing with Neural Nets,http//www.cse.iitb.ac
  .in/pb/Teaching.html.

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NP-complete NP-hard

• When a decision version of a combinatorial
  optimization problem is proved to belong to the
  class of NP-complete problems, which includes
  well-known problems such as satisfiability,traveli
  ng salesman, the bin packing problem, etc., then
  the optimization version is NP-hard.

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NP-complete NP-hard

• Is there a tour with length less than k" is
  NP-complete
• It is easy to determine if a proposed
  certificate has length less than k
• The optimization problem
• "what is the shortest tour?", is NP-hard
  Since there is no easy way to determine if a
  certificate is the shortest.

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Path lengths
Concurrent Neural Network
Self Organizing Maps