Title: Neural Network to solve Traveling Salesman Problem
1Neural Network to solve Traveling Salesman Problem
- Amit Goyal 01005009
- Koustubh Vachhani 01005021
- Ankur Jain 01D05007
2Roadmap
- 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
3Background
- 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
4Associative 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
5Hopfield Network
- Recurrent network
- Feedback from output to input
- Fully connected
- Every neuron connected to every other neuron
6Hopfield 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
7Computation
- 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
8Modes 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.
9Stability
- 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
10Procedure
- 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.
11Weight 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.
12Multiple 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
13Energy
- 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.
14Energy 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
15Continued
- ?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.
16Applications of Hopfield Nets
- Hopfield nets are applied for Optimization
problems. - Optimization problems maximize or minimize a
function. - In Hopfield Network the energy gets minimized.
17Traveling Salesman Problem
- Given a set of cities and the distances between
them, determine the shortest closed path passing
through all the cities exactly once.
18Traveling 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
19Hopfield 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
20Hopfield 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
21Determination 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
22Energy 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
23Energy 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
-
24Energy 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
25Energy 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
26Weight 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
27Observation
- 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)
28Observation (cont..)
- Local minima
- Energy function full of dips, valleys and local
minima - Speed
- Fast due to rapid computational capacity of
network
29Concurrent 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.
30Objective 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
31Cont
- 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
32Solution
- 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.
33Minimization 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
34Minimization of energy function
- Differentiating the energy
-
- dUki/dt - dE/ dski - dEi/ dski -
?dEerror/ dski - d?/dt dE/ d? Eerror
- ski tanh(aUki) , a const.
35Implementation
- 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
36Comparison 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.
37Comparison 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.
38Result
- 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.
39Shortest path generated
Concurrent Neural Network (2127 km)
Self Organizing Maps (1311km)
40Behavior in terms of probability
Concurrent Neural Network
Self Organizing Maps
41Conclusion
- 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.
42References
- 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.
43References
- 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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45NP-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.
46NP-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.
47Path lengths
Concurrent Neural Network
Self Organizing Maps