Tabu Search: A brief introduction

1
Tabu Search A brief introduction

• Partha Sarathi Dasgupta

2
Solving Combinatorial Optimization Problems

• constructive methods
• iterative methods
• specially applicable when no procedure is
• known to directly get an optimal solution

3
A simple and inefficient way

• S solution space
• N(i) neighborhood of solution i
• f(i) cost of solution i
• (1) start with an initial solution i in S
• (2) find a best j in N(i)
• best j f(j) lt f(k) for all k in N(i)
• (3) if f(j) gt f(i) then stop. Else set i j go
  to (2)

Why inefficient?
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Improvements to the simple method

• keep track of the route of most recent search
  steps
• restrict choice of j to a subset of N(i)
• have a dynamic neighborhood structure
• caveat no guarantee of hitting the best solution

5
Improved method - I

• (1) start with an initial solution i in S
• (2) generate a subset V of solution in N(i)
• (3) find a best j in V and set i j
• best j f(j) lt f(k) for all k in N(i)
• (4) if f(j) gt f(i) then stop. Else set i j go
  to (2)

V 1 ?
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Further improvements .. and Problems

• Move generating new solution(s) from an
  existing one
• non-improving moves to be considered (why?)
• guided choice in neighborhood
• Variable V
• risk of accepting non-improving moves
• cycling of solutions
• store last p solutions in memory

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Improved method - II

• i_best index of the best solution
• k iteration count
• N(i,k) neighborhood of solution i at iteration
  k
• V subset of N(i,k)
• (1) start with an initial solution i in S
  i_bestik0
• (2) set k k 1 and generate a subset V of
  N(i,k)
• (3) find a best j in V set i j
• (4) if f(i) lt f(i_best) then set i_best i
• (5) if stopping condition is met then stop. Else
  go to (2)

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Stopping conditions

• N(i,k1) empty
• k gt maximum number of iterations allowed
• number of iterations gt a specified number
• evidence available that optimum solution obtained

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Forbidden solutions - Tabu

• N(i,k1) N(i) - tabu solutions (tabu list)
• Advantages
• partially prevent cycles
• tabu list of last T solutions gt avoid cycles of
  length lt T
• Alternative way maintain tabu of last k moves
• tabu setting
• recency-based
• frequency-based

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Relaxing Tabu status Aspiration criteria

• Drawback of tabu status
• some regions of search space may remain unvisited
• Aspiration criteria
• means of overruling tabu status of some solutions
• useful for exploring potential regions of search
  space
• Example
• overrule tabu status when tabu solutions look
  attractive

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Tabu search

• (1) start with an initial solution i in S
  i_bestik0
• (2) set k k 1 and generate a subset V of
  N(i,k)
• for each solution in V
• either one of the tabu conditions is violated
• or at least an aspiration criteria is met
• (3) find a best j in V set i j
• (4) if f(i) lt f(i_best) then set i_best i
• (5) update tabu conditions / tabu list
• (6) if stopping condition is met then stop. Else
  go to (2)

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Applying Tabu search N-queens problem

• Placing N-queens on a N x N chessboard s.t. no
  two queens are in same row / same column / same
  diagonal

Q
Q
Q
Q
collisions 2
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Applying Tabu search N-queens problem

• N 7
• Modeling as a permutation of column numbers

1
7
Row numbers
4
5
3
6
7
1
2
Column numbers
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Applying Tabu search N-queens problem

• Defining a move

1
7
4
5
3
6
7
1
2
Swapping queens 2 and 6
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Applying Tabu search N-queens problem

• Tabu list

2
3
4
5
6
7
1
Remaining tabu tenure for pair (2,6)
3
2
3
4
5
6
16
Applying Tabu search 7-queens problem
Q
Q
Q
Q
Q
Q
Q
collisions 4
4
5
3
6
7
1
2
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Applying Tabu search N-queens problem

• Iteration 0

Top 5 candidates
2
3
4
5
6
7
swap
value
1
1
7
-2

2
2
4
-2
3
4
5
3
6
7
1
2
2
6
-2
4
5
6
-2
5
1
5
-1
6
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Applying Tabu search N-queens problem

• Iteration 1

Top 5 candidates
2
3
4
5
6
7
swap
value
3
1
2
4
-1

2
1
6
0
3
4
5
3
6
1
7
2
2
5
0
4
1
2
1
5
1
3
1
6
19
Applying Tabu search N-queens problem

• Iteration 5

Top 5 candidates
2
3
4
5
6
7
swap
value
1
1
1
3
-1
T
2
5
6
-1
3
3
6
2
7
4
1
5
5
7
0
T
4
3
1
6
0
2
5
1
7
2
6
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Applying Tabu search use of frequency counts
1
3
6
2
7
5
4
Top 5 candidates
1
7
swap
value
1
3
1
6
0
T
1
3
1
2
5
1
5
1
1

3
4
2
2
7
1
1
2
3
7
1
4
3
1
7
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Applying Tabu search Steiner Problem in graphs

• Given
• Connected undirected graph G(V, E)
• A set X of terminals (X subset of V)
• Non-negative weight function on the edges E

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Applying Tabu search Steiner Problem in graphs

• Neighborhood structure
• S subset of V \ X
• T(S) steiner tree using S MST on S U X
• Finding optimum steiner tree finding a subset
  (S) of steiner
• nodes (V\X) such that the cost of MST is
  minimum
• Neighbors of T(S) all possible MSTs obtained
  by
• inserting new steiner nodes
• deleting existing steiner nodes
• swapping steiner nodes

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Applying Tabu search Steiner Problem in graphs

• Defining Moves
• inserting a new non-terminal node (insertion
  move)
• deleting a non-terminal node from steiner tree
  (elimination move)
• concurrent insertion and deletion of nodes

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Applying Tabu search Steiner Problem in graphs
Procedure Tabu_steiner compute initial solution
/ S set of steiner nodes / Best set of
steiner nodes S S While (iterations_without_imp
rovement lt max_iterations_without_improvement)
s lt- local_search / return best node for move
/ if s is member of S, then S S - s if s
is not a member of S, then S S s insert s
into tabulist L for tabu_tenure iterations
Contd.
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Applying Tabu search Steiner Problem in graphs
Procedure Tabu_steiner (contd.) if (cost(T(S)
lt cost(T(S)) then S lt- S
iterations_without_improvement 0
interations_for_diversification 0 else
iterations_without_improvement
interations_for_diversification endif if
(interations_for_diversification lt
max_interations_for_diversification) then S
S else start with new initial solution
iterations_for_diversification 0
endif endwhile
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Applications of Tabu search

• Traveling salesman
• Vehicle Routing
• Graph partitioning / coloring
• Layout planning
• DNA sequencing

Not an exhaustive list
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References

• Glover, F Tabu search Part I, ORSA journal
  of computing, 1, 1989, 190-206.
• Glover, F Tabu search Part II, ORSA
  journal of computing, 2, 1990, 4-32.