1 combinatorial optimization : an instance s : solutions set f : s → cost function to minimize...
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1
COMBINATORIAL OPTIMIZATION
< s , f > : an instance
s : Solutions Set
f : s → Cost function to minimize (Max)
Find s* S s.t.
f ( s* ) f ( s ) , s S ( MIN)
or
f ( s* ) f ( s ) , s S ( MAX)
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Local Search ( LS)
Neighborhood structure : - i solution- N : S →
i→ N ( i ) S N ( i ) = ¨ near ¨ to i solutions
ĩ is a local minimum iff ( i ) f ( j ) , j N ( i )
ĩ is a local maximum iff ( i ) f ( j ) , j N ( i )
S
2
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Local Search Algorithm
Define a neighborhood N ( ) Initial solution = Find a solution ΄ N ( )
improving the cost : = ´ If ´ does not exist STOP ( local optimum)
0sss
s
ss s
s
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The local search algorithm
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Examples
The Traveling Salesman Problem
2 - opt , 3 – opt , . . . , k – opt
2 - exchange
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Examples
The Bipartitioning of a weighted
graph G ( V , E , W ) , = 2 n.
Find partitions A , B of V with
= and
Minimizing f ( A , B ) =
V
A B
AB
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Graph Bipartitioning
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Search strategies in LS
First improvement
Best improvement
Worst improvement
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The Quadratic Assignment Problem (QAP)
n locations : distance n facilities: flow fij
π(i)=k: facility i location k
minimize the local cost
Min f dik i ki k
n
( ) ( ), 1
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The Quadratic Assignment Problem (QAP)
n locations : distance n facilities: flow fij
π(i)=k: facility i location k
minimize the total cost
Min f dik i ki k
n
( ) ( ), 1
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The QAP 2-exchange:
π=(π(1),...,π(i),...,π(j),...π(n))πij=(π(1),...,π(j),...,π(i),...π(n))
N(π)=(n*(n-1))/2
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The QAP: an example
π=3 1 2 5 4 6
2-exch.
πij=3 4 2 5 1 6
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5
1
3
4
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5
1
4
3
2
locations
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Traveling Salesman Problem (2-exchange)
ij
ij
D= d adj. matrix
0 1 0 . . . . . 0 1
1 0 1 0 . . . . . 0
0 1 0 1 0 . . . . .
0 0 1 0 1 0 . . . .
. . 0 1 0 1 0 . . . F= f
. . . 0 1 0 1 0 . .
. . . . 0 1 0 1 0 0
. . . . . 0 1 0 1 0
0 . . . . . 0 1 0 1
1 0 . . . . . 0 1 0
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Bipartitioning
weighted graph G(V,E)
2-exchange
A BV
D = d adj.matrixij
2
1bgdi
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Particular cases(Bipartitioning)
F = fO U
U O
U = u i, j = 1,...,n
2 otherwise
ij
ij
di
di
FHG
IKJ
RS|T|1
0
,
h/2
h/2
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K-densest and k-lightest
Graph G V, E V n
consider m n m = 1,..., n
"Find a sub - graph G V ,E with V m
and E maximum minimum
NP - complete
- maximum clique
- maximum Independent set
G
b gb g
b gb g
,
,
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Results (2-exchange)
ij
1 2 n
1 0F= D= d
0 0
d d ... d degrees of G
m
n-m
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The Local Search: The MIS
example : The maximum Independent Set problem in a graph G(V,E)
"Find V V s. t. u, V u, " 1
2
3
4
1 3,l q a solution
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The MIS by the Local Search
Solution coding :
Function :
x x x x
x V
x V
1 2 i n
i i
i i
, ,..., ,...,
1
0
Max x
= u, with u, V
ii=1
n
,
m r
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Neighborhood : FLIP
x x x x x current sol.
x x x x x neighboor sol.
1 2 3 i n
1 2 3 i n
, , ,..., ,...,
, , ,..., ,...,
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LS Drawbacks
Local optimum “good“ neighborhoods exploration strategies Performances guarantee ? Parallelization ?
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