Ant Colony Optimization and Constraint Programming by Christine Solnon

By Christine Solnon

Ant colony optimization is a metaheuristic which has been effectively utilized to quite a lot of combinatorial optimization difficulties. the writer describes this metaheuristic and stories its potency for fixing a few tough combinatorial difficulties, with a selected concentrate on constraint programming. The textual content is geared up into 3 parts.

The first half introduces constraint programming, which supplies excessive point positive factors to declaratively version difficulties through constraints. It describes the most present ways for fixing constraint delight difficulties, together with whole tree seek ways and metaheuristics, and exhibits how they are often built-in inside of constraint programming languages.

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Definition of a constraint A constraint may be defined by a simple enumeration of the tuples of values that belong to the relation. 2. The constraint (x, y) ∈ {(0, 1), (0, 2), (1, 2)} constrains x to be assigned to 0 or 1 and y to be assigned to 1 or 2 if x = 0, and to 2 if x = 1. In a dual way, we may define a constraint by enumerating the tuples of values that do not belong to the relation; this is usually the case when there are less tuples that do not belong to the relation than tuples that do.

This depth-first exploration is very easy to implement with a recursive function. However, it may explore many unnecessary assignments when failure is due to a poor choice made much higher in the search tree. 2. e. X = {x1 , x2 , . . 1: simpleBacktrack((X, D, C), A) Input: a CSP (X, D, C) a (partial or complete) assignment A for (X, D, C) Precondition: A is consistent Postrelation: If A cannot be extended to a solution of (X, D, C) then return ∅, otherwise return a solution A of (X, D, C) such that A ⊆ A 1 begin 2 if var(A) = X then /* All variables are assigned ⇒ A is a solution */ 3 return A 4 else 5 Choose a variable xi ∈ X such that xi ∈ var(A) 6 for each value v ∈ D(xi ) do 7 if A ∪ {(xi , v)} is consistent then 8 Sol ← simpleBacktrack((X, D, C), A ∪ {(xi , v)}) 9 if Sol = ∅ then return Sol /* No value of D(xi ) allows A to be extended to a solution */ return ∅ 10 11 end – domains are defined by D(x1 ) D(xi ) D(xn−1 ) = D(xn ) = {n, n + 1} = {i, i + 1}, ∀i ∈ {2, .

N}, ∀j ∈ {1, . . , n}, i = j ⇒ xi + i = xj + j; - queens must be in different falling diagonals: ∀i ∈ {1, . . , n}, ∀j ∈ {1, . . , n}, i = j ⇒ xi − i = xj − j. 1 for four queens corresponds to the following assignment: A = {(x1 , 2), (x2 , 4), (x3 , 1), (x4 , 3)}. 4. Third CSP model The second model ensures that no queen may be captured by three subsets of binary difference constraints: the first subset ensures that all queens are in different rows, the second that they are in different rising diagonals and the third that they are in different falling diagonals.

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