6 Conclusion and future research
We find that a suitable restriction can increase efficacy of GAs. Restriction itself is not an operator. It can be added to other operators to make them more efficient. Restriction had better to be a necessary condition of the best solution, but they also work even though they are not. We can design other restrictions on GAs for TSP or other problems.
We can design another restriction for Euclidean TSP, whose cost of edge is exactly the distance of two cities. There is a necessary condition of Euclidean TSP:
Suppose the path P is the best solution of an Euclidean TSP. Then each pair of different edges of P do not intersect.
P roof : Suppose a path P = (A
1...A
n) is the best solution of an Euclidean TSP.
If P has two edges A
iA
i+1, A
jA
j+1(i < j) that intersect.
We can construct a new path Q = (A
1...A
iA
jA
j−1...A
i+1A
j+1...A
n)
A
A
A
A A
i
j
j
i +1
+1 1
A
A
A
A A
i
j
j
i +1
+1 1
