Add 35.2-5 problem

C35-Approximation-Algorithms/35.2-5.md

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+# Exercise 35.2-5
+
+*** 
+Suppose that the vertices for an instance of the traveling-salesman problem are points in the plane and that the cost *c(u,v)* is the euclidean distance between points *u* and *v*. Show that an optimaol tour never crosses itself.
+
+## Solution
+
+Cost function based on euclidean distance satisfies the triangle-inequality, s.t.:
+
+``` c(u,v) <= c(u,w) + c(w,v) ```
+
+Assume: The tour corsses itself.
+
+Let *(u,v)* and *(w,x)* be corssing edges, *u -> v -> w -> x* the assumed optimal tour and P the crossing point of *(u,v)* and *(w,x)*.
+
+Based on the triangle-inequality its: 
+```c(x,v) <= c(x,P) + c(P,v) ```
+
+Thus we can derive:
+
+```c(u,P) + c(P,w) + c(x,P) + c(P,v) + c(u,w) + c(w,v) >= c(u,w) + c(x,v) * c(u,x) + c(w,v)```
+
+Based on our definition its ```c(u,P) + c(P,v) = c(u,v)``` and ```c(x,P) + c(P,w) = c(x,w)```
+
+Therefore its ``` c(u,v) + c(v,w) >= c(u,w) + c(x,v)``` which denotes in combination with (u,x) and (v,w) a shorter tour then the original one. Contradiction. q.e.d