General Travelling Salesman Problem

Given an input $G=(V,d)$ which is a complete weighted graph, where $d:V\times V\to \mathbb{Q}$ is a distance function. The output is the shortest Hamiltonian cycle visiting all vertices. Formalising this problem, we find:

$$\begin{array}{rl}S& =\{\text{all possible routes }H\}\\ h(H)& =\text{length of}(H)=\sum _{(x,y)\in H}d(x,y)\end{array}$$