There are two famous NP optimization problems :(adsbygoogle = window.adsbygoogle || []).push({});

1) traveling salesman : find the shortest path in a graph, through all vertices. Without boundary conditions, this is : [tex] min_{\sigma\in Perm(\mathbb{N}_n)}\sum_{i=1}^{N-1}a_{\sigma(i),\sigma(j)}[/tex] where there are [tex]n[/tex] cities, and the distance between cities [tex]i[/tex] and [tex]j[/tex] is [tex]a_{ij}[/tex]..and [tex]\sigma[/tex] is just a permutation of the cities order.

2) coin-changing problem : the less coins you need to make up to a given sum S in a given coin system [tex] \left(v_i\right)_{i=1}^k[/tex] : [tex]\tilde{\min}_{n} \left(\sum_{i=1}^n v_{f(i)}=S\right)[/tex] where [tex] f : \mathbb{N}_n\rightarrow \mathbb{N}_k [/tex] with [tex]v_f(i)[/tex] is giving the index of the ith coin needed. [tex]\tilde{min}[/tex] gives the minimal n, If n does not exist, then it is [tex]\infty[/tex].

I thinnk there should exist a forumla betwen those two problems : the coin system is equivalent to the distances between cities (kind of constraint), but I cannot firgure it out..could somebody help ?

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Optimization problems

Can you offer guidance or do you also need help?

**Physics Forums | Science Articles, Homework Help, Discussion**