Find minimal cost of eulerian path,which is contain given edges in undirected graph(Algorithm)

Question

How to find in undirected, weighted graph the minimal eulerian path?(This path must contain the given edges)

The weight of the edges is the sum of the 2 points(exmpl:edge 4-9 weight=4+9=13) for all edges.

example: With 6 nodes(N),and with 5 edges(E):

(1-5)
(6-1)
(5-5)
(2-4)
(2-4)

solution: We must add 2 edges,to minimal eulerian path:

1-2
1-2

In this example 3rd node is isolated,but that's not problem. Target: An eulerian path,which is contain all the Start-edges. In this example we can with the 2 complementary edges(1-2)(1-2) an Euler path do: 5->5->1->2->4->2->1->6. So we visited all the Start-edges,with minimal complementary edges,and we use all edges just once.

What is the best algorithm to find, when 1<N,E<100000 ,and must run in 0.01 second?