Finding the shortest path in a graph without any negative prefixes

Question

Find the shortest path from source to destination in a directed graph with positive and negative edges, such that at no point in the path the sum of edges coming

Answer 1

Peter de Rivaz pointed out in a comment that this problem includes HAMILTONIAN PATH as a special case. His explanation was a bit terse, and it took me a while to figure out the details, so I've drawn some diagrams for the benefit of others who might be struggling. I've made this post community wiki.

I'll use the following graph with six vertices as an example. One of its Hamiltonian paths is shown in bold.

Given an undirected graph with n vertices for which we want to find a Hamiltonian path, we construct a new weighted directed graph with n² vertices, plus START and END vertices. Label the original vertices v_i and the new vertices w_ik for 0 ≤ i, k < n. If there is an edge between v_i and v_j in the original graph, then for 0 ≤ k < n−1 there are edges in the new graph from w_ik to w_j(k+1) with weight −2^j and from w_jk to w_i(k+1) with weight −2ⁱ. There are edges from START to w_i0 with weight 2ⁿ − 2ⁱ − 1 and from w_i(n−1) to END with weight 0.

It's easiest to think of this construction as being equivalent to starting with a score of 2ⁿ − 1 and then subtracting 2ⁱ each time you visit w_ij. (That's how I've drawn the graph below.)

Each path from START to END must visit exactly n + 2 vertices (one from each row, plus START and END), so the only way for the sum along the path to be zero is for it to visit each column exactly once.

So here's the original graph with six vertices converted to a new graph with 38 vertices. The original Hamiltonian path corresponds to the path drawn in bold. You can verify that the sum along the path is zero.