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

As Kaganar notes, we basically have to make some assumption in order to get a polytime algorithm. Let's assume that the edge lengths are in {-1, 1}. Given the graph, construct a weighted context-free grammar that recognizes valid paths from source to destination with weight equal to the number of excess 1 edges (it generalizes the grammar for balanced parentheses). Compute, for each nonterminal, the cost of the cheapest production by initializing everything to infinity or 1, depending on whether there is a production whose RHS has no nonterminal, and then relaxing n - 1 times, where n is the number of nonterminals.