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Imagine you are given a weighted undirected complete graph with n nodes with non-negative weights Cij, where for i = j Cii = 0, and for i != j Cij > 0. Suppose you have to find the maximal shortest path between any two nodes i and j. Here, you can easily use Floyd-Warshall, or use Dijkstra n times, or whatever, and then just to find the maximum among all n^2 shortest paths.

Now assume that Cij are not constant, but rather can take two values, Aij and Bij, where 0 <= Aij <= Bij. We also have Aii = Bii = 0. Assume you also need to find the maximal shortest path, but with the constraint that m edges must take value Bij, and other Aij. And, if m > n^2, then all edges are equal to Bij. But, when finding shortest path i -> p1 -> ... -> pk -> j, you are intesrested in the worst case, in the sense that on that path you need to choose those edges to take value of Bij, such that path value is maximal if you have fixed nodes on its direction.

For example, a if you have path of length four i-k-l-j, and, in optimal solution on that path only one weight is changed to Bij, and other take value of Aij. And let m1 = Bik + Akl + Alj, m2 = Aik + Bkl + Alj, m3 = Aik + Akl + Blj, the value of that path is max{m1, m2, m3}. So, among all paths between i and j, you have to choose one such that maximal value (described as in this example) is minimal (which is a variant of definition of shortest path). And you have to do it for all pairs i and j.

You are not given the constraint how many you need to vary on each path, but rather value of m, a number of weights that should be varied in the complete graph. And the problem is to find the maximum value of the shortest path, as described.

Also, my question is: is this NP-hard problem, or there exists some polynomial solution?

user3800794
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  • Should the path contain each vertex at most once or is it allowed to visit a vertex twice ot more times? – kraskevich Jul 03 '14 at 10:56
  • At most once. However, if you find a shortest path that contains some vertex more than one time, I think it is impossible to be the shortest since edges are non-negative. – user3800794 Jul 03 '14 at 10:59
  • Are the m B-edges allowed to vary with each choice of (start vertex, end vertex) or not? If yes, then is your question equivalent to the following? Find the vertex pair {u, v} maximising f(u, v), where f(u, v) is the minimum value of g(P) over all paths P from u to v, and g(P) is the maximum path length achievable by setting min(m, nEdges(P)) edges in P to their B-lengths and the rest to their A-lengths. – j_random_hacker Jul 03 '14 at 11:38
  • If the m B-edges are not supposed to vary, then is it equivalent to: Find the vertex pair {u, v} maximising h(u, v), where h(u, v) is the maximum value of a(u, v, G') across all graphs G' having the same edges as the input graph G and with exactly m of these edges assigned their B-lengths and the rest assigned their A-lengths, and a(u, v, G') is the minimum path length from u to v in G'? – j_random_hacker Jul 03 '14 at 11:39

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