Let G = (N, A) be a network with n nodes and m arcs, and integral capacities (u_

Question

Let G = (N, A) be a network with n nodes and m arcs, and integral capacities (u_e: e A). For two nodes s, t N, denote by P_s, t the set of directed paths from s to t in G. Consider the following linear program: maximize sigma_P P_s, t f P subject to simga_e P: P P_s, t f P lessthanorequalto u_e, Forall e A, f_p greaterthanorequalto 0, Forall P P_s, t What does this LP compute? Show that this LP can be solved in polynomial time, and give an upper bound (as tight as possible) on the running time needed.

Explanation / Answer

You can compute the facets of N in polynomial time. It is easy to see that s,t is a face of N if and only if there exists a set of facets of N, A = {F1, . . . , Fk} such that P= T i=1...k Fi . This can be computed in polynomial time by letting N be the set of all facets which contain P.

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