Question

1. Let G = (V, E) be a graph, let s, t be vertices of G, and suppose each edge e has a nonnegative weight $c_e$. We know that an st-cut $\delta(U)$ intersects every st-path. (a) Suppose S is a set of edges that contains at least one edge from every st-path. Show that there exists an st-cut $\delta(U)$ that is contained in the edges of S. (b) The weight of an st-cut $\delta(U)$ is defined as $c(\delta(U)) := \sum_{e \in \delta(U)} c_e$. Find an IP formulation for the problem of finding an st-cut of minimum weight, where we have a binary variable for every edge and a constraint for every st-path. (c) Show that the formulation given in (b) is correct.

          1. Let G = (V, E) be a graph, let s, t be vertices of G, and suppose each edge e has a nonnegative weight $c_e$. We know that an st-cut $\delta(U)$ intersects every st-path.
(a) Suppose S is a set of edges that contains at least one edge from every st-path. Show that there exists an st-cut $\delta(U)$ that is contained in the edges of S.
(b) The weight of an st-cut $\delta(U)$ is defined as
$c(\delta(U)) := \sum_{e \in \delta(U)} c_e$.
Find an IP formulation for the problem of finding an st-cut of minimum weight, where we have a binary variable for every edge and a constraint for every st-path.
(c) Show that the formulation given in (b) is correct.

1 let g v e be a graph let s t be vertices of g and suppose each edge e has a nonnegative weight ce we know that an st cut deltau intersects every st path a suppose s is a set of edges that 22547

Added by James C.

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