4. Collaborative: We define the Escape Problem as follows. We are given a direct

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4. Collaborative: We define the Escape Problem as follows. We are given a directed graph G V, E) (picture a network of roads.) A certain collection of vertices X C V are designated as populated vertices, and a certain other collection S c V are designated as safe vertices. (Assume that X and S are disjoint.) In case of an emergency, we want evacuation routes from the populated vertices to the safe vertices. A set of evacuation routes is defined as a set of paths in G such that (i) each vertex in X is the tail of one path, (ii) the last vertex on each path lies in S, and (iii the paths do not share any edges. Such a set of paths gives way for the occupants of the populated vertices to "escape" to S without overly congesting any edge in G (a) [20 pointsl Given G, X, and S, show how to decide in polynomial time whether such a set of evacuation routes exists (b) [20 pointsl Suppose we have exactly the same problem as in (a), but we want to enforce an even stronger version of the "no congestion" condition (iii). Thus we change ii) to say, "the paths do not share any vertices." With this new condition, show how to decide in polynomial time whether such a set of evacuation routes exists. Also provide an example with the same G, X, and S in which the answer is 'yes" to the question in (a) but "no" to the question in (b)

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Proof;

( ? ) assume the ?ow complex has a max-?ow value of | X | . (It cannot be added than | X | since the cut straightening out the source s has competence | X | .) Because all edge capacity are integral, the integrality theorem tells us with the aim of there exists an primary ?ow f .

This ?ow f have f ( e ) ? { 0 , 1 } for any rim e other than limits from S to t .

Similar to our evidence of Manger’s theorem, trace a saunter of edges with ?ow values a initial from s . This walk has to beginning at t due to ?ow destruction constraints.

eradicate all cycles from this amble and we get the rest X,Y -path.

Then,

situate reduce all ?ow values on limits of this walk by 1, and we get a new feasible ?ow with rate | X | - 1. Repeat the method recursively until we get | X | path.

( ? ) If a set of migration route exists, we simply set all ?ow values on edges of these routes to be 1, ?ow morals of edges from the font s to each vertex in X to 1, and ?ow principles of an edge ( v,t ) from S to t to be the number of mass going away routes which conclusion at t .

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