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)

Explanation / Answer

ANSWER

Proof

( ? ) reason the ?ow net has a max-?ow cost of | X | .

(It cannot be supplementary than | X | since the cut unscrambling the source s has capacity | X | .) since all edge capacity are integral, the integrality theorem tells us that there exists an basic ?ow f .

This ?ow f has f ( e ) ? { 0 , 1 } for every edge e other than limits from S to t .

analogous to our proof of Menger’s theorem, map out a walk of edges in the midst of ?ow morals a starting from s .

This walk has to finish at t due to ?ow protection constraints. take away all cycles on or after this walk and we get the ?rst X,Y -path.

Then, set reduce all ?ow principles on edges of this walk by 1, and we get a novel feasible ?ow with value | X | - 1. Repeat the process recursively until we obtain | X | paths.

( ? ) If a set of mass departure routes exists, we plainly set all ?ow values on edges of these route to be 1, ?ow values of limits from the source s to every vertex in X to 1, and ?ow values of an rim ( v,t ) from S to t to be the figure of mass going away routes which end at t .

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