SUNY Buffalo State College CSE 531a5-sol

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Solution to CSE 531 Homework Assignment 5 Prepared by Hung Q. Ngo∗ November 22, 2007 Problem 1. We define the Escape Problem as follows. We are given a directed graph G = (V, E) (picture a netwo... rk of roads). A certain collection of nodes X ⊂ V are designated as populated nodes, and a certain other collection S ⊂ V are designated as safe nodes. (Assume that X and S are disjoint.) In case of an emergency, we want evacuation routes from the populated nodes to the safe nodes. A set of evacuation routes is defined as a set of paths in G so that (i) each node in X is the starting point of one path, (ii) the last node on each path lies in S, and (iii) the paths do not share any edges. Such a set of paths gives a way for the occupants of the populated nodes to “escape” to S, without overly congesting any edge in G. (a) Given G, X, and S, show how to decide in polynomial time whether such a set of evacuation routes exists. (b) 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): we change (iii) to say “the paths do not share any nodes.” With this new condition, show how to decide in polynomial time whether such a set of evacuation routes exists. (c) 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). Solution Sketch. (a) Construct a flow network D as shown in Figure 1. All edge capacities are set to 1. Only the edges from S to t are of capacities |X| [Show More]

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