IEOR 6614, Spring 2012 : Homework 4

General Instructions

1. Please review the course information.
2. You must write down with whom you worked on the assignment. If this changes from problem to problem, then you should write down this information separately with each problem.
3. Numbered problems are all from the textbook Network Flows .

Problems

1. We define a shortest path algorithm to be oblivious if given a graph G=(V,E), with vertices numbered 1..n, it decides to a execute a series of relax statements based only the values n = |V| and m = |E|, and not based on the particular structure of the graph. For example, Bellman-Ford algorithm is oblivious, but Dijkstra's algorithm is not.

   Prove the following statement: Any oblivious algorithm that correctly computes single source shortest paths for all graphs G, must have a worst-case running time of Ω(nm).

2. Problem 5.51, 5.52. Bit-scaling for shortest paths
3. Problem 4.46. Constrained shortest paths.
4. Problem 4.34, Problem 4.36. K-shortest paths.
5. On pg. 94, there is a linear program for computing single source-shortest paths. Suppose that you have solved this linear program optimally and have a solution where the variables all take on integral values.
   1. Explain how to use the solution to the linear program to output a shortest path tree.
   2. Take the dual of the linear program. Give an explanation of what the dual solution represents.

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• Homework 3
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