# ClassW04ApproxAlgs/Hwk2

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homework 2, due Tuesday 1/13/2004

Work in groups of 1-3, but each person turn in your own write-up, stating with whom they worked.

1. Write up a careful proof of one of the the three theorems above: Euler's theorem, the shortcutting theorem, or Konig-Egervary theorem. If you are working in a group of 2-3 people, each person should choose a different theorem to prove and write up, and take primary responsibility for that theorem (with help and feedback from the others).
2. (problem 3.3 in text) The Directed Steiner Tree problem is the following.
Given a directed graph G=(V,E) with non-negative edge costs, a root vertex r, and a subset of vertices S, find a minimum-cost tree in G rooted at r and containing all vertices in S. (A tree in a directed graph is a collection of directed edges such that those edges together with their reversals contain no cycle of length > 2. The tree is rooted at r if there is a directed path in the tree from each vertex in the tree to r.)
The Set Cover Problem is, given a collection of sets, to choose some of those sets so that every element is in some chosen set. The goal is to minimize the number of chosen sets. Unless P=NP, there is no o(log n)-approximation algorithm for set cover, where n is the number of elements in the problem instance.
Using this latter fact, show that (unless P=NP) there is no o(log n)-approximation algorithm for the Directed Steiner tree problem, by giving an approximation-preserving reduction from Set Cover to Directer Steiner tree.

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