 
Summary: CMPSCI 611: Advanced Algorithms
Micah Adler
Problem Set 2 Out: October 10, 2000
Due: October 17, 2000
1. a) Consider the following optimization problem:
INPUT: An undirected graph G = (E; V ), a partition of the set of edges E into subsets E 1 ; E 2 ; : : : ; E k ,
and a weight function for the edges w : E ! <.
OUTPUT: A maximum weight subset of the edges S E such that there is at most one edge in S
contained in each subset E i (i.e., 8i; jS \ E i j 1j).
Dene a subset system that describes the set of valid (but not necessarily maximum weight) solutions
to this problem. Prove that this subset system is a matroid.
b) Give an eÆcient algorithm for this problem. Analyze its running time, and demonstrate that it
always provides an optimal solution.
c) Given an undirected graph G = (E; V ), and a subset of the vertices R, consider the problem of
determining whether or not there exists a spanning tree T for G such that every vertex in R is a leaf
of the tree T . Use your matroid from part (a) to demonstrate that there exists a polynomial time
algorithm that decides whether such a tree T exists or not. You do not have to describe the algorithm
or its running time.
2. Problem 222 from [CLR] page 459.
3. Consider the following paragraphing problem:
