Introduction to Algorithms
Problem Set 9
CS 482 Spring 2006
Due: Friday, April 21
Please hand in each problem on separate sheets with your name and netID on each.
If a problem requires multiple sheets, please staple the sheets for that problem together
Reading:
Chapters 10.1, 10.2, 11.1  11.4, 11.8.
Question 1
Show that for any fixed constant
k
≥
2, determining whether or not an undirected graph
G
has a
spanning tree
T
with exactly
k
leaves is NPComplete.
Question 2
Suppose we have an undirected graph
G
= (
V, E
) with edges costs
c
e
≥
0 for each edge
e
∈
E
, and a
set of nodes
S
⊆
V
. Done. Recall that a Steiner tree for
S
is a tree
T
in
G
that spans all nodes in
S
.
As you’ve seen in class, in general, finding the minimum cost Steiner tree is NPComplete.
(a)
Give a polynomial time algorithm to find the minimum cost Steiner tree in the case that

S

= 3.
(b)
Give a 2approximation for the problem of finding a minimum cost Steiner tree (for any set
S
).
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 Spring '06
 WEXLER
 Algorithms, Graph Theory, Steiner, minimum cost Steiner, cost Steiner tree

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