hw5.pdf - Fall 2021 CS 583: Approximation Algorithms...

Fall 2021, CS 583: Approximation Algorithms
Homework 5
Due: 11/30/2021 in Gradescope
Instructions and Policy: Each student should write up their own solutions independently.
You need to indicate the names of the people you discussed a problem with; ideally you
should discuss with no more than two other people. You may be able to find solutions to
the problems in various papers and books but it would defeat the purpose of learning if copy
them. You should cite all sources that you use and write in your own words.
Read through all the problems and think about them and how they relate to what we covered
in the lectures. Solve as many problems as you can. Please submit solutions to Problem 1,
6 and at least two other problems. Some problems are closely related so it may benefit you
to solve them together or view them as parts of an extended problem.
Please write clearly and concisely - clarity and brevity will be rewarded. Refer to known
facts as necessary.
Problem 1 We have mostly seen edge-weighted network design problems in undirected
graphs. As we discussed in lecture, directed network design problems are hard to approxi-
mate. Undirected node-weighted network design problems fall in between. Here we consider
node-weighted Steiner tree problem. The input is a graph G = ( V, E ) with non-negative
node weights w : V → R
+
, and a set of k terminals S ⊆ V . The goal is to find a Steiner tree
T = ( X, F ) for the terminals (that is, S ⊆ X ) with minimum node weight. We will assume
that the weight of the terminals is 0 without loss of generality since every terminal has to
be included in the tree. We saw that Steiner tree in edge-weighted graphs admits a sim-
ple 2-approximation via the MST heuristic, and improved approximations are also known.
In contrast the node-weighted problem is harder. This problem will walk you through an
approach to approximate it via Set Cover type ideas.
• Describe a reduction from Set Cover on m sets and n elements to node-weighted Steiner
tree with k = n . Informally argue that an α ( k )-approximation for node-weighted
Steiner tree implies an α ( n )-approximation for Set Cover. This should convince you
that node-weighted Steiner tree does not admit a better than ln k approximation.
• Given a set of terminals, a spider is a star-like tree with a center vertex c and k legs
where each leg is a c - t path for some terminal t (the center itself can be a terminal).
Thus, all the leaves are terminals if k > 1. If k = 1 then we require that the center
to be a terminal and in this case the spider is simply a path connecting two terminals.
See figure. Argue that any minimal Steiner tree ( X, F ) contains a collection of node-
disjoint spiders that together include all but one terminal from S .
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