This is the reading assignment for
Matroids, Secretary Problems, and Online Mechanisms
http://immorlica.com/pubs/matroidSecretaries.pdf
Bring the write-ups to class on Wednesday, April 6. Don't worry about
wordsmithing or formatting or anything. All the writing combined shouldn't be
more than a page.
PART I
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SECTION 1:
Skim this, skipping any words you don't know (this will likely be 80% of the words).
Write 3-4 sentences answering: "Why did the authors write this paper? What was
their main motivation? Why do they care about matroids and secretary problems?"
SECTION 2:
You can read this somewhat more carefully, while skipping anything relating to
game theory or auctions or economics.
For Domains 1-5: think through why each domain is an example of a matroid. Do
not spend more than five minutes trying to answer this for any given domain --
the proofs are quite difficult. But it is worth trying to reason through as far
as you can to make sure you have a good intuition for what matroids look
like. No need to write anything up.
Write 1 sentence answering: Why does Domain 2 not contradict the negative
example Prof. Roughgarden gave in class? (Namely, that the set of matchings in a
bipartite graph is not a matroid.)
SECTION 3.2:
Write a summary of the algorithm and proof in a couple of sentences, as if you
were telling them to a friend.
Write quick answers to the following:
Where in the proof do we use the first property of matroids, namely that a
subset of an independent set is an independent set?
What would happen if we picked j uniformly from 1 to k, rather than uniformly in
logspace?
Optional warm-up exercise, that may help with understanding the proof of Theorem 3.2:
Let M be a matroid of rank k.
Consider the following algorithm for picking an independent set:
Initialize the set B to the empty set.
For each element x_i of M:
Add x_i to B if B \cup {x_i} is independent.
Prove that the size of B at the end of the algorithm is k, regardless of the
order in which the elements are presented.
PART II
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We will assign each of you a color by email. Make sure you understand your
sections well enough to be explain them to your classmates.
RED:
Read the Section 3 intro, Section 3.1, Appendix A.1, and Appendix A.2.
Summarize the four sections (including the main ideas in each proof, when
applicable) in a few sentences each.
GREEN:
Read Section 4, ignoring the AGT stuff (e.g. skip Theorem 4.2).
Summarize the algorithm and the proof in a few sentences.
Read Section 5 up to the statement of Theorem 5.1. No need to write anything,
but you should be able to explain the high level idea for the reduction from
graphic matroids to transversal matroids.
BLUE:
Read Section 6, ignoring the AGT stuff (e.g. skip the last paragraph).
Summarize the algorithm and the proof in a few sentences. Feel free to take
Theorems 6.2 and 6.3 on faith, even if you have no idea how to prove them.