18.997 Topics in Combinatorial Optimization
February 3rd, 2004
Lecture 1
Lecturer: Michel X. Goemans
1
Scribe: Nick Harvey
Nonbipartite Matching
Our rst topic of study is matchings in graphs which are not necessarily bipartite. We begin with
some relevant
18.997 Topics in Combinatorial Optimization
9 March 2004
Matroids
Lecturer: Michel X. Goemans
Scribe: Bridget Eileen Tenner
For reference, see Chapter 39 of Schrijvers book.
Denition 1 A matroid M = (S, I ) is a nite ground set S together with a collectio
18.997 Topics in Combinatorial Optimization
April 6th, 2004
Lecture 15
Lecturer: Michel X. Goemans
1
Scribe: Supratim Deb
Matroid Matching
The Matroid matching Problem: Given a matroid M = (S, I ), let E be a set of pairs on S .
The matroid matching probl
April 8, 2004
18.997 Topics in Combinatorial Optimization
Lecture 16
Lecturer: Michel X. Goemans
Scribe: Jonathan Kelner
This lecture is about jump systems. While they are briey discussed in chapter 41 of Schrijvers
book, they are not covered extensively.
April 27, 2004
18.997 Topics in Combinatorial Optimization
Lecture 20
Lecturer: Michel X. Goemans
1
Scribe: Jan Vondrk
a
k -arc-connected orientations
We continue the discussion of how a 2k -edge-connected graph can be oriented so that the resulting
digra
April 22th, 2004
18.997 Topics in Combinatorial Optimization
Lecture 19
Lecturer: Michel X. Goemans
1
Scribe: Ben Recht
Special cases of submodular ows
We saw last time that orientation of a 2k -edge connected graph into a k -arc connected digraph and
the
March 4, 2004
18.997 Topics in Combinatorial Optimization
Lecture 8
Lecturer: Michel X. Goemans
Scribe: Constantine Caramanis
This lecture covers the proof of the Bessy-Thomass Theorem, formerly known as the Gallai
e
Conjecture. Also, we discuss the cycli
18.997 Topics in Combinatorial Optimization
March 2, 2004
Lecture 7
Lecturer: Michel X. Goemans
1
Scribe: Jan Vondrk
a
Gallais Conjecture
In this lecture, we will be concerned with graph coverings by a collection of paths or cycles. The
goal will be to co
18.997 Topics in Combinatorial Optimization
February 10th, 2004
Lecture 3
Lecturer: Michel X. Goemans
Scribe: Dan Stratila
In this lecture we will cover:
1. Topics related to Edmonds-Gallai decompositions ([Sch03], Chapter 24).
2. Factor critical-graphs a
18.997 Topics in Combinatorial Optimization
February 5, 2004
Lecture 2
Lecturer: Michel X. Goemans
Scribe: Robert Kleinberg
In this lecture, we will:
Present Edmonds algorithm for computing a maximum matching in a (not necessarily bipartite) graph G.
Us
February 12, 2004
18.997 Topics in Combinatorial Optimization
Lecture 4
Lecturer: Michel X. Goemans
Scribe: Constantine Caramanis
This lecture covers: the Matching polytope, total dual integrality, and Hilbert bases.
1
The Matching Polytope and Total Dual
18.997 Topics in Combinatorial Optimization
February 24th, 2004
Lecture 7
Lecturer: Michel X. Goemans
Scribe: Ben Recht
In this lecture, we investigate the relationship between total dual integrality and integrality of
polytopes. We then use a theorem on
18.997 Topics in Combinatorial Optimization
February 26th, 2004
Lecture 6
Lecturer: Michel X. Goemans
Scribe: Joungkeun Lim
Last time, we saw that the matching polytope was dened by:
x( (v ) 1 v V
x(E (S ) |S | , for |S | odd
2
x 0.
One may wonder whether
18 March 2004
18.997 Topics in Combinatorial Optimization
Lecture 12
Lecturer: Michel X. Goemans
Scribe: Vahab S. Mirrokni
Last time, we stated the following theorem by Edmonds and Lawler about the maximum independent set common to two matroids.
Theorem 1
18.997 Topics in Combinatorial Optimization
March 4, 2004
Lecture 8
Lecturer: Michel X. Goemans
Scribe: Constantine Caramanis
This lecture covers the proof of the Bessy-Thomass Theorem, formerly known as the Gallai
e
Conjecture. Also, we discuss the cycli
18.997 Topics in Combinatorial Optimization
April 1st, 2004
Lecture 14
Lecturer: Michel X. Goemans
Scribe: Mohamed Mostagir
In this lecture, we continue with more results on matroid union, as well as tie together some
loose ends from the past couple of le
18.997 Topics in Combinatorial Optimization
February 5, 2004
Lecture 2
Lecturer: Michel X. Goemans
Scribe: Robert Kleinberg
In this lecture, we will:
Present Edmonds algorithm for computing a maximum matching in a (not necessarily bipartite) graph G.
18.997 Topics in Combinatorial Optimization
February 10th, 2004
Lecture 3
Lecturer: Michel X. Goemans
Scribe: Dan Stratila
In this lecture we will cover:
1. Topics related to Edmonds-Gallai decompositions ([Sch03], Chapter 24).
2. Factor critical-graphs a
18.997 Topics in Combinatorial Optimization
February 12, 2004
Lecture 4
Lecturer: Michel X. Goemans
Scribe: Constantine Caramanis
This lecture covers: the Matching polytope, total dual integrality, and Hilbert bases.
1
The Matching Polytope and Total Dual
18.997 Topics in Combinatorial Optimization
February 24th, 2004
Lecture 5
Lecturer: Michel X. Goemans
Scribe: Ben Recht
In this lecture, we investigate the relationship between total dual integrality and integrality of
polytopes. We then use a theorem on
18.997 Topics in Combinatorial Optimization
February 26th, 2004
Lecture 6
Lecturer: Michel X. Goemans Last time, we saw that the matching polytope was dened by: x( (v ) 1 v V x(E (S ) |S | , for |S | odd 2 x 0. Scribe: Joungkeun Lim
One may wonder whethe
18.997 Topics in Combinatorial Optimization
18 March 2004
Lecture 12
Lecturer: Michel X. Goemans
Scribe: Vahab S. Mirrokni
Last time, we stated the following theorem by Edmonds and Lawler about the maximum independent set common to two matroids.
Theorem 1
18.997 Topics in Combinatorial Optimization
March 30, 2004
Lecture 13
Lecturer: Michel X. Goemans
Scribe: Constantine Caramanis
Last lecture we covered matroid intersection, and dened matroid union. In this lecture we review
the denitions of matroid inter
18.997 Topics in Combinatorial Optimization
March 2, 2004
Lecture 7
Lecturer: Michel X. Goemans
1
Scribe: Jan Vondrk
a
Gallais Conjecture
In this lecture, we will be concerned with graph coverings by a collection of paths or cycles. The
goal will be to co
18.997 Topics in Combinatorial Optimization
16 March 2004
Lecture 11
Lecturer: Michel X. Goemans
Scribe: Fumei Lam
Let M1 = (S, I1 ), M2 = (S, I2 ) be two matroids on common ground set S with rank functions
r1 and r2 . Many combinatorial optimization prob
18.997 Topics in Combinatorial Optimization
9 March 2004
Matroids
Lecturer: Michel X. Goemans
Scribe: Bridget Eileen Tenner
For reference, see Chapter 39 of Schrijvers book.
Denition 1 A matroid M = (S, I ) is a nite ground set S together with a collectio
18.997 Topics in Combinatorial Optimization
March 11, 2004
Lecture 10
Lecturer: Michel X. Goemans
Scribe: Nicole Immorlica
Matroid theory was rst formalized in 1935 by Whitney [5] who introduced the notion as an
attempt to study the properties of vector s
18.997 Topics in Combinatorial Optimization
April 6th, 2004
Lecture 15
Lecturer: Michel X. Goemans
1
Scribe: Supratim Deb
Matroid Matching
The Matroid matching Problem: Given a matroid M = (S, I ), let E be a set of pairs on S .
The matroid matching probl
February 3rd, 2004
18.997 Topics in Combinatorial Optimization
Lecture 1
Lecturer: Michel X. Goemans
1
Scribe: Nick Harvey
Nonbipartite Matching
Our rst topic of study is matchings in graphs which are not necessarily bipartite. We begin with
some relevant