Lecture 11 Notes: Metric Spaces
p
Let 1 p , and recall the definition of the metric space :
p
For 1 p <, =
p
sequences a = (an)n
=1
in R such that n=1 an <
;
whereas consists of all those sequences a = (an) n=1 such that supnN an <. We
p
defined the p
Lecture 1 Notes: Intermediate Vertexs
Our first topic of study is matchings in graphs which are not necessarily bipartite. We
begin with some relevant terminology and definitions. A matching is a set of edges that
share no endvertices. A vertex v is cover
Lecture 10 Notes: Graphic Matroids
Matroid theory was first formalized in 1935 by Whitney [5] who introduced the notion
as an attempt to study the properties of vector spaces in an abstract manner. Since then,
matroids have proven to have numerous applica
Lecture 4 Notes: Polytopes
This lecture covers: the Matching polytope, total dual integrality, and Hilbert bases.
1 The Matching Polytope and Total Dual Integrality
In this section we introduce the matching polytope as the convex hull of incidence vector
Lecture 5 Notes: Dual Programs
1 Total Dual Integrality
Consider the linear program defined as
max
c x
s.t.
Ax b
(1)
where A and b are rational and the associate dual program
min
y b
s.t.
A y=c
(2)
y 0
Definition
vectors c the
1The system of inequalitiesA
Lecture 6 Notes: Ordered Sets
x(v)1v V
S

x(E(S) 2
, forSodd
x 0.
S
One may wonder whether we need all blossom inequalities x(E(S)
2
. In other words,
which of these inequalities define facets of the polytope and are essential in the description.

Lecture 2 Notes: Recapitulation
1
Recapitulation
Recall the following
essential defitions and facts from the last lecturematching.A in an
undirected
graph G is a set of edges, no two
of which share a common endpoint. Givenad a matcgrphing
if it is the end
Lecture 9 Notes: Matroid Theory
I
S
) is
Definition 1AmatroidM= (S,
I 2 , known
a
as theindependent sets, satisfying
1. If I I and JI then J I.
2. If I, J I and J>I, then
indicates union.
ground Sset ogether with a collection of sets
finite
the follow
Lecture 8 Notes: Bessy Theorem
This lecture covers the proof of the BessyThomasse Theorem, formerly known as the Gallai
Conjecture. Also, we discuss the cyclic stable set polytope, and show that it is totally dual integral (TDI)
(see lecture 5 for more o