Name: '\oL-r'z*:
Extra-Credit Quiz - Math
409
1. A stable set ^9 in a graph G : (V,E) is a set of vertices such that there
vertices in ^9.
(a) List all the stable sets in the graph
4(3
axe no edges be
Problem Set 4: Totally Unimodular Matrices
(1) Which of the following matrices are totally unimodular. You must justify your answer.
(a)
1 1
1
1
(b)
1 1 0
0 1 1
1 0 1
(c)
1 1 1
0
1
0
0
1
0
1
0 1
1. Easy Interger LPs
Consider the linear integer program
IP minimize cT x
subject to Ax = b, 0 x , and x Zn ,
where c Rm , A Zmn , and b Zm . The LP relaxation of IP is the linear program
IP relax min
Problem Set 6: Flows in a Digraph
(1) Apply the ow augmenting path algorithm to solve the max-ow problems given by
the capacitated digraphs below. In each graph the source node is the node number 1
an
Problem Set 2: The Branch and Bound Algorithm
(1) Solve the following integer knapsack problems by the branch and bound algorithm.
(a)
maximize 8x1 + 11x2 + 6x3
subject to 5x1 + 7x2 + 4x3 14
x cfw_0,
v2
e52
e12
e24
e32
v1
v4
v5
e35
e12
e13
e45
v3
Figure 1. A graph with 5 vertices.
1. Graphs, Digraphs, and Networks
1.1. The Basics. A graph is a mathematical structure comprised of two classes of obj
i fgf p x x x y p fe 1PtQQqswte j tol D8lz1o1su xheswh2d v v v v v r 2z w tv tz w v 2x v v w v 2 w tz tu w 8u 2z v v v v w v x v w tv w tz 2z 8u v v w v u 2z w tz t w 2x v sedpDsi| t v v v v w p f
Problem Set 1: Modeling Integer Programming Problems
(1) Suppose that you are interested in choosing to invest in one or more of 10 investment
opportunities. Use 0-1 variables to model the following l
1. Introduction to Discrete Optimization
In nite dimensional optimization we are interested in locating solutions to the problem
P : minimize
xX
subject to
f0 (x)
x .
where X is the variable space (or
lcn yb fff b b b m m TpxTy2gfqTejpgte EgrfhB n rf xv f rd r q rt W t vj t r q v g Ev frGg1Tp y ph Tr x f i r y g g v r kreshpppst2pep#f12hi t %[email protected] pBTEx f f4fqx krpgtr fhTiqgq jfkr#eptv jB f i
Discrete Optimization
Spring 2017
Thomas Rothvoss
B
D
A
C
E
Last changes: March 24, 2017
2
Contents
1 Introduction to Discrete Optimization
1.1 Algorithms and Complexity . . . . . . . . .
1.1.1 Comple
EXAMPLE OF TECHNIQUE 2
Given X =(matchings, paths, trees, cycles, etc.), let P = conv(X). Let Q = cfw_x :
Ax b. We want to show that P = Q. Showing that P Q is usually easy. The
other way can be trick
Math 409 Midterm exam
May 4, 2015
INSTRUCTIONS READ THIS NOW
Name:
NetID:
OFFICIAL
USE ONLY
This test has 5 problems worth a total of 50 points. Look over your test
booklet right now. If you find any
Math 409 Midterm exam
May 4, 2015
Name:
NetID:
INSTRUCTIONS READ THIS NOW
OFFICIAL
USE ONLY
This test has 5 problems worth a total of 50 points. Look over your test
booklet right now. If you find any
Math and Your Love Life
Annie Raymond
University of Washington
March 21, 2016
The stable marriage problem
Annie Raymond (University of Washington)
Math and Your Love Life
March 21, 2016
2 / 15
The sta
Massachusetts Institute of Technology
Michel X. Goemans
18.433: Combinatorial Optimization
March 1, 2015
3. Linear Programming and Polyhedral Combinatorics
Summary of what was seen in the introductory
5. Matroid optimization
5.3.1
April 8, 2015
8
Span
The following definition is also motivated by the linear algebra setting.
Definition 5.1 Given a matroid M = (E, I) and given S E, let
span(S) = cfw_
Massachusetts Institute of Technology
Michel X. Goemans
18.433: Combinatorial Optimization
March 30, 2015
4. Lecture notes on flows and cuts
4.1
Maximum Flows
Network flows deals with modelling the fl
5. Matroid optimization
Exercise 5-6.
April 8, 2015
5
Let M = (E, I) be a matroid. Let k N and define
Ik = cfw_X I : |X| k.
Show that Mk = (E, Ik ) is also a matroid. This is known as a truncated matr
Massachusetts Institute of Technology
Michel X. Goemans
18.433: Combinatorial Optimization
February 4th, 2015
1. Lecture notes on bipartite matching
Matching problems are among the fundamental problem
1. Lecture notes on bipartite matching
1.1.1
February 4th, 2015
6
Halls Theorem
Halls theorem gives a necessary and sufficient condition for a bipartite graph to have a
matching which saturates (or ma
Massachusetts Institute of Technology
Michel X. Goemans
18.433: Combinatorial Optimization
February 4th, 2015
2. Lecture notes on non-bipartite matching
Given a graph G = (V, E), we are interested in
3. Linear Programming and Polyhedral Combinatorics
March 1, 2015
13
hibit n affinely independent permutations (and prove that they are affinely independent).)
Exercise 3-12. A stable set S (sometimes,
3. Linear Programming and Polyhedral Combinatorics
March 1, 2015
10
Theorem 3.10 If the face associated with aTi x bi for i I< is not a facet then the
inequality is redundant.
And this one shows that
Massachusetts Institute of Technology
Michel X. Goemans
18.433: Combinatorial Optimization
April 8, 2015
5. Matroid optimization
5.1
Definition of a Matroid
Matroids are combinatorial structures that
e l o nl o n l o 2n a l o q o n l o | o l o l o n o o
e l o
$ " ! 43210)('&%#
y Eer rq rq tkyasr f s r s r gVRfuVs2Q w z w vh X TR reatkr xr o q f l n l d f n f 2n 2pl l l X j h d 2e `RgkXeWi agsr