AMS 550.666: Combinatorial Optimization
Solutions to Homework Problems - Week I
For any digraph D = (V, E ) with source r and sink s, (D) denotes the length of the shortest
path from r to s in D and (D) denotes the set of edges cfw_e E : e is on SOME shor
AMS 550.666: Combinatorial Optimization
Homework Problems - Week X
Let P = cfw_x Rn : Ax b. PI will denote the convex hull of P Zn . We say that a valid
inequality w x t for PI has Chvtal rank k if k is the smallest natural number such that w x t
a
has a
AMS 550.666: Combinatorial Optimization
Homework Problems - Week IX
1. Let w = (w1 , w2 , . . . , wn ) Zn . We observed in class that if t Z, the hyperplane cfw_x Rn :
w x = t does not contain any integral points x Zn . Does this mean that w x t always
co
AMS 550.666: Combinatorial Optimization
Homework Problems - Week XII
A graphic matroid is a matroid that can be represented as the set of forests in a graph.
1. Let (X, I ) be a matroid. Two elements x, y X are called parallel if cfw_x, y is a circuit. S
AMS 550.666: Combinatorial Optimization
Homework Problems - Week II
In the following G = (V, E ) is an undirected graph. For a subset A V , oc(V \ A) denotes the
number of odd connected components in G \ A.
1. (Problem 3.16 from textbook) Find the maximum
AMS 550.666: Combinatorial Optimization
Homework Problems - Week III
In the following G = (V, E ) is an undirected graph. For a subset A V , oc(V \ A) denotes the
number of odd connected components in G \ A.
1. Show that the following two statements are e
AMS 550.666: Combinatorial Optimization
Homework Problems - Week IV
For any two vectors x, y Rd , x y means the inequality holds for each component: xi yi
for all i = 1, . . . , d. Similarly, x > y means xi > yi for all i = 1, . . . , d.
For the following
AMS 550.666: Combinatorial Optimization
Homework Problems - Week V
For the following problems, A, M Rmn will be m n matrices, and b Rm . An ane
subspace is the set of solutions to a a system of linear equations, i.e., cfw_x Rn : Ax = b is an ane
subspace
AMS 550.666: Combinatorial Optimization
Homework Problems - Week VI
We say w x t is a valid inequality for a polyhedron P if w x t for all x P . We say
that w x t is equivalent to w x t if there exists > 0 such that w = w and t = t.
We say w x t is a fa
AMS 550.666: Combinatorial Optimization
Midterm Solutions
There are 5 questions on this test.
You have to return the test to me or Steve by 4:30 pm on Tuesday, October 29,
2013. No answers will be accepted after this deadline. You can choose to send
you
AMS 550.666: Combinatorial Optimization
Midterm
There are 5 questions on this test.
You have to return the test to me or Steve by 4:30 pm on Tuesday, October 29,
2013. No answers will be accepted after this deadline. You can choose to send
your answers