CME 305: Discrete Mathematics and Algorithms
Instructor: Reza Zadeh (rezab@stanford.edu)
HW#1 Due at the beginning of class Thursday 01/21/16
1. Prove that at least one of G and G is connected. Here, G is a graph on the vertices of
G such that two vertice

CME 305: Discrete Mathematics and Algorithms
Instructor: Reza Zadeh (rezab@stanford.edu)
HW#4 Due at the beginning of class Thursday 03/13/14
1. We are given n jobs that each take one unit of processing time. All jobs are available
at time 0, and job j ha

CME 305: Discrete Mathematics and Algorithms
Instructor: Professor Amin Saberi (saberi@stanford.edu) HW#2 Due 02/11/11
1. In this problem we use the well known simplex algorithm to prove the strong duality theorem for linear programs. For matrix A Rmn and

CME 305: Discrete Mathematics and Algorithms
Instructor: Professor Amin Saberi (saberi@stanford.edu) HW#2 Solutions
1. Let T be a spanning tree of a graph G with an edge cost function c. We say that T has the cycle property if for any edge e T , c(e ) c(e

CME 305: Discrete Mathematics and Algorithms
Instructor: Reza Zadeh (rezab@stanford.edu)
HW#2 Due at the beginning of class Thursday 02/05/15
1. (Kleinberg Tardos 7.27) Some of your friends with jobs out West decide they really
need some extra time each d

CME 305: Discrete Mathematics and Algorithms
Instructor: Reza Zadeh (rezab@stanford.edu)
HW#2 Due at the beginning of class Thursday 02/06/14
1. Exhibit a graph G = (V, E) where there are an exponential (in |V | = n, the number
of nodes) number of minimum

CME 305: Discrete Mathematics and Algorithms
Instructor: Professor Amin Saberi (saberi@stanford.edu) HW#1 Solutions
1. (Lovasz, Pelikan, and Vesztergombi 7.3.9) Prove that at least one of G and G is connected. Here, G is a graph on the vertices of G such

CME 305: Discrete Mathematics and Algorithms
TA: Kun Yang (kunyang@stanford.edu)
Workshop#2 on Friday 02/21/14
There are some common techniques to approach a discrete math problem. Let us go over
these techniques and discuss how to apply them to new probl

CME 305 Problem Session 1
2/10/2014
1. Recall the definition of a bipartite graph. Let G(V, E) be a graph and (A, B) be a partition of V . We
say that G is bipartite if all edges in E have one end-point in A and the other in B. More precisely,
for all (u,

CME 305: Discrete Mathematics and Algorithms
Instructor: Professor Amin Saberi (saberi@stanford.edu)
Midterm 02/28/10
Problem 1. Show that a graph has a unique minimum spanning tree if, for every cut of the
graph, the edge with the smallest cost across th

CME 305: Discrete Mathematics and Algorithms
Instructor: Reza Zadeh (rezab@stanford.edu)
Midterm Review Session 02/10/15
1. Hamiltonian Paths and Cycles
(a) Show that determining whether a graph contains a Hamiltonian Path is at least
as hard as determini

CME 305: Discrete Mathematics and Algorithms
Instructor: Reza Zadeh (rezab@stanford.edu)
Problem Session #1 02/10/15
1. (Kleinberg Tardos 11.10) Suppose you are given an n by n grid graph G with
vertex weights w(v) 0 that are all distinct and integer. The

CME 305: Discrete Mathematics and Algorithms
Instructor: Professor Amin Saberi (saberi@stanford.edu)
Midterm 02/17/11
This exam is closed notes/books/laptops. You will have until the end of class to
ask any questions clarifying any of the problems. You ma

Discrete Mathematics and Algorithms Qualifying Exam
September 2012
1. Imagine n cars, each of which travels at a different maximum speed. Initially, the cars
are queued in uniform random order at the start of a semi-infinite, one lane highway.
Each car dr

CME 305: Discrete Mathematics and Algorithms
TA: Kun Yang (kunyang@stanford.edu)
Workshop#2 on Friday 02/21/14
There are some common techniques to approach a discrete math problem. Let us go over
these techniques and discuss how to apply them to new probl

ICME QUALIFYING EXAM
DISCRETE MATHEMATICS AND ALGORITHMS
Let G(V, E) be a connected d-regular graph, v0 V (G), and assume that at each node,
the ends of the edges incident with the node are labelled 1, 2, d. A traverse sequence
(for this graph, starting p

CME 305: Discrete Mathematics and Algorithms
TA: Kun Yang (kunyang@stanford.edu)
1. Consider an optimization version of the Hitting Set Problem defined as follows. We
are given a set A = cfw_a1 , ., an and a collection B1 , ., Bm of subsets of A. Also, e

CME 305: Discrete Mathematics and Algorithms
Instructor: Reza Zadeh (rezab@stanford.edu)
Problem Session #1 02/11/15
1. (Kleinberg Tardos 11.10) Suppose you are given an n by n grid graph G with
vertex weights w(v) 0 that are all distinct and integer. The

CME 305: Discrete Mathematics and Algorithms
Instructor: Reza Zadeh (rezab@stanford.edu)
Midterm Review Session 02/10/15
Note that these solutions are compact and only provide the key ideas in answer of the
question. They should not be considered model so

Discrete Mathematics and Algorithms Qualifying Exam
June 2013
1. (30 points) Consider the following problem: Given n items with sizes a1 , a2 , an all
in (0, 1], find a packing in unit size bins that minimizes the number of bins used.
(a) Prove that the f

CME 305 Problem Session 1
2/10/2014
1. Recall the definition of a bipartite graph. Let G(V, E) be a graph and (A, B) be a partition of V . We
say that G is bipartite if all edges in E have one end-point in A and the other in B. More precisely,
for all (u,

CME 305: Discrete Mathematics and Algorithms
1
Computation and Intractability
In this series of lecture notes, we have discussed several problems for which there exist polynomial time
algorithms for producing solutions min-cut, shortest s-t path, whether

CME 305: Discrete Mathematics and Algorithms
1
Random Walks and Electrical Networks
Random walks are widely used tools in algorithm design and probabilistic analysis and they have numerous
applications. Given a graph and a starting vertex, select a neighb

arXiv:0803.0929v4 [cs.DS] 18 Nov 2009
Graph Sparsication by Eective Resistances
Daniel A. Spielman
Program in Applied Mathematics and
Department of Computer Science
Yale University
Nikhil Srivastava
Department of Computer Science
Yale University
November

An O(log n/ log log n)-approximation Algorithm for the
Asymmetric Traveling Salesman Problem
Arash Asadpour
Michel X. Goemans
Aleksander Madry
Amin Saberi
Shayan Oveis Gharan
Abstract
theorem [24] shows that if the tree chosen in the rst step is
We consid

CME 305: Discrete Mathematics and Algorithms
Instructor: Reza Zadeh (rezab@stanford.edu)
HW#1 Due at the beginning of class Thursday 01/21/16
1. Prove that at least one of G and G is connected. Here, G is a graph on the vertices of
G such that two vertice

CME 305: Discrete Mathematics and Algorithms
Instructor: Reza Zadeh (rezab@stanford.edu)
HW#2 Due at the beginning of class Thursday 02/04/16
1. (Kleinberg Tardos 7.27) Some of your friends with jobs out West decide they really
need some extra time each d

CME 305: Discrete Mathematics and Algorithms
Instructor: Reza Zadeh (rezab@stanford.edu)
HW#2 Due at the beginning of class Thursday 02/05/15
1. (Kleinberg Tardos 7.27) Some of your friends with jobs out West decide they really
need some extra time each d

CME 305: Discrete Mathematics and Algorithms
Instructor: Reza Zadeh (rezab@stanford.edu)
HW#3 Due at the beginning of class Thursday 02/25/16
1. Consider a model of a nonbipartite undirected graph in which two particles (starting
at arbitrary positions) f

CME 305: Discrete Mathematics and Algorithms
Instructor: Reza Zadeh (rezab@stanford.edu)
HW#3 Due at the beginning of class Thursday 02/26/15
1. Consider a model of a nonbipartite undirected graph in which two particles (starting
at arbitrary positions) f