The smallest networks on which the Ford-Fulkerson
maximum ow procedure may fail to terminate
Uri Zwick
July 11, 1993
Abstract
It is widely known that the Ford-Fulkerson procedure for nding the maximum ow in a network
need not terminate if some of the capa

End-Term Examination
Mathematical Programming with Applications to Economics
Total Score: 50; Time: 3 hours
Please write clearly with precise arguments. Avoid unnecessary elaboration.
1. In some stage of the simplex method of a linear program, the followi

International Journal of Game Theory (1994) 23:75-83
Strategy-Proofness and the Strict Core in a Market with
Indivisibilities 1
JINPENG MA
Department of Economics, SUNY at Stony Brook, NY 11794, USA
Abstract." We show that, in markets with indivisibilitie

Linear Programming and Vickrey Auctions
Sushil Bikhchandani
Sven de Vries
Rakesh V. Vohra
James Schummer
May 15, 2001
Abstract
The Vickrey sealed bid auction occupies a central place in auction
theory because of its eciency and incentive properties. Imple

Graph Theory and Cayleys Formula
Chad Casarotto
August 10, 2006
Contents
1 Introduction
1
2 Basics and Denitions
1
3 Cayleys Formula
4
4 Pr fer Encoding
u
5
5 A Forest of Trees
7
1
Introduction
In this paper, I will outline the basics of graph theory in a

Mathematical Programming with
Applications to Economics
Debasis Mishra
January 4, 2015
Schedule: Every Monday and Wednesday 9:30 AM - 11:30 AM.
Aim: The course will cover some fundamental concepts of mathematical programming,
graph theory, and discrete op

Theory of Linear Programming
Debasis Mishra
March 19, 2014
1
Introduction
Optimization of a function f over a set S involves nding the maximum (minimum) value of
f (objective function) in the set S (feasible set). Properties of f and S dene various types

End-Term Examination
Mathematical Programming with Applications to Economics
Total Score: 60
1. Suppose G = (N, E, w) is a weighted strongly connected directed graph with w : E
R. Denote the shortest path from node i to node j in G as s(i, j). Show that

Theory of Linear Programming
Debasis Mishra
April 6, 2011
1
Introduction
Optimization of a function f over a set S involves nding the maximum (minimum) value of
f (objective function) in the set S (feasible set). Properties of f and S dene various types
o

15-210 (Spring 2013)
Parallel and Sequential Data Structures and Algorithms Lecture 13
Lecture 13 Shortest Weighted Paths II
Parallel and Sequential Data Structures and Algorithms, 15-210 (Spring 2013)
Lectured by Umut Acar February 26, 2013
What was cove

Basic Graph Theory
with Applications to Economics
Debasis Mishra
February 6, 2014
1
What is a Graph?
Let N = cfw_1, . . . , n be a nite set. Let E be a collection of ordered or unordered pairs of
distinct 1 elements from N. A graph G is dened by (N, E). T

Basic Graph Theory
with Applications to Economics
Debasis Mishra
February 23, 2011
1
What is a Graph?
Let N = cfw_1, . . . , n be a nite set. Let E be a collection of ordered or unordered pairs of
distinct 1 elements from N. A graph G is dened by (N, E).

Integer Programming and Submodular Optimization
Debasis Mishra
April 15, 2014
1
Integer Programming
1.1 What is an Integer Program?
Suppose that we have a linear program
maxcfw_cx : Ax b, x 0
(P)
where A is an m n matrix, c an n-dimensional row vector, b

Theory of Integer Programming
Debasis Mishra
April 20, 2011
1
1.1
Integer Programming
What is an Integer Program?
Suppose that we have a linear program
maxcfw_cx : Ax b, x 0
(P)
where A is an m n matrix, c an n-dimensional row vector, b an m-dimensional c

Introduction to Convex Sets
with Applications to Economics
Debasis Mishra
March 5, 2014
1
Convex Sets
A set C Rn is called convex if for all x, y C, we have x + (1 )y C for all [0, 1].
The denition says that for any two points in set C, all points on the

Introduction to Convex Sets
with Applications to Economics
Debasis Mishra
March 21, 2011
1
Convex Sets
A set C Rn is called convex if for all x, y C, we have x + (1 )y C for all [0, 1].
The denition says that for any two points in set C, all points on the

Theory of Mechanism Design - Mid Term Examination
September, 2014; Duration: 2 hours; Total marks: 45.
Explain your answers clearly, but avoid unnecessary elaboration.
Notations and concepts are as dened in the classnote.
1. Let A be the set of alternativ

Final Examination
Mathematical Programming with Applications to Economics
Total Score: 100
Note: All matrix multiplications are done by taking appropriate transposes, which is not
shown in notations. You may draw the gures in pencil but use a pen otherwis

End-Term Examination
Mathematical Programming with Applications to Economics
Total Score: 50
1. Consider an undirected graph G = (N, E), where N is the set of vertices and E is the
set of edges. A matching of G is a subset of edges S E such that no two ed

A short proof of the BergeTutte Formula and the
GallaiEdmonds Structure Theorem
Douglas B. West
Abstract
We present a short proof of the BergeTutte Formula and the GallaiEdmonds
Structure Theorem from Halls Theorem.
The fundamental theorems on matchings i

Mid-Term Examination Key
Mathematical Programming with Applications to Economics
Total Score: 45
1. Let G = (N, E) be an undirected graph which satises the property that for any
i, j N with cfw_i, j E, we have degree(i) + degree(j) |N| 1. Show that G is
/

End-Term Examination - Spring 2014
Mathematical Programming with Applications to Economics
Total Score: 45; Time: 3 hours
1. While solving for the optimal solution of a linear program, we encountered the following
dictionary in the second phase of the sim

Mid-Term Examination
Mathematical Programming with Applications to Economics
Total Score: 50
1. Show that if every cycle of an undirected graph has even number of edges then it is a
bipartite graph. (5 marks)
Answer: Suppose we have an undirected graph in

Mid-Term Examination Key
Mathematical Programming with Applications to Economics
Total Score: 100
Note: All matrix multiplications are done by taking appropriate transposes, which is not
shown in notations. You may draw the gures in pencil but use a pen o

Mid-Term Examination - Spring 2014
Mathematical Programming with Applications to Economics
Total Score: 45; Time: 3 hours
1. Let G = (N, E) be a directed graph. Dene the indegree of a vertex i N as the
number of edges that are coming into i. Formally,
ind

Mid-Term Examination
Mathematical Programming with Applications to Economics
Total Score: 100
Note: All matrix multiplications are done by taking appropriate transposes, which is not
shown in notations. You may draw the gures in pencil but use a pen other

Mid-Term Examination Solution
Mathematical Programming with Applications to Economics
Total Score: 50; Time: 3 hours
1. Consider an undirected graph G = (N, E) such that every vertex has degree greater
than 1. Is it possible that G contains no cycles? Exp

Mid-Term Examination (Answer Sketch)
Mathematical Programming with Applications to Economics
Total Score: 50; Time: 3 hours
1. Consider an undirected graph G with n vertices and e edges. Suppose G is connected.
Show the following.
(a) e n 1. (2 marks)
Ans

Mid-Term Examination
Mathematical Programming with Applications to Economics
Total Score: 50; Time: 3 hours
1. Consider an undirected graph G = (N, E) such that every vertex has degree greater
than 1. Is it possible that G contains no cycles? Explain your

Mid-Term Examination
Mathematical Programming with Applications to Economics
Total Score: 50
1. Show that if every cycle of an undirected graph has even number of edges then it is a
bipartite graph. (5 marks)
2. Suppose G = (N, E, w) is a strongly connect