Birla Institute of Technology & Science, Pilani  Hyderabad
Mathematical Programming
CPE 360

Winter 2012
The smallest networks on which the FordFulkerson
maximum ow procedure may fail to terminate
Uri Zwick
July 11, 1993
Abstract
It is widely known that the FordFulkerson procedure for nding the maximum ow in a network
need not terminate if some of the capa
Birla Institute of Technology & Science, Pilani  Hyderabad
Mathematical Programming
CPE 360

Winter 2012
EndTerm 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
Birla Institute of Technology & Science, Pilani  Hyderabad
Mathematical Programming
CPE 360

Winter 2012
International Journal of Game Theory (1994) 23:7583
StrategyProofness 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
Birla Institute of Technology & Science, Pilani  Hyderabad
Mathematical Programming
CPE 360

Winter 2012
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
Birla Institute of Technology & Science, Pilani  Hyderabad
Mathematical Programming
CPE 360

Winter 2012
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
Birla Institute of Technology & Science, Pilani  Hyderabad
Mathematical Programming
CPE 360

Winter 2012
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
Birla Institute of Technology & Science, Pilani  Hyderabad
Mathematical Programming
CPE 360

Winter 2012
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
Birla Institute of Technology & Science, Pilani  Hyderabad
Mathematical Programming
CPE 360

Winter 2012
EndTerm 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
Birla Institute of Technology & Science, Pilani  Hyderabad
Mathematical Programming
CPE 360

Winter 2012
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
Birla Institute of Technology & Science, Pilani  Hyderabad
Mathematical Programming
CPE 360

Winter 2012
15210 (Spring 2013)
Parallel and Sequential Data Structures and Algorithms Lecture 13
Lecture 13 Shortest Weighted Paths II
Parallel and Sequential Data Structures and Algorithms, 15210 (Spring 2013)
Lectured by Umut Acar February 26, 2013
What was cove
Birla Institute of Technology & Science, Pilani  Hyderabad
Mathematical Programming
CPE 360

Winter 2012
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
Birla Institute of Technology & Science, Pilani  Hyderabad
Mathematical Programming
CPE 360

Winter 2012
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).
Birla Institute of Technology & Science, Pilani  Hyderabad
Mathematical Programming
CPE 360

Winter 2012
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 ndimensional row vector, b
Birla Institute of Technology & Science, Pilani  Hyderabad
Mathematical Programming
CPE 360

Winter 2012
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 ndimensional row vector, b an mdimensional c
Birla Institute of Technology & Science, Pilani  Hyderabad
Mathematical Programming
CPE 360

Winter 2012
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
Birla Institute of Technology & Science, Pilani  Hyderabad
Mathematical Programming
CPE 360

Winter 2012
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
Birla Institute of Technology & Science, Pilani  Hyderabad
Mathematical Programming
CPE 360

Winter 2012
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
Birla Institute of Technology & Science, Pilani  Hyderabad
Mathematical Programming
CPE 360

Winter 2012
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
Birla Institute of Technology & Science, Pilani  Hyderabad
Mathematical Programming
CPE 360

Winter 2012
EndTerm 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
Birla Institute of Technology & Science, Pilani  Hyderabad
Mathematical Programming
CPE 360

Winter 2012
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
Birla Institute of Technology & Science, Pilani  Hyderabad
Mathematical Programming
CPE 360

Winter 2012
MidTerm 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
/
Birla Institute of Technology & Science, Pilani  Hyderabad
Mathematical Programming
CPE 360

Winter 2012
EndTerm 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
Birla Institute of Technology & Science, Pilani  Hyderabad
Mathematical Programming
CPE 360

Winter 2012
MidTerm 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
Birla Institute of Technology & Science, Pilani  Hyderabad
Mathematical Programming
CPE 360

Winter 2012
MidTerm 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
Birla Institute of Technology & Science, Pilani  Hyderabad
Mathematical Programming
CPE 360

Winter 2012
MidTerm 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
Birla Institute of Technology & Science, Pilani  Hyderabad
Mathematical Programming
CPE 360

Winter 2012
MidTerm 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
Birla Institute of Technology & Science, Pilani  Hyderabad
Mathematical Programming
CPE 360

Winter 2012
MidTerm 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
Birla Institute of Technology & Science, Pilani  Hyderabad
Mathematical Programming
CPE 360

Winter 2012
MidTerm 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
Birla Institute of Technology & Science, Pilani  Hyderabad
Mathematical Programming
CPE 360

Winter 2012
MidTerm 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
Birla Institute of Technology & Science, Pilani  Hyderabad
Mathematical Programming
CPE 360

Winter 2012
MidTerm 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