Mathematical Programming and Matching

K. Subramani LDCSEE, West Virginia University, Morgantown, WV {ksmani@csee.wvu.edu}

1 General Information

(a) Instructor: K. Subramani.

(b) Meeting Times: Mon 10 : 00 am - 11 : 00 am.

Location: 749 ESB-E.

(c) Contact Information: 749 ESB, ksmani@csee.wvu.edu.

(d) Office Hours: Tu, Th - 10 : 00 am - 11 : 00 am.

(e) Textbook - [IC93] and [KV00].

(f) Assessment:

(a) Presentations (2) - You will be required to present allotted material. The presentation will be graded on the basis of clarity, comprehensiveness and effectiveness. The presentations will be worth 40% of your grade.

(b) Assignments - You will be given several assignments during the course. These assignments will be worth 60% of your grade.

A maximum of 5 bonus points will be awarded for class performance.

(g) Grade Boundaries (a) A: 80 and up

(b) B: 65 − 79

(c) C: 50 − 64

(d) D: 45 − 49

(e) F: 0 − 44

(h) Grading policy - If you have any questions about the grading, you must contact the instructor within two days of your paper being returned.

(i) Course Objectives - The objectives of this course are as follows:

(a) To ensure that students have a thorough grasp of the fundamentals of linear and integer programming.

(b) To apply linear and integer programming techniques to Matching theory.

(j) Learning Outcomes - Upon successful completion of this course, the student should be able to: (a) Model real-world problems using linear and integer programs.

(b) Solve programs using the Simplex method.

(c) Grasp the fundamentals of matching theory.

2 Syllabus Sketch

2.1 Mathematical Preliminaries

Vector Addition, Multiplication of a Vector by a Scalar,Vector Multiplication, Norm of a Vector, Special Vector Types, Linear Dependence and Independence, Spanning Sets and Bases, Matrix Addition, Multiplication by a Scalar, Matrix Multiplication, Special Matrices, Determinants, The Inverse of a Matrix, The Rank of a Matrix, The Solution of Simultaneous Linear Equations, A Unique Solution of A · x⃗ = b⃗, An Infinite Number of Solutions to A · x⃗ = b⃗.

2.2 Overview

The Linear Decision Model, Applications of Linear Programming.

2.3 The Conventional Linear Programming Model

Models and Model Types, General Guidelines in Model Building, Definitions, Basic Steps in Linear Programming Model Formulation, The General Form of the Linear Programming Model, Assumptions of the Linear Programming Model, Examples of Linear Programming Model Formulation Model Validity.

2.4 Foundations of the Simplex Method

Converting a Linear Program into Standard Form, Graphical Solution of 2-dimensional Linear Programs, Convex Sets and Polyhedral Sets, Extreme Points, Extreme Directions, and Optimality, Basic Feasible Solutions and Extreme Points.

2.5 The Simplex Method: Tableau and Computation

Algebra of the Simplex Method, The Simplex Method in Tableau Form, Finding the initial Basic Feasible Solution, Unrestricted Variables and Variables with Negative Lower Bounds, Degeneracy and Cycling.

2.6 Duality and Sensitivity Analysis

Formulation of the Linear Programming Dual, Relationships in Duality, Geometric Interpretation of Optimality Conditions, Economic Interpretations of the Dual, The Dual Simplex Algorithm.

2.7 Integer Programming

Introduction, Graphical Solution of 2-Dimensional Integer Programs, Computational Complexity, Formulating Integer Programing Problems, Branch-and-Bound Enumeration, Implicit Enumeration, Cutting Plane Methods.

2.8 Matching

Minimum cost flows, Maximum matching, Weighted matching, Non-bipartite matching, b-matching and T -joins.

I would like to reiterate that this is a sketch of the topics that we will be covering. For various reasons, I may choose to drop a mentioned topic or cover a new topic. In such cases, advance notice will be given.

3 Academic Integrity Statement

The integrity of the classes offered by any academic institution solidifies the foundation of its mission and cannot be sacrificed to expediency, ignorance, or blatant fraud. Therefore, I will enforce rigorous standards of academic integrity in all aspects and assignments of this course. For the detailed policy of West Virginia University regarding the definitions of acts considered to fall under academic dishonesty and possible ensuing sanctions, please see the Student Conduct Code

at http://www.arc.wvu.edu/admissions/integrity.html  . Should you have any questions about possibly

improper research citations or references, or any other activity that may be interpreted as an attempt at academic dishonesty, please see me before the assignment is due to discuss the matter.

4 Social Justice Statement

West Virginia University is committed to social justice. I concur with that commitment and expect to foster a nurturing learning environment, based upon open communication, mutual respect and non-discrimination. Our University does not discriminate on the basis of race, sex, age, disability, veteran status, religion, sexual orientation, color or national origin. Any suggestions to further such a positive and open environment in this class will be appreciated and given serious consideration. If you are a person with a disability and anticipate needing any type accommodation, in order to participate in this class, please advise me of the same and make appropriate arrangements with Disability Services (293 − 6700).

If you feel that you are being treated inappropriately or unfairly in any way, please feel free to bring your concerns to my attention; rest assured that doing so will not prejudice the grading process. In return, I expect you to behave professionally and ethically.

References

[IC93] James P. Ignizio and Tom P. Cavalier. Linear Programming. Prentice Hall, 1993.

[KV00] B. Korte and J. Vygen. Combinatorial Optimization. Number 21 in Algorithms and Combinatorics. Springer-Verlag, New York, 2000.