Graph Theory, Discrete Mathematics and Optimization

A.Y. 2019/2020
Overall hours
MAT/09 SECS-S/06
Learning objectives
This course aims at introducing modern and advanced mathematical techniques useful for understanding and modeling big data structures and algorithms.
Expected learning outcomes
At the end of the course the student will be able to formalize real world problems in mathematical terms and to solve simple exercises related with linear algebra, graph theory, Markov Chains, Optimization and Decision Theory.
Course syllabus and organization

Single session

Lesson period
First trimester
Prerequisites for admission
Prerequisites for this course include a good knowledge of the mathematical tools presented in Calculus I and Linear Algebra courses and in an introductory course in Probability and Statistics, with particular reference to the following arguments: real functions of one and two variables; derivatives; partial derivatives; integrals; vectors and matrices; operations between matrices; determinants; eigenvalues and eigenvectors; probability measures; events; random variables; distributions.
Assessment methods and Criteria
The final examination consists of a written exam during which the student must solve some exercises in the format of open-ended and/or multiple answer questions, with the aim of assessing the student's knowledge and ability to solve problems related with the arguments of the course. The duration of the written exam will be proportional to the number of exercises assigned, also taking into account the nature and complexity of the exercises themselves (however, the duration is usually of 1.5 hours). In place of a single written exam given during the first examination session, the student may choose instead to take 3 midterm exams.
The outcomes of the tests will be available in the SIFA service through the UNIMIA portal and on the web site of the course.
Module Graph Theory and Discrete Mathematics
Course syllabus
Vectors, Matrices, Vector spaces, Vector equations, Linear systems, Linear models, Row Reduction and Echelon Forms, Linear Independence, Linear Transformations, the Matrix of a Linear Transformation, Matrix algebra, Determinants, Inner Product, norm, Orthogonality, Eigenvectors and Eigenvalues,
Graph theory: definitions and fundamental concepts. Directed and undirected graphs.

Probability, events and Combinatorics. Finite and discrete time Markov chains. Random walks. Graph theory: fundamental algorithms. Mining social network graphs.
Teaching methods
Face-to-face lectures, tutorials
Teaching Resources
David C. Lay, Steven R. Lay and Judi J. McDonald, Linear Algebra and Its Applications, Pearson, 2016
K. Ruohonen, Graph Theory, Course notes. (Chapters 1,2,3)
O. Haggstrom, Finite Markov Chains and Algorithmic Applications, London Mathematical Society, 2003. (Chapters1,2,3).…
J. Leskovec, A. Rajaraman, J.Ullman, Mining of Massive Datasets, Cambridge University Press, last edition (available online at ) - (Chapter 10).
Lecture notes of the teachers on the module 1 web site:
Module Optimization
Course syllabus
1.Introduction to complex decision problems: case studies; formal definitions.
2.Mathematical Programming: Karush-Kuhn-Tucker conditions.
3.Multi-objective Programming: Paretian case; Multi-Attribute Utility Theory; Analytic Hierarchy Process; ELECTRE Methods.
4.Uncertain Programming: decision making under ignorance; decision making under risk; decision theory.
Teaching methods
face to face lectures
Teaching Resources
Lecture notes on the module 2 web site:
Module Graph Theory and Discrete Mathematics
Lessons: 40 hours
Module Optimization
MAT/09 - OPERATIONS RESEARCH - University credits: 6
Lessons: 40 hours
Professor: Cordone Roberto
By appointment
DI - Via Celoria 10, Milan
Appointment by email
Office or online (by videocall)