### 2021　Combinatorial Algorithms

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Undergraduate major in Mathematical and Computing Science
Instructor(s)
Sumita Hanna  Yamashita Makoto
Course component(s)
Lecture    (ZOOM)
Day/Period(Room No.)
Mon5-6(W321)  Thr5-6(W321)
Group
-
Course number
MCS.T322
Credits
2
2021
Offered quarter
4Q
Syllabus updated
2021/3/19
Lecture notes updated
-
Language used
Japanese
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### Course description and aims

This course will cover representative combinatorial optimization problems and their methods to solve the problems. Real world problems are modeled as optimization problems by extracting the vital constraints and objectives. It is practically important to solve the optimization problems efficiently. For this end, we need to investigate and utilize a specific property and structure of each problem. This course describes fundamental "combinatorial" optimization problems and their algorithms using their own properties. In addition, this course explains the theory on combinatorial structures and algorithms.

The aim of this course is to give knowledge on fundamental combinatorial optimization problems and their algorithms, and give mathematical tools to argue the performance of the algorithms.

### Student learning outcomes

By the end of this course, students will be able to:
1) Recognize fundamental combinatorial optimization problems appearing in real world situations.
2) Have a vague idea of methods to solve those combinatorial optimization problems.
3) Explain roughly the performance of methods from a theoretical perspective.

### Keywords

combinatorial optimization, algorithm, online optimization, dynamic programming, modeling

### Competencies that will be developed

 ✔ Specialist skills Intercultural skills Communication skills Critical thinking skills Practical and/or problem-solving skills

### Class flow

In each class we focus on one combinatorial optimization problem. Students learn its applications and basic methods to solve the problem.

### Course schedule/Required learning

Course schedule Required learning
Class 1 Introduction to combinatorial optimization Evaluation of the grade will be explained
Class 2 Shortest path problem
Class 3 Maximum flow problem
Class 4 Minimum cost flow problem
Class 5 Maximum matching problem
Class 6 Stable marriage problem
Class 7 Matroid
Class 8 Review of Classes 1-7 and advanced topics
Class 9 Knapsack problem and approximation algorithms
Class 10 Knapsack problem and dynamic programming
Class 11 Traveling salesman problem
Class 12 Submodular function maximization problem
Class 13 Online problems
Class 14 Secretary problem Final report

### Out-of-Class Study Time (Preparation and Review)

To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.

### Textbook(s)

We do not assign textbooks. We will use course materials that are based on the following books.

### Reference books, course materials, etc.

B. Korte and J. Vygen, "Combinatorial Optimization: Theory and Algorithms", Springer, 2018.
A. Schrijver, "Combinatorial Optimization: Polyhedra and Efficiency", Springer, 2003.
M. Shigeno, "Network Optimization and Algorithms （ネットワーク最適化とアルゴリズム)", Asakura-Shoten, 2010. (Japanese)
Y. Kawahara and K. Nagano, "Submodular Optimization and Machine Learning (劣モジュラ最適化と機械学習)", Kodansya, 2015. (Japanese)

All course materials will be uploaded on OCW-i.

### Assessment criteria and methods

Students will be assessed on the final report and quizzes.

### Related courses

• MCS.T302 ： Mathematical Optimization

### Prerequisites (i.e., required knowledge, skills, courses, etc.)

No prerequisites are necessary, but enrollment in the related course (Mathematical Optimization) is desirable.