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.
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.
combinatorial optimization, algorithm, online optimization, dynamic programming, modeling
|✔ Specialist skills||Intercultural skills||Communication skills||Critical thinking skills||Practical and/or problem-solving skills|
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|
|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 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|
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.
We do not assign textbooks. We will use course materials that are based on the following books.
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.
Students will be assessed on the final report and quizzes.
No prerequisites are necessary, but enrollment in the related course (Mathematical Optimization) is desirable.