### Course Meeting Times

Lectures: 2 sessions / week, 1.5 hours / session

### Description

In this graduate-level course, we will be covering advanced topics in combinatorial optimization. We will start with non-bipartite matchings and cover many results extending the fundamental results of matchings, flows and matroids. The emphasis is on the derivation of purely combinatorial results, including min-max relations, and not so much on the corresponding algorithmic questions of how to find such objects (although we will be discussing a few algorithmic issues, such as minimizing submodular functions). The course will rely heavily on the recent 3-volume textbook by Lex Schrijver on Combinatorial Optimization. The intended audience consists of Ph.D. students interested in optimization, combinatorics, or combinatorial algorithms.

### Prerequisites

Students taking this course should have had prior exposure to combinatorial optimization, for example, by taking 18.433 (Combinatorial Optimization ) or a similar course. This course assumes knowledge of bipartite matchings, spanning trees, and similar basic notions in combinatorial optimization.

### Grading

The grade assigned is based on students preparing scribe notes of the lectures and on class participation. The job of the scribe is to prepare a good set of lecture notes based on what was covered in lecture and additional readings.

### Textbook

Schrijver, Alexander. *Combinatorial Optimization: Polyhedra and Efficiency.* New York, NY: Springer-Verlag, 2003. ISBN: 3540443894.

The textbook is the 3-volume book, but we will be covering only some of the 83 chapters. This is a marvelous book with lots of results, references and concise proofs. I will assume familiarity with many basic results in combinatorial optimization.

### Topics

Here is a preliminary (and partial) list of topics to be discussed:

- Ear decompositions
- Nonbipartite matching
- Gallai-Milgram and Bessy-Thomasse theorems on partitioning/covering graphs by directed paths/cycles
- Minimization of submodular functions
- Matroid intersection, Polymatroid intersection
- Jump systems
- Matroid union
- Matroid matching, path matchings
- Packing trees and arborescences
- Packing directed cuts and the Lucchesi-Younger theorem
- Submodular flows and the Edmonds-Giles theorem
- Graph orientation
- Connectivity tree and connectivity augmentation
- Multicommodity flows