# MA252 Combinatorial Optimisation

**Lecturer****: Vadim Lozin**

**Term(s):** Term 2

**Status for Mathematics students:** List A for mathematics

**Commitment:** 30 lectures.

**Assessment:** 2 hour exam

**Prerequisites:** No formal prerequisites. Students who have taken CS126 Design of Information Structures, CS137 Discrete Mathematics and its Applications 2, MA241 Combinatorics or CS260 Algorithms will find it helpful, but no background from these modules will be assumed.

**Leads To: **This module may be useful for students interested in taking MA241 Combinatorics, MA3J2 Combinatorics II or MA4J3 Graph Theory, but it is not a formal prerequisite for them.

**Content:** The focus of combinatorial optimisation is on finding the "optimal" object (i.e. an object that maximises or minimises a particular function) from a finite set of mathematical objects. Problems of this type arise frequently in real world settings and throughout pure and applied mathematics, operations research and theoretical computer science. Typically, it is impractical to apply an exhaustive search as the number of possible solutions grows rapidly with the "size" of the input to the problem. The aim of combinatorial optimisation is to find more clever methods (i.e. algorithms) for exploring the solution space.

This module provides an introduction to combinatorial optimisation. Our main focus is on several fundamental problems arising in graph theory and linear programming and algorithms developed to solve them. These include problems related to shortest paths, minimum weight spanning trees, linear programming, matchings, network flows, cliques, colourings, dynamic programming, multicommodity flows and matroids. We will also discuss "intractible" (e.g. NP-hard) problems.

**Preliminary Schedule:**

A preliminary schedule for the 2018-2019 edition of the module can be found here.

**Main Reference:**

- A. Schrijver,
*A Course in Combinatorial Optimization*, Unpublished Lecture Notes, 2017. Available through the link.^{1}

**Other Resources:**

- D. Bertsimas, J. N. Tsitsiklis,
*Introduction to linear optimization*, Athena Scientific, c1997 - A. Bondy and U. S. R. Murty,
*Graph Theory with Applications*, Elsevier, 1976. Available through the link. - M. Goemans,
*Combinatorial Optimisation*, Unpublished Lecture Notes. Available through the link (scroll down after clicking the link).^{1} - B. Korte and J. Vygen,
*Combinatorial Optimization: Theory and Algorithms*, Springer, 6th Edition, 2018. E-book available through the Warwick Library; click the link. - A. Schrijver,
*Advanced Graph Theory and Combinatorial Optimization*, Unpublished Lecture Notes. Available for free through the link.^{1} - W.J. Cook, William H. Cunningham, W. R. Pulleybank, and A. Schrijver,
*Combinatorial Optimization*, Wiley-Interscience Series in Discrete Mathematics, 1998. - C.H. Papadimitriou and K. Steiglitz
*Combinatorial**Optimization: Algorithms and Complexity**Optimization: Algorithms and Complexity*, Dover Publications, 1998. - D. P. Williamson,
*Mathematical Programming I*, Unpublished Lecture Notes. Available through the link (scroll down after clicking the link).^{1}

^{1}Permission to use these unpublished lecture notes for this module was generously granted by the author.