# Linear Programming and Combinatorial Optimization

This is a basic course for undergraduate students.## Organization

During the Summer Semester 2019-2020 the lectures are scheduled on Mondays at 2:00pm in lecture room T6 at Troja. Tutorials are held at the same time in on Tuesdays in lecture room S11 at Mala Strana.## Course Requirements (Exam etc.)

The final grade for the course will be based on an exam at the end of the semester. You must obtain a "pass" in the tutorial to be able to take the exam for this course. Details can be found here.## Syllabus

The following is a list of Books and other material relevant to the lectures. The list will be updated as needed.

Understanding and Using Linear Programming (UULP) | by Jiří Matoušek and Bernd Gaertner |

Lectures on Polytopes (LoP) | by Guenter M. Ziegler |

Combinatorial Optimization (CO) | by Cook, Cunningham, Pulleyblank and Schrijver |

Combinatorial Optimization: Polyhedra and Efficiency (COPE) | Alexander Schrijver |

A First Course in Combinatorial Optimization (AFCCO) | Jon Lee |

Notes on LP duality and Applications | Link |

Notes on Maximum matching problem | Link |

### Material Covered in the Lectures

**February 17:**Example of a Linear Program; General form; basic terminology: objective function, constraints, feasible solution, optimal solution, feasible/infeasible, bounded/unbounded; Modeling problems as LP; Examples: Diet optimization, Network flow; Integer Programming; Example: Knapsack problem.

**Recommended reading:**Chapters 1,2 of UULP.

**February 24:**Integer Program; \(0/1\)-Integer Program; LP relaxation; Applications of LP relaxations: (Serendipitous Integrality) Perfect Matchings in bipartite graphs, (Approximation by rounding) \(2\)-approximation by rounding LP optimum for Vertex Cover, (Limits) Independent Set.

**Recommended reading:**Chapter 3 of UULP.

**March 02:**Combinations and Hulls: Convex, Conic, Affine; Basics of Polyhedral theory: halfspaces, polyhedron, polytope, valid inequality, face, vertex, extreme point; Minkowski-Weyl theorem for polyhedra (without proof); MW theorem for polytopes (partial proof); Proof (Sketch) that if \(P\) is a bounded polyhedron (polytope) then there exists a finite set \(S\) with \(P=\mathrm{conv}(S)\).

**Recommended reading:**Chapters 0, 1, 2 of LoP. Section 4.3 of UULP.

**March 09:**Pointed polyhedron; Non-empty pointed polyhedra have at least one vertex; For linear optimization over non-empty pointed polyhedra, if optimum exists, it is attained at a vertex; The feasible region of an LP in Equational form is pointed if non-empty; Basic feasible solutions; (feasible) Basis; Correspondence between basic feasible solutions and vertices; Multiple basis might correspond to a single basic feasible solution; Simplex method: Geometric version; Basic problems: Finding an initial vertex, choosing an edge to follow, stopping criteria.

**Recommended reading:**Chapter 4 of UULP.

**Due to the emergency, future lectures will be in a self-study style. More details and weekly instructions are available on the Moodle page of the course.**