Linear Programming and Combinatorial Optimization
This is a basic course for undergraduate students.Organization
During the Summer Semester 2023-2024 the lectures are scheduled on Wednesdays at 09:00 in S3 in Mala Strana. Tutorials are held on Thursdays at 15:40 in S11.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. .Syllabus
This course covers the following topics:- Linear and Integer Programming.
- LP Duality
- Simplex Algorithm
- Ellipsoid Algorithm
- Matroids
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 28: Example of a Linear Program; General form; basic terminology: objective function, constraints, feasible solution, optimal solution, feasible/infeasible, bounded/unbounded; Modeling problems as LP; Example: Network flow; Integer Programming; 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.
Recommended reading: Chapters 1,2,3 of UULP.
March 06: Equational form of an LP; Basic feasible solutions; For LPs in equational form feasibility and boundedness imply existence of optimal basic feasible solutions.
Recommended reading: Chapters 4 of UULP
March 13: Combinations and Hulls: Convex, Conic, Affine; Basics of Polyhedral theory: halfspaces, polyhedron, polytope, valid inequality, face, proper face, vertex, extreme point; vertex = extreme point; Minkowski-Weyl theorem (without proof); Correspondence between basic feasible solutions and vertices; Simplex method: Geometric version; Basic problems: Finding an initial vertex, choosing an edge to follow, stopping criteria.
Recommended reading: Chapters 0, 1 of LoP; Chapter 4 of UULP <>
March 20: The Simplex Method; Examples including for exceptional cases: degeneracy, unboundedness, infeasibility; Simplex Tableu; Pivot rules: the problem of cycling; Bland's rule; Bland's rule avoids cycling (without proof); Simplex method is not known to terminate in polynomial time (without proof); LP Duality; Conic combinations of valid inequalities yield valid inequalities that may provide upper bound on objective value; Dual LP;
Recommended reading: Chapter 5,6 of UULP.
March 27: LP Duality; Weak Duality; Strong duality (proof using simplex method).
Recommended reading: Chapter 6 of UULP.