### Algorithms Beyond the Worst Case - M2 - 6EC

[FALL 2020]

Prerequesites
- Mathematical maturity
- Basic knowledge of discrete algorithms, see e.g., Ahuja, Magnanti and Orlin, “Network Flows”, Chapters 2-6; or Cormen, Leiserson, Rivest, Stein, "Introduction to Algorithms”, Parts IV and VI.
- Basic knowledge of probability theory: any textbook in your bachelor on probability theory, e.g., Grimmett and Stirzaker, “Probability and Random Processes”, Chapters 1-5.
- Familiarity with randomized algorithms and the probabilistic method in combinatorics will be helpful, but is not strictly necessary.
- Knowledge of discrete optimization is a plus.

Aim of the course
Understanding and applying methods to analyze algorithms for which the classical worst-case analysis fails to explain the performance.

Traditionally, the classical analysis of algorithms focuses on the worst-case behavior. E.g., what is the worst case running time, or the worst case approximation or competitive ratio. In many settings, this gives too pessimistic insights and does not reflect empirical observations. A classic example is the Simplex Algorithm for Linear Programming, which takes exponential time is worst case, but works spectacularly fast in practice.

In recent years, there has been increasing focus on exploring more realistic models for various problems and developing a robust theory for the design and analysis of algorithms. This has led to both beautiful mathematical
ideas and new algorithmic insights. A nice example is the smoothed analysis framework developed by Spielman and Teng to explain the empirical behavior of Simplex.

This course will focus on methods and techniques to rigorously analyze algorithms beyond the classical worst-case analysis, with special focus on heuristics that work well in practice. We will consider a wide variety of  algorithmic problems spanning different areas such as classic NP-Hard problems, online algorithms, compressed sensing and machine learning. These problems will be explored both using probabilistic approaches such as  average-case analysis, smoothed analysis and semi-random input models and deterministic approaches such as local-search, resource augmentation, stability and perturbation resilience, refined analysis based on input parameters such as conditions numbers or non-degeneracy conditions.

Lecturers
Bodo Manthey (UT), Tjark Vredeveld (UM).