The first session of this talk will take place on Tuesday, September 1 from 3:00 pm – 4:00 pm; the second session of this talk will take place on Wednesday, September 2 from 3:00 pm – 4:00 pm; the fourth session of this talk will take place on Friday, September 4 from 3:00 pm – 4:00 pm.

We expect that NP-hard problems can be solved only by algorithms that need exponential time in the worst case. However, there is still a lot that can be said about the complexity of such problems. The study of exponential algorithms aims to find the best possible form of exponential running time that can be achieved. Parameterized complexity analyzes the running time of an algorithm as a function of the input size n and some parameter k of the input. The goal is to determine the optimal dependence on the parameter. Of particular interest is the case when only a multiplicative factor of the running time depends on the parameter k, that is, the problem is fixed-parameter tractable (FPT).

The mini-course will survey a selection of algorithmic results, highlighting some nontrivial algorithmic techniques for NP-hard problems. Then we show how the (Strong) Exponential-Time Hypothesis and other conjectures can be used to give lower bounds that, in many cases, tightly match the algorithmic results.