## Special Colloquium

Daniel Lokshtanov

University of Bergen

Coping with NP-hardness

**Abstract:**The vast majority of optimization problems arising in practical applications are NP-hard. Thus it is difficult to over-state the importance of understanding how to handle NP-hard problems algorithmically. When a problem is NP-hard, this means that (unless P=NP) there cannot exist an algorithm that solves all instances of the problem optimally using time polynomial in the size of the instance. However, this does not mean that algorithms stand powerless in the face of NP-hardness. Indeed, an NP-hard problem may still admit algorithms of the following kind. - An algorithm that solves all instances of the problem optimally in time O(1.001^n), or even O(2^sqrt(n)), where n is the size of the input instance. Such algorithms are studied in the field of exact exponential time algorithms, and may still be useful even for moderate size worst-case input instances. - An algorithm that solves all instances of the problem within a factor 1.01 of optimality using time polynomial in the size of the instance. In many cases, being just 1% worse than the optimal solution is acceptable. Such algorithms are studied in the field of approximation algorithms. - An algorithm that solves structurally simple instances of the problem optimally using time polynomial in the size of the instance. This is very useful if we know that the instances that we actually want to solve are structurally simple. Algorithms of this kind are studied in the fields of restricted input and parameterized algorithms. - An algorithm that works in polynomial time and reduces possibly large yet structurally simple instances to equivalent small instances. This can be very useful provided that the output instance is so small that a solution can be found by brute force. The performance of such algorithms is studied in the field of kernelization. In this talk I will give a sample of each of the above fields, viewed through the lens of my contributions.

Tea will be served after the talk.

Monday February 5, 2018 at 3:00 PM in SEO 636