# Algorithmic Enumeration: Output-sensitive, Input-Sensitive, Parameterized, Approximative

## Motivation

About fifty years ago, NP-completeness became the lens through which computer science views computationally hard (decision and optimization) problems. In the last decades, various new approaches to solve NP-hard problems exactly have attracted a lot of attention, among them parameterized and exact exponential-time algorithms, typically dealing with decision and optimization problems.

While optimization is ubiquitous in computer science and many application areas, relatively little is known about enumeration within the `Algorithms and Complexity' community. Fortunately, there has been important algorithmic research dedicated to enumeration problems in various fields of computer science, as, e.g., artificial intelligence and data mining, in natural sciences, engineering, and social sciences.

Enumeration problems require to list all wanted objects of the input as, e.g., particular subsets of the vertex or edge set of a given graph, or particular satisfying assignments of logical expressions. Contrary to decision, optimization, and even counting problems, the output length of enumeration problems is often exponential in the size of the input and cannot be neglected. This motivates the classical approach in enumeration, now called output-sensitive, which measures running time in (input and) output length, and asks for output-polynomial algorithms and algorithms of polynomial delay. This approach has been studied since a long time and has produced its own important open questions, among them the question whether the minimal transversals of a hypergraph can be enumerated in output-polynomial time. This longstanding and challenging question has triggered a lot of research. It is open for more than fifty years and most likely the best known open problem in algorithmic enumeration.

Recently, as a natural extension to research in exact exponential-time algorithms, a new approach, called input-sensitive, which measures the running time in the input length, has found growing interest. Due to the number of objects to enumerate (in the worst case), the corresponding algorithms have exponential running time. So far branching algorithms are a major tool. Input-sensitive enumeration is strongly related to lower and upper combinatorial bounds on the maximum number of objects to be enumerated for an input of given size. Such bounds can be achieved via input-sensitive enumeration algorithms but also by the use of combinatorial (non-algorithmic) means.

The area of algorithmic enumeration is in a nascent state though has a huge potential due to theoretical challenges and practical applications. While output-sensitive enumeration has a long history, input-sensitive enumeration has been initiated only recently. Natural and promising approaches like using parameterized or approximative approaches have not been explored yet to their full capacities.

Principal goals of our Dagstuhl Seminar are to increase the visibility of algorithmic enumeration within (theoretical) computer science and to contribute to establishing it as an area of `Algorithms and Complexity'. The seminar will bring together researchers within the algorithms community, other fields of computer science and computer engineering, as well as researchers working on enumeration problems in other application areas, in particular biology. Besides the people already working with enumeration, we invited researchers from other fields of computer science. In particular, we invited researchers who are interested in parameterized complexity, approximation, and different aspects of counting problems. The aim is to accelerate developments and discuss new directions including algorithmic tools and hardness proofs.