PArameterized Complexity and Kernelization for ENUMeration

Algorithms play crucial roles in many aspects of the lives of billions of people worldwide. Many of the problems we wish to solve, in industry and

academia, are NP-hard and it is expected that no polynomial-time algorithm exists to obtain an optimal solution for them. Nevertheless, they are

solved millions of times on a daily basis. Solving them would be unfeasible without the use of preprocessing techniques, which significantly reduce

running times and are often necessary to solve a problem. Explaining why these methods work in practice and designing new ones that come with

performance guarantees is a great challenge in Theoretical Computer Science. In the framework of Parameterized Complexity, they are modeled

through kernelization, which uses an additional measurement of the problem's structure (the parameter) to output a small equivalent instance that

can be quickly solved.

However, there will usually exist several optimal solutions, regardless of the optimality criterion, and drawing conclusions from a single one may be

misleading. Knowing more about the set of optimal solutions is thus necessary in many scenarios and can be formalized through enumeration

problems. Unlike decision problems, very little is known about preprocessing for enumeration problems. In the recently defined enumeration kernel,

solutions to the reduced instance are used to partition and efficiently list the solution set of the input. Through this project, the researcher will

design and implement novel parameterized algorithms and kernels for enumeration problems, and build the lower-bound theory required to

separate problems between those that admit polynomial enumeration kernels and those that do not. The designed kernels will be some of the

earliest enumeration kernels, while the lower-bound theory will be a fundamental part of Parameterized Complexity, allowing researcher's to

identify problems that do not admit efficient preprocessing and focus their efforts on problems that do.