Periodic Reporting for period 1 - PACKENUM (PArameterized Complexity and Kernelization for ENUMeration)
Berichtszeitraum: 2024-07-08 bis 2026-07-07
Preprocessing is a ubiquitous technique in algorithm design, aimed at reducing complex problem instances to simpler ones that can be solved more efficiently. While it is a core tool when dealing with optimization problems, preprocessing has not seen the same level of success on enumeration problems, where the goal is now to generate all viable solutions instead of only one. These problems naturally appear in various fields, including network design, data mining, and bioinformatics. Despite the practical importance of these problems, the theoretical understanding of preprocessing for enumeration is still in its early stages.
To remedy this situation, the PACKENUM project (“PArameterized Complexity and Kernelization for ENUMeration”) was established to investigate preprocessing for enumeration problems under the lens of parameterized complexity theory. In this setting, preprocessing is modeled through the kernelization algorithms (or kernels), which transforms a given instance into an equivalent but smaller one whose size depends only on a chosen parameter, providing formal performance guarantees for the preprocessing algorithm; Intuitively, the smaller the output size, the better the kernel, with polynomial-sized kernels being defined as the efficient ones. Overall, PACKENUM goals were to develop a coherent theoretical foundation for enumerative kernelization and to provide concrete algorithmic advances. Its impact is to be materialized by both providing new examples of enumeration kernels and setting new research directions on the subject for the parameterized complexity and enumeration algorithms communities.
To remedy this situation, the PACKENUM project (“PArameterized Complexity and Kernelization for ENUMeration”) was established to investigate preprocessing for enumeration problems under the lens of parameterized complexity theory. In this setting, preprocessing is modeled through the kernelization algorithms (or kernels), which transforms a given instance into an equivalent but smaller one whose size depends only on a chosen parameter, providing formal performance guarantees for the preprocessing algorithm; Intuitively, the smaller the output size, the better the kernel, with polynomial-sized kernels being defined as the efficient ones. Overall, PACKENUM goals were to develop a coherent theoretical foundation for enumerative kernelization and to provide concrete algorithmic advances. Its impact is to be materialized by both providing new examples of enumeration kernels and setting new research directions on the subject for the parameterized complexity and enumeration algorithms communities.
The project’s main outcomes were theoretical contributions to the understanding of kernelization for enumeration problems. initially, we developed enumeration kernels for structural parameterizations of Matching Multicut, a generalization of the Matching Cut problem, the problem upon which much of enumerative kernelization literature is built upon. Afterwards, we identified important gaps in the literature; in particular, little to no work was done on enumeration variants of the core problems in parameterized complexity: Vertex Cover and Feedback Vertex Set. We immediately set out to address these limitations. For the former, we developed the first linear kernel, improving upon the best known quadratic kernel and matching the best known sizes for decision kernelization. For the latter problem, we designed the first known enumeration kernel, but here we only obtain a cubic guarantee on the size, while the best known decision kernels are quadratic. These kernels, however, are much more complex than their decision counterparts.
During the development of the aforementioned results, we observed a key limitation in the current model for enumeration kernels. To overcome them, we introduced the concept of polynomial-delay kernels, a more robust model for enumerative kernelization that preserves all the properties of a sound kernelization model while simplifying its application to enumeration problems. Using this model, we produced new results for enumeration variants of the well-known problems Vertex Cover and Hitting Set, as well as a framework that translates some decision kernels into enumeration kernels.
Using the latter, we not only significantly simplified our previous result for Vertex Cover, but also showed a dichotomy for structural parameterizations on minor-closed graph classes, matching the known bounds for decision kernelization. For the other problem, we developed the first-known enumeration kernel, also matching the best bounds in the decision literature.
Beyond kernelization, we have investigated classical and parameterized algorithms for variants of the Minimal Dominating Set problem on chordal bipartite graphs, obtaining novel algorithms and theoretical lower bounds.
Finally, PACKENUM also enabled the work on questions outside of enumeration. In particular, novel algorithms for the Vertex-Disjoint Paths problem and results in structural graph theory were obtained as a direct result of the project.
During the development of the aforementioned results, we observed a key limitation in the current model for enumeration kernels. To overcome them, we introduced the concept of polynomial-delay kernels, a more robust model for enumerative kernelization that preserves all the properties of a sound kernelization model while simplifying its application to enumeration problems. Using this model, we produced new results for enumeration variants of the well-known problems Vertex Cover and Hitting Set, as well as a framework that translates some decision kernels into enumeration kernels.
Using the latter, we not only significantly simplified our previous result for Vertex Cover, but also showed a dichotomy for structural parameterizations on minor-closed graph classes, matching the known bounds for decision kernelization. For the other problem, we developed the first-known enumeration kernel, also matching the best bounds in the decision literature.
Beyond kernelization, we have investigated classical and parameterized algorithms for variants of the Minimal Dominating Set problem on chordal bipartite graphs, obtaining novel algorithms and theoretical lower bounds.
Finally, PACKENUM also enabled the work on questions outside of enumeration. In particular, novel algorithms for the Vertex-Disjoint Paths problem and results in structural graph theory were obtained as a direct result of the project.
As an area still in early stages of development, few enumeration kernels were known and not many open questions were readily available to be worked on. By providing numerous new kernelization examples and raising key questions for the area, PACKENUM has paved the way for the growth of the field and of the community. In particular, the new kernelization model seems like a much better target for the development of a rigorous lower bound theory, which would allow practitioners to more easily decide if their problem of interest does not admit a polynomial-sized enumeration kernel.