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.
