A more versatile model for enumerative kernelization: a case study for Vertex Cover

Abstract: Enumerative kernelization is a recent promising at the intersection of parameterized complexity and enumeration algorithms, with two proposed models. The first, known as enum-kernels and due to Creignou et al., was too permissive, leading to constant-sized kernels for every problem solvable with FPT-delay. To remedy this, Golovach et al. proposed the polynomial-delay enumeration kernelization model that, while addressing the shortcoming of the previous one, appears to be too strict, which we believe is a central reason for the slow development of the area. In this paper, we propose a new model for enumeration kernels, which we have called polynomial-delay (PD) kernels. It is more flexible than Golovach et al.'s kernels while still preserving their qualities; informally, it allows us to ignore bad'' solutions of the compressed instance when producing the solution set of the input instance, but still requires that thegood'' solutions are lifted with polynomial-delay. After discussing the main properties of our model, we design a generic framework for vertex-subset problems to adapt decision kernels into PD kernels of the same size. We showcase our model's versatility and the framework's expressive power on the \textsc{Enum Vertex Cover} problem, where we want to list all vertex covers of size at most $k$ of a given graph. We generalize the kernelization dichotomy by Bougeret et al. about the existence of polynomial kernels for \textsc{Vertex Cover} parameterized by the vertex deletion distance to a minor-closed graph class, as well as by the solution size or feedback vertex number. The second one, in particular, is significantly simpler than the known kernel, requiring only a few lines for its lifting algorithm. Beyond our framework, we also show how to generalize to the enumeration setting the kernel of Bougeret et al. for the vertex-deletion distance to $c$-treedepth.

• The paper introduces a PD kernel model that flexibly maps compressed solutions to full enumerations with guaranteed polynomial delay.
• It adapts decision kernel methods to efficiently enumerate vertex covers, achieving linear or cubic kernel sizes under different parameterizations.
• The work establishes a practical framework that reduces algorithmic complexity and paves the way for further advancements in combinatorial enumeration.

A Versatile Model for Enumerative Kernelization: An Analysis for Vertex Cover

Introduction and Context

Enumerative kernelization extends classical parameterized preprocessing—kernelization—into the domain of enumeration problems, where the goal is not merely to decide, but to efficiently list all solutions of a combinatorial problem parameterized by some parameter $k$. While decision kernelization has matured, enumeration kernelization is less developed, with prior models having either excessive flexibility (leading to meaningless kernel bounds) or being too restrictive to accommodate practical algorithm design. This paper develops an intermediate model tailored for enumerative problems, specifically focusing on the enumeration of vertex covers up to size $k$.

Prior Models and their Limitations

The earliest general-purpose model, the so-called enum-kernel of Creignou et al. [creignou2017enum], allowed arbitrary lifting procedures with fpt-delay, yielding constant-size kernels for any problem with such an algorithm. However, as Golovach et al. [golovach2022refined] observed, this model's permissiveness invalidated kernel size as a complexity measure. Their refinement required the lifting procedure to map every solution of the compressed instance to at least one output solution via a polynomial-delay algorithm, but this made kernel construction intricate even for simple base problems like Enum Vertex Cover and excluded algorithms that use failure states as a control structure.

The Polynomial-Delay Kernel (PD Kernel) Model

This work introduces polynomial-delay kernels (PD kernels), which relax Golovach et al.'s requirement: a solution $Y$ of the compressed instance may be ignored by the lifting algorithm (i.e., it may map to an empty set of solutions), as long as all "good" solutions (those relevant for the input) are lifted with polynomial delay and the nontrivial union of these preimages covers the solution space of the input. This permits enumeration algorithms to exploit failure states or prune unpromising search paths, a necessary property for advanced enumeration strategies.

Formally, a PD kernel consists of a polynomial-time compression algorithm $A_1$ and a polynomial-delay lifting algorithm $A_2$ whose domains can be partial with respect to solutions of the kernelized instance. Importantly, the union over all $Y$ of $A_2$ is required to partition precisely the solution set of the original instance.

Meta-Theoretical Properties

Key properties of PD kernels, proved in this work:

• Equivalence to FPT-delay Enumeration: A parameterized enumeration problem admits an FPT-delay algorithm if and only if it admits a PD kernel (Theorem 3.2). Thus, polynomial-size PD kernels precisely capture the enumerative tractability of the problem.
• Preservation of Kernel Size as a Complexity Measure: Under standard complexity assumptions, a problem admits a constant-size PD kernel if and only if it can be solved with polynomial delay. Thus, PD kernels enable meaningful kernel size analyses for enumeration.
• Generalization of Previous Notions: PD kernels generalize strong enumeration kernels, as the latter require non-empty lifting for every compressed solution and are thus a special case of PD kernels.

Framework for Adapting Decision Kernels

The authors develop a systematic framework to adapt known decision kernels for graph vertex-subset problems to enumeration PD kernels. This is based on three conditions:

1. Good Decision Kernel: The decision kernel preserves a suitable 'core' set allowing a partition of solutions by their intersection with the core, such that these traces correspond between input and compressed instances.
2. Trace Decidability: There exists a choosing algorithm to identify, for each good trace, a canonical kernel solution representing each "good" solution class.
3. Polynomial-Delay Lifting: For any good trace, one can enumerate all corresponding input solutions with polynomial delay.

When these conditions are met, the enumeration problem admits a PD kernel of size given by the base decision kernel.

Case Study: Enum Vertex Cover and Structural Parameterizations

Applying this framework, the authors revisit the paradigmatic Vertex Cover enumeration problem under various parameterizations, and demonstrate that the transition from decision to enumeration kernelization is feasible and algorithmically manageable in the PD-kernel model.

• Parameterization by Solution Size: The classical crown decomposition-based decision kernel can be adapted to yield a linear-vertex PD kernel with an extremely simple lifting algorithm (vs. unwieldy lifting for prior "strong" kernels).
• Parameterization by Feedback Vertex Set (FVS) Size: The cubic-size FVS kernelization of Jansen and Bodlaender [vc_fvs] is shown to satisfy the conditions for PD kernelization, enabling similarly efficient enumeration.
• Vertex-deletion Distance to Treedepth or Bridgedepth: The authors provide fully worked adaptations for deletion distance to constant-bounded treedepth or bridgedepth, generalizing the dichotomy for polynomial kernels under minor-closed targets [bridgedepth]. Specifically, they prove:

  Enum Vertex Cover admits a polynomial-sized PD kernel parameterized by the vertex-deletion distance to a minor-closed class $\mathcal{F}$ if and only if $\mathcal{F}$ has bounded bridgedepth (Theorem 1.1).

  This is exactly the dichotomy known for the decision version, now transferred to enumeration with only moderate technical effort, thanks to the flexibility of the PD kernel definition.

• Bikernelization: The framework supports compositional or bikernel strategies where the kernelization lands in a different (annotated) problem, provided there are appropriate reductions (permissive PPTs) with bounded or polynomial bad solutions that can be handled in the lifting algorithm.

Numerical and Algorithmic Strengths

• Kernel Size: The paper systematically matches the best known decision kernel sizes for all parameterizations analyzed. For example, they obtain $O(k)$-vertex kernels for size, $k$0 for feedback vertex set of size $k$1, and $k$2 for modulator size to treedepth-$k$3 graphs.
• Lifting Complexity: The lifting algorithms within this model benefit from striking simplicity, in stark contrast to prior work—for instance, only a few lines suffice in the crown decomposition case.
• Expressive Power: The dichotomy result shows that the model is expressive enough to characterize precisely the tractable cases in the parameterized enumeration of vertex cover under all major structurally-defined parameterizations.

Implications and Future Directions

Practically, these results substantially lower the barrier to designing enumeration kernels for important combinatorial problems, by permitting realistic algorithmic mechanisms—e.g., failure pruning and more flexible mappings—within provable kernel size and delay bounds. This should greatly facilitate the integration of kernelization into integer programming and combinatorial enumeration software.

Theoretically, the dichotomy extension underscores the tight relationship between decision and enumeration kernelization for a broad class of graph problems. The techniques may be extendable to other subset enumeration problems (e.g., Feedback Vertex Set), potentially yielding better-than-known kernel bounds in those cases.

Open directions include:

• Quadratic or Linear Kernels for Harder Problems: Improving enumeration kernel sizes for problems like Feedback Vertex Set. The framework permits such exploration by leveraging suitable decision kernels.
• Lower Bound Theory: The field currently lacks enumeration-specific lower bounds; practically, all lower bounds are inherited from decision kernels. Developing direct enumeration lower bounds would sharpen our understanding of the limits of this methodology.
• Extension to Non-Subset or Non-Vertex Problems: Applying or generalizing this approach to problems outside the current scope, possibly by enhancing the model's compositionality or redefining equivalence classes of solutions/cores.

Conclusion

The polynomial-delay kernel (PD kernel) model presented here achieves a nuanced balance between flexibility and tractability in enumerative kernelization, restoring kernel size as a meaningful measure while enabling powerful and implementable algorithms. By providing a systematic bridge between decision and enumeration kernelization for a range of graph problems, the work provides a foundation for a unified theory and practical toolkit for efficient parameterized enumeration. This is especially salient given the demonstrated extension of the decision kernelization dichotomy for vertex cover to the enumerative setting. The results set the stage for further theoretical and practical breakthroughs in algorithmic enumeration.