Polynomial-Delay Enumeration Kernel

• Polynomial-delay enumeration kernels are techniques that compress parameterized problems into reduced instances while preserving a partition of the original solution space.
• The lifting algorithm efficiently reconstructs original solutions with polynomial delay, ensuring each non-empty kernel solution represents an equivalence class.
• These kernels are applied in graph problems such as Matching Cut, d-Cut, and Long Path, demonstrating effectiveness in practical enumeration tasks.

A polynomial-delay enumeration kernel is a kernelization framework for parameterized enumeration in which an instance $x$ with parameter $\kappa(x)$ is reduced in polynomial time to a bounded-size instance $y$, and the solutions of $x$ are recovered from the solutions of $y$ with polynomial delay rather than merely FPT-delay. In current usage, the term covers a small family of closely related models—fully-polynomial enumeration kernels, polynomial-delay enumeration kernels, strong PD kernels, and PD kernels—whose common purpose is to make kernel size meaningful for enumeration while preserving output-sensitive reconstruction of the original solution space (Golovach et al., 2021, Bougeret et al., 25 Apr 2026). The framework has moved from abstract complexity theory to concrete kernelizations for problems such as Matching Cut, $d$-Cut, Long Path, and Enum Vertex Cover (Komusiewicz et al., 28 Feb 2025, Komusiewicz et al., 2023).

1. Formal setting

A parameterized enumeration problem is a triple $\Pi^\kappa = (L,\Sol,\kappa)$, where $L\subseteq \Sigma^*$ is decidable, $\Sol(x)$ is a finite computable solution set, and $\Sol(x)\neq \emptyset$ if and only if $\kappa(x)$0 (Bougeret et al., 25 Apr 2026). Enumeration complexity is measured either by total FPT time or by delay, where delay means preprocessing time, time between consecutive outputs, and postprocessing time.

In the refined kernel models, a compression algorithm $\kappa(x)$1 takes $\kappa(x)$2 and $\kappa(x)$3, runs in time polynomial in $\kappa(x)$4, and outputs an instance $\kappa(x)$5 such that

$\kappa(x)$6

for some computable function $\kappa(x)$7, while preserving emptiness: $\kappa(x)$8 The difference between the models lies in the lifting algorithm $\kappa(x)$9, which receives $y$0, $y$1, their parameters, and a kernel solution $y$2, and must produce a subset $y$3 (Bougeret et al., 25 Apr 2026, Golovach et al., 2021).

For the PD-kernel model introduced later, $y$4 must enumerate $y$5 with delay bounded by a polynomial in

$y$6

and the non-empty sets $y$7 must form a partition of $y$8. This means that some kernel solutions may be ignored entirely, but every original solution must appear in exactly one non-empty lifted set (Bougeret et al., 25 Apr 2026). In the earlier strong model, every kernel solution must lift to a non-empty set.

Two complexity characterizations are central. First, a parameterized enumeration problem admits an FPT-delay algorithm if and only if it admits a strong polynomial-delay enumeration kernel in the sense of Golovach et al. (Golovach et al., 2021). Second, the later PD-kernel model preserves the same equivalence with FPT-delay, and a problem admits polynomial-delay enumeration if and only if it admits a constant-sized PD kernel (Bougeret et al., 25 Apr 2026).

2. Evolution of kernel notions

The theory developed in three stages. The first stage was the enum-kernel model of Creignou et al.; the second stage was the refinement to fully-polynomial and polynomial-delay enumeration kernels; the third stage was the PD-kernel model, which relaxes the strong non-emptiness requirement on lifted sets (Golovach et al., 2021, Bougeret et al., 25 Apr 2026).

The central criticism of enum-kernels is that they are too permissive. Because the lifting algorithm is allowed FPT-delay, any problem with an FPT-delay algorithm can be given a constant-sized enum-kernel by pushing essentially all work into the lifting phase. As a result, kernel size ceases to convey structural information (Bougeret et al., 25 Apr 2026). The refined models were introduced precisely to prevent this collapse.

The refined landscape can be summarized as follows.

Model	Lifting requirement	Treatment of kernel solutions
Enum-kernel	FPT-delay	$y$9 may be empty
Fully-polynomial enumeration kernel	Polynomial time	Every $x$0 is non-empty
Strong PD kernel	Polynomial delay	Every $x$1 is non-empty
PD kernel	Polynomial delay	$x$2 may be empty

In the terminology of the 2026 model, the polynomial-delay enumeration kernels of Golovach et al. are renamed strong PD kernels, and the new PD kernels are their relaxation. The decisive change is that PD kernels allow “bad” kernel solutions: kernel solutions $x$3 for which $x$4. Only the non-empty lifted sets must partition $x$5 (Bougeret et al., 25 Apr 2026). This relaxation is presented as the main reason the new model is more versatile in practice, because it permits compression strategies that introduce spurious kernel solutions.

A common misconception is that allowing empty lifted sets is merely cosmetic. The literature treats it as substantive. In the strong model, every kernel solution must correspond to at least one original solution, which complicates many natural constructions. In the PD model, one canonical kernel solution per relevant trace or equivalence class is sufficient; other kernel solutions can be discarded during lifting (Bougeret et al., 25 Apr 2026).

3. Compression, traces, and lifting

The standard architecture of a polynomial-delay enumeration kernel is not just “reduce and regenerate.” It is usually built around a structural partition of the original solution space, a canonical description of each part, and a lifting routine that enumerates exactly the members of one part. The generic framework developed for vertex-subset problems makes this explicit through three components: a core, a choice rule for canonical kernel solutions, and a per-trace enumeration algorithm (Bougeret et al., 25 Apr 2026).

In that framework, the decision kernel provides a reduced instance $x$6 together with a core $x$7, where $x$8 and $x$9 is injective. A trace is the intersection of a solution with the core. The essential requirement is that every trace realized by some original solution is also realized by at least one kernel solution. A polynomial-time choosing algorithm $y$0 then selects exactly one canonical kernel solution for each good trace, and a polynomial-delay algorithm $y$1 enumerates all original solutions with that trace. The main theorem states that if these ingredients exist, then the enumeration problem admits a PD kernel of the same size as the underlying decision kernel (Bougeret et al., 25 Apr 2026).

This architecture explains why lifting is usually phrased as a partition: $y$2 The kernel does not merely preserve yes/no information; it preserves a decomposition of the solution set into liftable blocks.

The same design principle appears in older work on refined enumeration kernels for Matching Cut. There, the reduced graph stores one representative per equivalence class of cuts, and the lifting algorithm reconstructs all equivalent cuts in the original graph by expanding the representative locally, with either polynomial total time per kernel solution or polynomial delay, depending on the variant (Golovach et al., 2021). This suggests that polynomial-delay enumeration kernels are best viewed as compressed indexes of equivalence classes of solutions rather than as reduced instances in the ordinary decision-kernel sense.

4. Representative kernelizations

The theory is now supported by several explicit kernelizations. The most developed examples occur for cut problems, long-path enumeration, and vertex-subset problems.

Problem	Parameter	Kernel guarantee
ENUM MINIMAL MC	Vertex cover $y$3	Fully-polynomial kernel with $y$4 vertices
ENUM MC / ENUM MAXIMAL MC	Vertex cover $y$5	Polynomial-delay kernels with $y$6 vertices
ENUM MINIMAL MC / ENUM MC / ENUM MAXIMAL MC	Clique partition $y$7	Bijective kernels with $y$8 vertices
ENUM $y$9-CUT / ENUM MAX-$d$0-CUT	Vertex cover or neighborhood diversity	Polynomial-delay kernels of polynomial size
ENUM MIN-$d$1-CUT	Vertex cover or neighborhood diversity	Fully-polynomial kernels of polynomial size
ENUM LONG-PATH	Vertex cover $d$2	$d$3-delay kernel with $d$4 vertices
ENUM LONG-PATH	Dissociation number $d$5	$d$6-delay kernel with $d$7 vertices
ENUM LONG-PATH	Distance to clique $d$8	$d$9-delay kernel with $\Pi^\kappa = (L,\Sol,\kappa)$0 vertices
Enum Vertex Cover	Solution size $\Pi^\kappa = (L,\Sol,\kappa)$1	PD kernel of size $\Pi^\kappa = (L,\Sol,\kappa)$2

For Matching Cut, the 2021 framework shows that the refined notions are not merely definitional. Under the vertex-cover parameter, ENUM MINIMAL MC has a fully-polynomial enumeration kernel, whereas ENUM MC and ENUM MAXIMAL MC have only polynomial-delay enumeration kernels, all with $\Pi^\kappa = (L,\Sol,\kappa)$3 vertices (Golovach et al., 2021). The same paper gives a bijective enumeration kernel with $\Pi^\kappa = (L,\Sol,\kappa)$4 vertices when the input is accompanied by a clique partition of size at most $\Pi^\kappa = (L,\Sol,\kappa)$5, and a fully-polynomial kernel with at most $\Pi^\kappa = (L,\Sol,\kappa)$6 vertices for ENUM MINIMAL MC parameterized by modular width (Golovach et al., 2021). These constructions are representative because they make the partition property explicit: each kernel cut represents one equivalence class of original cuts, and lifting enumerates exactly that class.

The 2023 study of $\Pi^\kappa = (L,\Sol,\kappa)$7-cuts extends the same pattern beyond matching cuts. For fixed $\Pi^\kappa = (L,\Sol,\kappa)$8, when parameterized by vertex cover number or neighborhood diversity, the paper provides polynomial-delay enumeration kernelizations of polynomial size for ENUM $\Pi^\kappa = (L,\Sol,\kappa)$9-CUT and ENUM MAX-$L\subseteq \Sigma^*$0-CUT, and fully-polynomial enumeration kernels of polynomial size for ENUM MIN-$L\subseteq \Sigma^*$1-CUT. Under clique partition number, all three variants admit bijective enumeration kernels (Komusiewicz et al., 2023). This places bounded-degree cuts squarely inside the refined kernel framework rather than treating Matching Cut as an isolated case.

Long-path enumeration provides a graph-theoretic case study in polynomial-size polynomial-delay kernels. The 2025 paper on ENUM LONG-PATH gives a kernel with $L\subseteq \Sigma^*$2 vertices and $L\subseteq \Sigma^*$3 delay under the vertex-cover parameter, a kernel with $L\subseteq \Sigma^*$4 vertices and $L\subseteq \Sigma^*$5 delay under dissociation number, and a kernel with $L\subseteq \Sigma^*$6 vertices and $L\subseteq \Sigma^*$7 delay under distance to clique (Komusiewicz et al., 28 Feb 2025). The same paper also gives a cubic-size kernel for constant $L\subseteq \Sigma^*$8-component-order connectivity. Its lifting machinery is more elaborate than in cut problems: solutions are partitioned by signatures relative to a modulator, kernel paths represent equivalence classes, and only one canonical kernel path is allowed to generate the non-kernel members of its class (Komusiewicz et al., 28 Feb 2025).

The 2026 PD-kernel paper uses Enum Vertex Cover as a broader case study in model design rather than in a single problem family. It gives a PD kernel of size $L\subseteq \Sigma^*$9 for Enum Vertex Cover parameterized by the solution size $\Sol(x)$0, a cubic-size PD kernel under feedback vertex number, a polynomial-size PD kernel for deletion distance to fixed $\Sol(x)$1-treedepth, and a bridgedepth dichotomy: for a minor-closed graph class $\Sol(x)$2, Enum Vertex Cover parameterized by the size of a modulator to $\Sol(x)$3 admits a polynomial-size PD kernel if and only if $\Sol(x)$4 has bounded bridgedepth, assuming $\Sol(x)$5 (Bougeret et al., 25 Apr 2026). These results are especially important because they show that PD kernels can systematically inherit the size of decision kernels even when strong PD kernels are cumbersome.

5. Kernel-like mechanisms outside the formal model

The formal theory of polynomial-delay enumeration kernels interacts with a broader body of work on compressed enumeration, quotient enumeration, and amortized-to-worst-case delay conversion. Some of these results do not define parameterized kernels, but they behave as kernel-like mechanisms.

A clear example is the framework for enumerating equivalence classes of solutions in e-colored acyclic decomposable AND/OR graphs. There, solutions are compressed by contracting AND nodes, and the quotient objects $\Sol(x)$6 can be enumerated with delay $\Sol(x)$7, where $\Sol(x)$8 and $\Sol(x)$9 is the maximum size of an equivalence-class tree (Wang et al., 2020). The same work provides a restriction procedure $\Sol(x)\neq \emptyset$0 such that $\Sol(x)\neq \emptyset$1. From a kernel perspective, the trees in $\Sol(x)\neq \emptyset$2 function as canonical compressed representatives of entire solution classes, and $\Sol(x)\neq \emptyset$3 acts as a decompression handle (Wang et al., 2020).

A different but complementary connection comes from the relation between amortized and worst-case delay. The 2021 result $\Sol(x)\neq \emptyset$4 shows that any polynomial-space enumerator with polynomial amortized delay can be transformed into one with polynomial worst-case delay by geometric amortization, with a logarithmic overhead in the number of solutions (Capelli et al., 2021). This matters for enumeration kernels because many kernel-based generators first produce only incremental or amortized guarantees on the reduced instance. The theorem implies that, under polynomial-space conditions, this is not a fundamental obstacle to obtaining polynomial delay on top of the kernel.

These developments suggest that the formal kernel notion sits inside a larger ecosystem of compressed-state enumeration. In some problems the compression is parameterized and explicit; in others it is quotient-based, state-based, or amortization-based. The common theme is that a smaller structured object is enumerated first, and the original solutions are then reconstructed with controlled delay.

6. Limits, lower bounds, and open directions

The theory is constrained by both model-theoretic and problem-specific limits. At the model level, the failure modes are now well understood. Enum-kernels are too permissive because any FPT-delay problem can receive a constant-sized kernel; strong PD kernels are often too strict because every kernel solution must lift to a non-empty set; PD kernels were proposed precisely to occupy the middle ground (Bougeret et al., 25 Apr 2026). This three-way tension is central to the subject.

At the problem level, polynomial-size polynomial-delay kernels do not exist uniformly across parameters. For ENUM LONG-PATH, there is no polynomial-delay enumeration kernel of polynomial size when parameterized by the size of a largest connected component, unless $\Sol(x)\neq \emptyset$5 (Komusiewicz et al., 28 Feb 2025). For the Matching Cut family, ENUM MINIMAL MC, ENUM MAXIMAL MC, and ENUM MC do not admit polynomial-delay enumeration kernels of polynomial size when parameterized by treewidth or cliquewidth unless $\Sol(x)\neq \emptyset$6, and nonempty matching cut enumeration has no such kernel under modular width unless the same collapse occurs (Golovach et al., 2021). These results show that kernel existence is sharply parameter-sensitive.

There are also obstacles that arise before the formal kernel question is even posed. For large maximal matching enumeration, the literature gives polynomial-delay algorithms but no parameter-$\Sol(x)\neq \emptyset$7 enumeration kernel, and it identifies a concrete obstruction: the Maximal Matching Extension problem is NP-complete even on planar bipartite degree-3 graphs, which blocks standard extension-oracle-based kernel strategies (Kobayashi et al., 2021). This suggests that polynomial-delay enumeration can be available even when known kernelization techniques fail.

Open questions remain in both the formal and kernel-like settings. In the AND/OR-graph framework, it is explicitly left open whether polynomial-delay enumeration of equivalence classes remains possible without decomposability (Wang et al., 2020). In the PD-kernel model, the main program is to identify broader classes of decision kernels that can be converted into PD kernels without the heavy trace balancing required by strong PD kernels (Bougeret et al., 25 Apr 2026). More broadly, the field still lacks a complete map of which structural parameters support polynomial-size polynomial-delay kernels and which only support FPT-delay or kernel-like compression without a formal kernel theorem.

Taken together, these results define the current meaning of a polynomial-delay enumeration kernel: not merely a reduced instance, but a reduced instance whose solutions index a partition of the original solution space and support reconstruction with worst-case output sensitivity. That combination of structural compression and delay control is what distinguishes the notion from both classical decision kernels and from polynomial-delay enumeration in the absence of preprocessing.