Enumerative kernelization is a relatively recent and promising area sitting at the intersection of parameterized complexity and enumeration algorithms, with two main models being proposed. The first, known as enum-kernels and due to Creignou et al. [Theory Comput. Syst., 2017], was too permissive, leading to constant-sized kernels for every problem solvable with FPT-delay. To remedy this, Golovach et al. [J. Comput. Syst. Sci., 2022] 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 that the area has enjoyed so far. 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 the "good" 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 increased versatility and the expressive power of our framework on the Enum Vertex Cover problem, where we want to list all vertex covers of size at most k of a given graph. In particular, we manage to generalize the kernelization dichotomy by Bougeret et al. [SIAM J. Discrete Math., 2022] about the existence of polynomial kernels for Vertex Cover parameterized by the vertex deletion distance to a minor-closed graph class, as well as the solution size and feedback vertex number parameterizations. The second one, in particular, is significantly simpler than the kernel designed by Bougeret et al. [IPEC, 2025], 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. [Algorithmica, 2019] for the vertex-deletion distance to c-treedepth.

A matching is said to be disconnected if the saturated vertices induce a disconnected subgraph and induced if the saturated vertices induce a 1-regular graph. The disconnected and induced matching numbers are defined as the maximum cardinality of such matchings, respectively, and are known to be NP-hard to compute. In this paper, we study the relationship between these two parameters and the matching number. In particular, we discuss the complexity of two decision problems; first: deciding if the matching number and disconnected matching number are equal; second: deciding if the disconnected matching number and induced matching number are equal. We show that given a bipartite graph with diameter four, deciding if the matching number and disconnected matching number are equal is NP-complete; the same holds for bipartite graphs with maximum degree three. We characterize diameter three graphs with equal matching number and disconnected matching number, which yields a polynomial time recognition algorithm. Afterwards, we show that deciding if the induced and disconnected matching numbers are equal is co-NP-complete for bipartite graphs of diameter 3. When the induced matching number is large enough compared to the maximum degree, we characterize graphs where these parameters are equal, which results in a polynomial time algorithm for bounded degree graphs.

In the Directed Disjoint Paths problem (k-DDP), we are given a digraph and k pairs of terminals, and the goal is to find k pairwise vertex-disjoint paths connecting each pair of terminals. Bang-Jensen and Thomassen [SIAM J. Discrete Math. 1992] claimed that k-DDP is NP-complete on tournaments, and this result triggered a very active line of research about the complexity of the problem on tournaments and natural superclasses. We identify a flaw in their proof, which has been acknowledged by the authors, and provide a new NP-completeness proof. From an algorithmic point of view, Fomin and Pilipczuk [J. Comb. Theory B 2019] provided an FPT algorithm for the edge-disjoint version of the problem on semicomplete digraphs, and showed that their technique cannot work for the vertex-disjoint version. We overcome this obstacle by showing that the version of k-DDP where we allow congestion c on the vertices is FPT on semicomplete digraphs provided that c is greater than k/2. This is based on a quite elaborate irrelevant vertex argument inspired by the edge-disjoint version, and we show that our choice of c is best possible for this technique, with a counterexample with no irrelevant vertices when c ≤ k/2. We also prove that k-DDP on digraphs that can be partitioned into h semicomplete digraphs is W[1]-hard parameterized by k+h, which shows that the XP algorithm presented by Chudnovsky, Scott, and Seymour [J. Comb. Theory B 2019] is essentially optimal.

Enumerative kernelization is a recent and promising area sitting at the intersection of parameterized complexity and enumeration algorithms. Its study began with the paper of Creignou et al. [Theory Comput. Syst., 2017], and development in the area has started to accelerate with the work of Golovach et al. [J. Comput. Syst. Sci., 2022]. The latter introduced polynomial-delay enumeration kernels and applied them in the study of structural parameterizations of the Matching Cut problem and some variants. Few other results, mostly on Longest Path and some generalizations of Matching Cut, have also been developed. However, little success has been seen in enumeration versions of Vertex Cover and Feedback Vertex Set, some of the most studied problems in kernelization. In this paper, we address this shortcoming. Our first result is a polynomial-delay enumeration kernel with 2k vertices for Enum Vertex Cover, where we wish to list all solutions with at most k vertices. This is obtained by developing a non-trivial lifting algorithm for the classical crown decomposition reduction rule, and directly improves upon the kernel with 𝒪(k²) vertices derived from the work of Creignou et al. Our other result is a polynomial-delay enumeration kernel with 𝒪(k³) vertices and edges for Enum Feedback Vertex Set; the proof is inspired by some ideas of Thomassé [TALG, 2010], but with a weaker bound on the kernel size due to difficulties in applying the q-expansion technique.

A graph convexity on a finite graph G is a family C ⊆ 2 V(G) closed for intersections containing θ and V(G). The elements of C are said to be convex sets. Several graph convexities have been considered in the literature motivated by structural properties, analogies to geometry or applications. One of those is the cycle convexity, in which a set S ⊆ V(G) is convex if for every v ε G\S there is no cycle containing v in the subgraph induced by S U { v }. Several natural questions arise, when studying graph convexities. One of them is the problem of partitioning the vertex set of a graph on a given number k of convex sets. This problem has been previously addressed for different graph convexities. In this paper, we extend this line of research for the cycle convexity, by showing that the problem is NP-complete for every fixed k ≥ 2. On the positive side, we provide a linear-time algorithm for the case when the input graph is chordal.

Enumerating minimal dominating sets with polynomial delay in bipartite graphs is a long-standing open problem. To date, even the subcase of chordal bipartite graphs is open, with the best known algorithm due to Golovach, Heggernes, Kanté, Kratsch, Sæther, and Villanger running in incremental-polynomial time. We improve on this result by providing a polynomial delay and space algorithm enumerating minimal dominating sets in chordal bipartite graphs. Additionally, we show that the total and connected variants admit polynomial and incremental-polynomial delay algorithms, respectively, within the same class. This provides an alternative proof of a result by Golovach et al. for total dominating sets, and answers an open question for the connected variant. Finally, we give evidence that the techniques used in this paper cannot be generalized to bipartite graphs for (total) minimal dominating sets, unless P = NP, and show that enumerating minimal connected dominating sets in bipartite graphs is harder than enumerating minimal transversals in general hypergraphs.

A matching cut of a graph is a partition of its vertex set in two such that no vertex has more than one neighbor across the cut. The Matching Cut problem asks if a graph has a matching cut. This problem, and its generalization d-cut, has drawn considerable attention of the algorithms and complexity community in the last decade, becoming a canonical example for parameterized enumeration algorithms and kernelization. In this paper, we introduce and study a generalization of Matching Cut, which we have named Matching Multicut: can we partition the vertex set of a graph in at least $\ell$ parts such that no vertex has more than one neighbor outside its part? We investigate this question in several settings. We start by showing that, contrary to Matching Cut, it is NP-hard on cubic graphs but that, when $\ell$ is a parameter, it admits a quasi-linear kernel. We also show an $O(\ell^{\frac{n}{2}})$ time exact exponential algorithm for general graphs and a $2^{O(t \log t)}n^{O(1)}$ time algorithm for graphs of treewidth at most $t$. We then study parameterized enumeration aspects of matching multicuts. First, we generalize the quadratic kernel of Golovach et. al for Enum Matching Cut parameterized by vertex cover, then use it to design a quadratic kernel for Enum Matching (Multi)cut parameterized by vertex-deletion distance to co-cluster. Our final contributions are on the vertex-deletion distance to cluster parameterization, where we show an FPT-delay algorithm for Enum Matching Multicut but that no polynomial kernel exists unless NP $\subseteq$ coNP/poly; we highlight that we have no such lower bound for Enum Matching Cut and consider it our main open question.

We study the problem of reconfiguring one minimum $s$-$t$-separator $A$ into another minimum $s$-$t$-separator $B$ in some $n$-vertex graph $G$ containing two non-adjacent vertices $s$ and $t$. We consider several variants of the problem as we focus on both the token sliding and token jumping models. Our first contribution is a polynomial-time algorithm that computes (if one exists) a minimum-length sequence of slides transforming $A$ into $B$. We additionally establish that the existence of a sequence of jumps (which need not be of minimum length) can be decided in polynomial time (by an algorithm that also outputs a witnessing sequence when one exists). In contrast, and somewhat surprisingly, we show that deciding if a sequence of at most $\ell$ jumps can transform $A$ into $B$ is an $\textsf{NP}$-complete problem. To complement this negative result, we investigate the parameterized complexity of what we believe to be the two most natural parameterized counterparts of the latter problem; in particular, we study the problem of computing a minimum-length sequence of jumps when parameterized by the size $k$ of the minimum \stseps and when parameterized by the number of jumps $\ell$. For the first parameterization, we show that the problem is fixed-parameter tractable, but does not admit a polynomial kernel unless $\textsf{NP} \subseteq \textsf{coNP/poly}$. We complete the picture by designing a kernel with $\mathcal{O}(\ell^2)$ vertices and edges for the length $\ell$ of the sequence as a parameter.

A locating-dominating set D of a graph G is a dominating set of G where each vertex not in D has a unique neighborhood in D, and the Locating-Dominating Set problem asks if G contains such a dominating set of bounded size. This problem is known to be NP-hard even on restricted graph classes, such as interval graphs, split graphs, and planar bipartite subcubic graphs. On the other hand, it is known to be solvable in polynomial time for some graph classes, such as trees and, more generally, graphs of bounded cliquewidth. While these results have numerous implications on the parameterized complexity of the problem, little is known in terms of kernelization under structural parameterizations. In this work, we begin filling this gap in the literature. Our first result shows that Locating-Dominating Set is W[1]-hard when parameterized by the size of a minimum clique cover. We present an exponential kernel for the distance to cluster parameterization and show that, unless NP ⊆ coNP/poly, no polynomial kernel exists for Locating-Dominating Set when parameterized by vertex cover nor when parameterized by distance to clique. We then turn our attention to parameters not bounded by either of the previous two, and exhibit a linear kernel when parameterizing by the max leaf number; in this context, we leave the parameterization by feedback edge set as the primary open problem in our study.

An n-vertex graph is equitably k-colorable if there is a proper coloring of its vertices such that each color is used either [n/k] or [n/k] times. While classic Vertex Coloring is fixed parameter tractable under well established parameters such as pathwidth and feedback vertex set, equitable coloring is W[1]-hard. Little is known, however, about kernelization aspects of Equitable Coloring. In this work, we begin this investigation by first presenting a linear kernel for the parameter distance to clique, which contrasts with the quadratic kernel for Vertex Coloring under the same parameterization. Our main technical contribution is an OR-cross-composition from Multicolored Clique to Number List Coloring parameterized by vertex cover and number of colors which, along with two simple PPT reductions, implies that Equitable Coloring has no polynomial kernel under the same parameterization unless NP ⊆ coNP/poly.

In the Selective Coloring problem, we are given an integer k, a graph G, and a partition of V(G) into p parts, and the goal is to decide whether or not we can pick exactly one vertex of each part and obtain a k-colorable induced subgraph of G. This generalization of Vertex Coloring has only recently begun to be studied by Demange et al. [Theoretical Computer Science, 2014], motivated by scheduling problems on distributed systems, with Guo et al. [TAMC, 2020] discussing the first results on the parameterized complexity of the problem. In this work, we study multiple structural parameterizations for Selective Coloring. We begin by revisiting the many hardness results of Demange et al. and show how they may be used to provide intractability proofs for widely used parameters such as pathwidth, distance to co-cluster, and max leaf number. Afterwards, we present fixed parameter tractable algorithms when parameterizing by distance to cluster, or under the joint parameterizations treewidth and number of parts, and co-treewidth and number of colors. Our main contribution is a proof that, for every fixed k > 1, Selective Coloring does not admit a polynomial kernel when jointly parameterized by the vertex cover number and the number of parts, which implies that Multicolored Independent Set does not admit a polynomial kernel under the same parameterization.

A matching cut is a partition of the vertex set of a graph into two sets A and B such that each vertex has at most one neighbor in the other side of the cut. The Matching Cut problem asks whether a graph has a matching cut, and has been intensively studied in the literature. Motivated by a question posed by Komusiewicz et al. [IPEC 2018], we introduce a natural generalization of this problem, which we call d-Cut: for a positive integer d, a d-cut is a bipartition of the vertex set of a graph into two sets A and B such that each vertex has at most d neighbors across the cut. We generalize (and in some cases, improve) a number of results for the Matching Cut problem. Namely, we begin with an NP-hardness reduction for d-Cut on (2d+2)-regular graphs and a polynomial algorithm for graphs of maximum degree at most d+2. The degree bound in the hardness result is unlikely to be improved, as it would disprove a long-standing conjecture in the context of internal partitions. We then give FPT algorithms for several parameters: the maximum number of edges crossing the cut, treewidth, distance to cluster, and distance to co-cluster. In particular, the treewidth algorithm improves upon the running time of the best known algorithm for Matching Cut. Our main technical contribution, building on the techniques of Komusiewicz et al. [IPEC 2018], is a polynomial kernel for d-Cut for every positive integer d, parameterized by the distance to a cluster graph. We also rule out the existence of polynomial kernels when parameterizing simultaneously by the number of edges crossing the cut, the treewidth, and the maximum degree. Finally, we provide an exact exponential algorithm slightly faster than the naive brute force approach running in time O^*(2^n).

Learning to rank (L2R) algorithms use a labeled training set to generate a ranking model that can be later used to rank new query results. These training sets are very costly and laborious to produce, requiring human annotators to assess the relevance or order of the documents in relation to a query. Active learning (AL) algorithms are able to reduce the labeling effort by actively sampling an unlabeled set and choosing data instances that maximize the effectiveness of a learning function. But AL methods require constant supervision, as documents have to be labeled at each round of the process. In this paper, we propose that certain characteristics of unlabeled L2R datasets allow for an unsupervised, compression-based selection process to be used to create small and yet highly informative and effective initial sets that can later be labeled and used to bootstrap a L2R system. We implement our ideas through a novel unsupervised selective sampling method, which we call Cover, that has several advantages over AL methods tailored to L2R. First, it does not need an initial labeled seed set and can select documents from scratch. Second, selected documents do not need to be labeled as the iterations of the method progress since it is unsupervised (i.e., no learning model needs to be updated). Thus, an arbitrarily sized training set can be selected without human intervention depending on the available budget. Third, the method is efficient and can be run on unlabeled collections containing millions of query-document instances. We run various experiments with two important L2R benchmarking collections to show that the proposed method allows for the creation of small, yet very effective training sets. It achieves full training-like performance with less than 10% of the original sets selected, outperforming the baselines in both effectiveness and scalability.

We here propose a new method for expanding entity related queries that automatically filters, weights and ranks candidate expasion terms extracted from Wikipedia articles related to the original query. Our method is based on state-of-the-art tag recommendation methods that exploit heuristic metrics to estimate the descriptive capacity of a given term. Originally proposed for the context of tags, we here apply these recommendation methods to weight and rank terms extracted from multiple fields of Wikipedia articles according to their relevance for the article. We evaluate our method comparing it against three state-of-the-art baselines in three collections. Our results indicate that our method outperforms all baselines in all collections, with relative gains in MAP of up to 14% against the best ones.