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.

For a finite collection of connected graphs $\mathcal{F}$, the $\mathcal{F}$-MINOR-DELETION problem consists in, given a graph $G$ and an integer $\ell$, deciding whether $G$ contains a vertex set of size at most $\ell$ whose removal results in an $\mathcal{F}$-minor-free graph. We lift the existence of (approximate) polynomial kernels for $\mathcal{F}$-MINOR-DELETION by the solution size to (approximate) polynomial kernels parameterized by the vertex-deletion distance to graphs of bounded elimination distance to $\mathcal{F}$-minor-free graphs. This results in exact polynomial kernels for every family $\mathcal{F}$ that contains a planar graph, and an approximate polynomial kernel for PLANAR VERTEX DELETION. Moreover, combining our result with a previous lower bound, we obtain the following infinite set of dichotomies, assuming $NP \not\subseteq coNP/poly$: for any finite set $\mathcal{F}$ of biconnected graphs on at least three vertices containing a planar graph, and any minor-closed class of graphs $\mathcal{C}$, $\mathcal{F}$-MINOR-DELETION admits a polynomial kernel parameterized by the vertex-deletion distance to $\mathcal{C}$ if and only if $\mathcal{C}$ has bounded elimination distance to $\mathcal{F}$-minor-free graphs. For instance, this yields dichotomies for CACTUS VERTEX DELETION, OUTERPLANAR VERTEX DELETION, and TREEWIDTH-$t$ VERTEX DELETION for every integer $t \geq 0$. Prior to our work, such dichotomies were only known for the particular cases of VERTEX COVER and FEEDBACK VERTEX SET. We also provide lower bounds on the size of the kernels.

The paper deals with the Feedback Vertex Set problem parameterized by the solution size. Given a graph $G$ and a parameter $k$, one has to decide if there is a set $S$ of at most $k$ vertices such that $G-S$ is acyclic. Assuming the Exponential Time Hypothesis, it is known that FVS cannot be solved in time $2^{o(k)}n^{\mathcal{O}(1)}$ in general graphs. To overcome this, many recent results considered FVS restricted to particular intersection graph classes and provided such $2^{o(k)}n^{\mathcal{O}(1)}$ algorithms. In this paper we provide generic conditions on a graph class for the existence of an algorithm solving FVS in subexponential FPT time, i.e. time $2^{k^\varepsilon} \mathop{\rm poly}(n)$, for some $\varepsilon<1$, where $n$ denotes the number of vertices of the instance and $k$ the parameter. On the one hand this result unifies algorithms that have been proposed over the years for several graph classes such as planar graphs, map graphs, unit-disk graphs, pseudo-disk graphs, and string graphs of bounded edge-degree. On the other hand it extends the tractability horizon of FVS to new classes that are not amenable to previously used techniques, in particular intersection graphs of ``thin'' objects like segment graphs or more generally $s$-string graphs.

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.

For a fixed graph $H$, the $H$-SUBGRAPH HITTING problem consists in deleting the minimum number of vertices from an input graph to obtain a graph without any occurrence of $H$ as a subgraph. This problem can be seen as a generalization of VERTEX COVER, which corresponds to the case $H = K_2$. We initiate a study of $H$-SUBGRAPH HITTING from the point of view of characterizing structural parameterizations that allow for polynomial kernels, within the recently active framework of taking as the parameter the number of vertex deletions to obtain a graph in a "simple" class $C$. Our main contribution is to identify graph parameters that, when $H$-SUBGRAPH HITTING is parameterized by the vertex-deletion distance to a class $C$ where any of these parameters is bounded, and assuming standard complexity assumptions and that $H$ is biconnected, allow us to prove the following sharp dichotomy: the problem admits a polynomial kernel if and only if $H$ is a clique. These new graph parameters are inspired by the notion of $C$-elimination distance introduced by Bulian and Dawar [Algorithmica 2016], and generalize it in two directions. Our results also apply to the version of the problem where one wants to hit $H$ as an induced subgraph, and imply in particular, that the problems of hitting minors and hitting (induced) subgraphs have a substantially different behavior with respect to the existence of polynomial kernels under structural parameterizations.

In the $K_r$-Cover problem, given a graph $G$ and an integer $k$ one has to decide if there exists a set of at most $k$ vertices whose removal destroys all $r$-cliques of $G$. In this paper we give an algorithm for $K_r$-Cover that runs in subexponential FPT time on graph classes satisfying two simple conditions related to cliques and treewidth. As an application we show that our algorithm solves $K_r$-Cover in time * $2^{O_r\left (k^{(r+1)/(r+2)}\log k \right)} \cdot n^{O_r(1)}$ in pseudo-disk graphs and map-graphs; * $2^{O_{t,r}(k^{2/3}\log k)} \cdot n^{O_r(1)}$ in $K_{t,t}$-subgraph-free string graphs; and * $2^{O_{H,r}(k^{2/3}\log k)} \cdot n^{O_r(1)}$ in $H$-minor-free graphs.

We introduce a new kernelization tool, called rainbow matching technique, that is appropriate for the design of polynomial kernels for packing problems. Our technique capitalizes on the powerful combinatorial results of [Graf, Harris, Haxell, SODA 2021]. We apply the rainbow matching technique on two (di)graph packing problems, namely the TRIANGLE-PACKING IN TOURNAMENT problem (TPT), where we ask for a packing of k directed triangles in a tournament, and the INDUCED 2-PATH-PACKING (I2PP) where we ask for a packing of k induced paths of length two in a graph. The existence of a sub-quadratic kernels for these problems was proven for the first time in [Fomin, Le, Lokshtanov, Saurabh, Thomassé, Zehavi. ACM Trans. Algorithms, 2019], where they gave a kernel of

Many discrete optimization problems amount to selecting a feasible set of edges of least weight. We consider in this paper the context of spatial graphs where the positions of the vertices are uncertain and belong to known uncertainty sets. The objective is to minimize the sum of the distances of the chosen set of edges for the worst positions of the vertices in their uncertainty sets. We first prove that these problems are [Formula: see text]-hard even when the feasible sets consist either of all spanning trees or of all s – t paths. Given this hardness, we propose an exact solution algorithm combining integer programming formulations with a cutting plane algorithm, identifying the cases where the separation problem can be solved efficiently. We also propose a conservative approximation and show its equivalence to the affine decision rule approximation in the context of Euclidean distances. We compare our algorithms to three deterministic reformulations on instances inspired by the scientific literature for the Steiner tree problem and a facility location problem. History: Accepted by David Alderson, Area Editor for Network Optimization: Algorithms & Applications. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoc.2023.1276 .

We consider robust variants of the bin packing problem with uncertain item sizes. Specifically we consider two uncertainty sets previously studied in the literature. The first is budgeted uncertainty (the $U^\Gamma$ model), in which at most $\Gamma$ items deviate, each reaching its peak value, while other items assume their nominal values. The second uncertainty set, the $U^\Omega$ model, bounds the total amount of deviation in each scenario. We show that a variant of the Next-cover algorithm is a $2$ approximation for the $U^\Omega$ model, and another variant of this algorithm is a $2\Gamma$ approximation for the $U^\Gamma$ model. Unlike the classical bin packing problem, it is shown that (unless $\mathcal{P}=\mathcal{NP}$) no asymptotic approximation scheme exists for the $U^\Gamma$ model, for $\Gamma=1$. This motivates the question of the existence of a constant approximation factor algorithm for the $U^\Gamma$ model. Our main result is to answer this question by proving a (polynomial-time) $4.5$ approximation algorithm, based on a dynamic-programming approach.

We study the kernelization complexity of structural parameterizations of the Vertex Cover problem. Here, the goal is to find a polynomial-time preprocessing algorithm that can reduce any instance $(G,k)$ of the Vertex Cover problem to an equivalent one, whose size is polynomial in the size of a predetermined complexity parameter of $G$. A long line of previous research deals with parameterizations based on the number of vertex deletions needed to reduce $G$ to a member of a simple graph class $\mathcal{F}$, such as forests, graphs of bounded tree-depth, and graphs of maximum degree two. We set out to find the most general graph classes $\mathcal{F}$ for which Vertex Cover parameterized by the vertex-deletion distance of the input graph to $\mathcal{F}$ admits a polynomial kernelization. We give a complete characterization of the minor-closed graph families $\mathcal{F}$ for which such a kernelization exists. We introduce a new graph parameter called bridge-depth, and prove that a polynomial kernelization exists if and only if $\mathcal{F}$ has bounded bridge-depth. The proof is based on an interesting connection between bridge-depth and the size of minimal blocking sets in graphs, which are vertex sets whose removal decreases the independence number.

In the Maximum Minimal Vertex Cover (MMVC) problem, we are given a graph G and a positive integer k, and the objective is to decide whether G contains a minimal vertex cover of size at least k. Motivated by the kernelization of MMVC with parameter k, our main contribution is to introduce a simple general framework to obtain lower bounds on the degrees of a certain type of polynomial kernels for vertex-optimization problems, which we call {lop-kernels}. Informally, this type of kernels is required to preserve large optimal solutions in the reduced instance, and captures the vast majority of existing kernels in the literature. As a consequence of this framework, we show that the trivial quadratic kernel for MMVC is essentially optimal, answering a question of Boria et al. [Discret. Appl. Math. 2015], and that the known cubic kernel for Maximum Minimal Feedback Vertex Set is also essentially optimal. On the positive side, given the (plausible) non-existence of subquadratic kernels for MMVC on general graphs, we provide subquadratic kernels on H-free graphs for several graphs H, such as the bull, the paw, or the complete graphs, by making use of the Erdős-Hajnal property in order to find an appropriate decomposition. Finally, we prove that MMVC does not admit polynomial kernels parameterized by the size of a minimum vertex cover of the input graph, even on bipartite graphs, unless NP ⊆ coNP / poly. This indicates that parameters smaller than the solution size are unlike to yield polynomial kernels for MMVC.

We study the kernelization complexity of structural parameterizations of the Vertex Cover problem. Here, the goal is to find a polynomial-time preprocessing algorithm that can reduce any instance $(G,k)$ of the Vertex Cover problem to an equivalent one, whose size is polynomial in the size of a pre-determined complexity parameter of $G$. A long line of previous research deals with parameterizations based on the number of vertex deletions needed to reduce $G$ to a member of a simple graph class $\mathcal{F}$, such as forests, graphs of bounded tree-depth, and graphs of maximum degree two. We set out to find the most general graph classes $\mathcal{F}$ for which Vertex Cover parameterized by the vertex-deletion distance of the input graph to $\mathcal{F}$, admits a polynomial kernelization. We give a complete characterization of the minor-closed graph families $\mathcal{F}$ for which such a kernelization exists. We introduce a new graph parameter called bridge-depth, and prove that a polynomial kernelization exists if and only if $\mathcal{F}$ has bounded bridge-depth. The proof is based on an interesting connection between bridge-depth and the size of minimal blocking sets in graphs, which are vertex sets whose removal decreases the independence number.

A knot in a directed graph G is a strongly connected subgraph Q of G with at least two vertices, such that no vertex in V(Q) is an in-neighbor of a vertex in V(G)\V(Q). Knots are important graph structures, because they characterize the existence of deadlocks in a classical distributed computation model, the so-called OR-model. Deadlock detection is correlated with the recognition of knot-free graphs as well as deadlock resolution is closely related to the Knot-Free Vertex Deletion (KFVD) problem, which consists of determining whether an input graph G has a subset S subseteq V(G) of size at most k such that G[V\S] contains no knot. Because of natural applications in deadlock resolution, KFVD is closely related to Directed Feedback Vertex Set. In this paper we focus on graph width measure parameterizations for KFVD. First, we show that: (i) KFVD parameterized by the size of the solution k is W[1]-hard even when p, the length of a longest directed path of the input graph, as well as kappa, its Kenny-width, are bounded by constants, and we remark that KFVD is para-NP-hard even considering many directed width measures as parameters, but in FPT when parameterized by clique-width; (ii) KFVD can be solved in time 2^{O(tw)} x n, but assuming ETH it cannot be solved in 2^{o(tw)} x n^{O(1)}, where tw is the treewidth of the underlying undirected graph. Finally, since the size of a minimum directed feedback vertex set (dfv) is an upper bound for the size of a minimum knot-free vertex deletion set, we investigate parameterization by dfv and we show that (iii) KFVD can be solved in FPT-time parameterized by either dfv+kappa or dfv+p. Results of (iii) cannot be improved when replacing dfv by k due to (i).

A tournament is a directed graph in which there is a single arc between every pair of distinct vertices. Given a tournament T on n vertices, we explore the classical and parameterized complexity of the problems of determining if T has a cycle packing (a set of pairwise arc-disjoint cycles) of size k and a triangle packing (a set of pairwise arc-disjoint triangles) of size k. We refer to these problems as Arc-disjoint Cycles in Tournaments (ACT) and Arc-disjoint Triangles in Tournaments (ATT), respectively. Although the maximization version of ACT can be seen as the linear programming dual of the well-studied problem of finding a minimum feedback arc set (a set of arcs whose deletion results in an acyclic graph) in tournaments, surprisingly no algorithmic results seem to exist for ACT. We first show that ACT and ATT are both NP-complete. Then, we show that the problem of determining if a tournament has a cycle packing and a feedback arc set of the same size is NP-complete. Next, we prove that ACT and ATT are fixed-parameter tractable, they can be solved in 2^{O(k log k)} n^{O(1)} time and 2^{O(k)} n^{O(1)} time respectively. Moreover, they both admit a kernel with O(k) vertices. We also prove that ACT and ATT cannot be solved in 2^{o(sqrt{k})} n^{O(1)} time under the Exponential-Time Hypothesis.

In the last years, kernelization with structural parameters has been an active area of research within the field of parameterized complexity. As a relevant example, Gajarský et al. (J Comput Syst Sci 84:219–242, 2017) proved that every graph problem satisfying a property called finite integer index admits a linear kernel on graphs of bounded expansion and an almost linear kernel on nowhere dense graphs, parameterized by the size of a c-treedepth modulator, which is a vertex set whose removal results in a graph of treedepth at most c, where $$c \ge 1$$ is a fixed integer. The authors left as further research to investigate this parameter on general graphs, and in particular to find problems that, while admitting polynomial kernels on sparse graphs, behave differently on general graphs. In this article we answer this question by finding two very natural such problems: we prove that Vertex Cover admits a polynomial kernel on general graphs for any integer $$c \ge 1$$ , and that Dominating Set does not for any integer $$c \ge 2$$ even on degenerate graphs, unless $$\text {NP} \subseteq \text {coNP}/\text {poly}$$ . For the positive result, we build on the techniques of Jansen and Bodlaender (Proceedings of the 28th symposium on theoretical aspects of computer science (STACS), volume 9 of LIPIcs, pp 177–188, 2011), and for the negative result we use a polynomial parameter transformation for $$c\ge 3$$ and an or-cross-composition for $$c = 2$$ . As existing results imply that Dominating Set admits a polynomial kernel on degenerate graphs for $$c = 1$$ , our result provides a dichotomy about the existence of polynomial kernels for Dominating Set on degenerate graphs with this parameter.

Given a tournament T and a positive integer k, the C_3-Packing-T asks if there exists a least k (vertex-)disjoint directed 3-cycles in T. This is the dual problem in tournaments of the classical minimal feedback vertex set problem. Surprisingly C_3-Packing-T did not receive a lot of attention in the literature. We show that it does not admit a PTAS unless P=NP, even if we restrict the considered instances to sparse tournaments, that is tournaments with a feedback arc set (FAS) being a matching. Focusing on sparse tournaments we provide a (1+6/(c-1)) approximation algorithm for sparse tournaments having a linear representation where all the backward arcs have "length" at least c. Concerning kernelization, we show that C_3-Packing-T admits a kernel with O(m) vertices, where m is the size of a given feedback arc set. In particular, we derive a O(k) vertices kernel for C_3-Packing-T when restricted to sparse instances. On the negative size, we show that C_3-Packing-T does not admit a kernel of (total bit) size O(k^{2-epsilon}) unless NP is a subset of coNP / Poly. The existence of a kernel in O(k) vertices for C_3-Packing-T remains an open question.

This work provides an analysis of checkpointing strategies for minimizing expected job execution times in an environment that is subject to processor failures. In the case of both sequential and parallel jobs, we give the optimal solution for exponentially distributed failure inter-arrival times, which, to the best of our knowledge, is the first rigorous proof that periodic checkpointing is optimal. For non-exponentially distributed failures, we develop a dynamic programming algorithm to maximize the amount of work completed before the next failure, which provides a good heuristic for minimizing the expected execution time. Our work considers various models of job parallelism and of parallel checkpointing overhead. We first perform extensive simulation experiments assuming that failures follow Exponential or Weibull distributions, the latter being more representative of real-world systems. The obtained results not only corroborate our theoretical findings, but also show that our dynamic programming algorithm significantly outperforms previously proposed solutions in the case of Weibull failures. We then discuss results from simulation experiments that use failure logs from production clusters. These results confirm that our dynamic programming algorithm significantly outperforms existing solutions for real-world clusters.

In this work we study the portfolio problem which is to find a good combination of multiple heuristics to solve given instances on parallel resources in minimum time. The resources are assumed to be discrete, it is not possible to allocate a resource to more than one heuristic. Our goal is to minimize the average completion time of the set of instances, given a set of heuristics on homogeneous discrete resources. This problem has been studied in the continuous case in [T. Sayag et al., 2006]. We first show that the problem is hard and that there is no constant ratio polynomial approximation unlessP=NPin the general case. Then, we design several approximation schemes for a restricted version of the problem where each heuristic must be used at least once. These results are obtained by using oracle with several guesses, leading to various tradeoff between the size of required information and the approximation ratio. Some additional results based on simulations are finally reported using a benchmark of instances on SAT solvers.