arXiv subjects

Petr A. Golovach

Publications and source records attributed to Petr A. Golovach.

At least 19 recordsLinked to original sources

arXiv2026

Packing Diverse Shortest Cycles

Bentert, Fomin, Golovach, Korhonen, Lochet, Panolan, Ramanujan, Saurabh, and Simonov (SODA 2025) initiated the parameterized study of Edge-Disjoint Shortest Cycle Packing: given a weighted graph $G$ and an integer $k$, decide whether $G$ contains $k$ edge-disjoint cycles of minimum weight. They showed that the problem admits an algorithm running in time $n^{O(k^6)}$ and asked whether it is fixed-parameter tractable or $W[1]$-hard parameterized by $k$. We resolve this question by proving that Edge-Disjoint Shortest Cycle Packing is $W[1]$-hard parameterized by $k$, even on unweighted subcubic graphs. The same lower bound also applies to the vertex-disjoint variant. For planar graphs, they provides a construction of a kernel with $O(k^2)$ vertices and an algorithm running in time $k^{O(k)} \cdot n^{O(1)}$, and explicitly asked whether the problem admits a single-exponential algorithm of running time $2^{O(k)} \cdot n^{O(1)}$. Rather than addressing this question in isolation, we introduce a more general framework, Diverse Shortest Cycle Coverage, which asks for $k$ shortest cycles that may overlap in a controlled way while maximizing the total weight of covered edges. This framework simultaneously captures edge-disjoint shortest cycle packing, the problem of finding diverse shortest cycles, and the problem of maximizing edge coverage by shortest cycles. Our main algorithmic result shows that Diverse Shortest Cycle Coverage can be solved on planar graphs in time $2^{O(k)} \cdot n^{O(1)}$, thereby giving a single-exponential algorithm for Edge-Disjoint Shortest Cycle Packing as a special case. The key idea of our algorithm is a structural analysis of the Laminar Shortest Cycles Tree, a tree-like decomposition that reveals a laminar interaction pattern among shortest cycles in planar graphs and enables an efficient dynamic programming algorithm.

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arXiv2026

Routing Multiple Agents Below the Sum of Distances

We study Transient Multiagent Pathfinding, a variant of the classical Multi-Agent Pathfinding problem in which a set of agents must be routed without collisions from designated start vertices to designated destination vertices in a graph. We analyze the problem within the above-and-below-guarantee paradigm of parameterized complexity. In particular, we consider the natural upper bound $L$, given by the sum of the shortest-path distances between pairs of agents' terminals (corresponding to sequential routing of the agents). The parameterization is given by the gap $ζ= L - λ$ between this bound and the target makespan $λ$, together with the number $k$ of agents. Our main result establishes fixed-parameter tractability for the combined parameter $k + ζ$. Matching lower bounds show that parameterization by $k$ alone is W[1]-hard, and that parameterization by $ζ$ alone is W[1]-hard when terminals are not required to be distinct. On the positive side, if all terminals are distinct, the problem becomes fixed-parameter tractable when parameterized solely by $ζ$. Finally, we show that Transient Multiagent Pathfinding is unlikely to admit a polynomial kernel when parameterized by $k + ζ$. Together, our results provide an almost complete characterization of the parameterized complexity landscape of the problem for the considered parameters.

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arXiv2026

Ultrametric Violation Distance: Polynomial Kernel and FPT Algorithm

In the Ultrametric Violation Distance problem, we are given a set of distances between $n$ points, and the goal is to modify the minimum number of distances so that the resulting set forms a valid ultrametric. In other words, the task is to fit an ultrametric to the given data, where the quality of the fit is measured by the $\ell_0$-norm of the error. While variants of this problem under the $\ell_\infty$ and $\ell_1$-norms have been well studied, the complexity of Ultrametric Violation Distance under the $\ell_0$-norm remained largely unexplored until recently. This changed with the work of Cohen-Addad, Fan, Lee, and Mesmay [FOCS 2022], who introduced a constant-factor approximation algorithm. Significant further progress on approximation algorithms was made in subsequent work by Charikar and Gao [SODA 2024], and by An, Kao, Lee, and Lee [FOCS 2025]. In this paper, we initiate a systematic study of Ultrametric Violation Distance from the perspectives of kernelization and fixed-parameter tractability (FPT). By the work of Fan, Gilbert, Raichel, Sonthalia, and Van Buskirk [SWAT 2020], the problem is known to be FPT when parameterized by the number of violated distances $k$. We show that the problem admits a kernel with $\mathcal{O}(k^2)$ points. Additionally, we present a single-exponential-time algorithm with running time $9^k \cdot n^{\mathcal{O}(1)}$, which is asymptotically tight.

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arXiv2026

An FPT Algorithm for Diverse Minimum s-t Cuts

We study the problem of finding a family of diverse minimum edge s-t cuts in a directed weighted graph G. Given integers k and d, the task is to decide whether G contains k minimum s-t cuts C_1, ..., C_k such that for any i,j in [k], the number of edges in the symmetric difference of C_i and C_j is at least d. For d being 1 or 2, the problem corresponds to counting minimum s-t cuts in G, which is #P-complete [Provan and Ball, SICOMP 1983]. The problem is also known to be NP-complete already for k = 3 [de Berg, López Martínez, Spieksma, ISAAC 2024]. Our main result shows that the problem is fixed-parameter tractable (FPT) when parameterized by the combined parameter k + d. The main ingredients of our FPT algorithm build on novel structural properties of diverse minimum s-t cuts and a non-trivial application of the flow-augmentation technique of Kim, Kratsch, Pilipczuk, and Wahlström [JACM 2025].

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arXiv2026

Finding irrelevant vertices in linear time on bounded-genus graphs

The irrelevant vertex technique provides a powerful tool for the design of parameterized algorithms for a wide variety of problems on graphs. A common characteristic of these problems, permitting the application of this technique on surface-embedded graphs, is the fact that every graph of large enough treewidth contains a vertex that is irrelevant, in the sense that its removal yields an equivalent instance of the problem. The straightforward application of this technique yields algorithms with running time that is quadratic in the size of the input graph. This running time is due to the fact that it takes linear time to detect one irrelevant vertex and the total number of irrelevant vertices to be detected is linear as well. Using advanced techniques, sub-quadratic algorithms have been designed for particular problems, even in general graphs. However, designing a general framework for linear-time algorithms has been open, even for the bounded-genus case. In this paper we introduce a general framework that enables finding in linear time an entire set of irrelevant vertices whose removal yields a bounded-treewidth graph, provided that the input graph has bounded genus. Our technique consists of decomposing any surface-embedded graph into a tree-structured collection of bounded-treewidth subgraphs where detecting globally irrelevant vertices can be done locally and independently. Our method is applicable to a wide variety of known graph containment or graph modification problems where the irrelevant vertex technique applies. Examples include the (Induced) Minor Folio problem, the (Induced) Disjoint Paths problem, and the $\mathcal{F}$-Minor-Deletion problem.

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arXiv2026

Clustering Permutations under the Ulam Metric: A Parameterized Complexity Study

Rank aggregation seeks a representative permutation for a collection of rankings and plays a central role in areas such as social choice, information retrieval, and computational biology. Two fundamental aggregation tasks are the center and median problems, which minimize the maximum and the total distance to the input permutations, respectively. While these problems are well understood under Kendall's tau and related distances, their parameterized complexity under the Ulam metric, an edit-distance-based metric on permutations, has remained largely unexplored. In this work, we initiate a systematic study of the parameterized complexity of rank aggregation under the Ulam metric. We consider both the center and median problems, as well as their generalizations to the $k$-center and $k$-median clustering settings, parameterized by the number of centers $k$ and the distance budget $d$ (corresponding to the maximum distance for center variants and the total distance for median variants). Both problems are known to be NP-hard already for $k=1$. We show that the Ulam $k$-center problem remains NP-hard when $d=1$, but is fixed-parameter tractable when parameterized by $k + d$. Our algorithm is based on a novel local-search framework tailored to the non-local nature of Ulam distances. We complement this by proving that no polynomial kernel exists for the $k+d$ parameterization unless NP $\subseteq$ coNP/poly. For the Ulam $k$-median problem parameterized by the total distance $d$, we establish W[1]-hardness and provide an XP algorithm. We also provide a polynomial kernel for the parameter $k + d$, which in turn yields a fixed-parameter tractable algorithm.

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arXiv2026

Identification to Subclasses of Chordal Graphs

An identification of two vertices $u$ and $v$ in a graph replaces them with a new vertex whose neighborhood is the union of the neighborhoods of $u$ and $v$. We study the {\sc ${\cal H}$-Identification} problem, which is to decide whether a given graph $G$ can be transformed (``identified'') to a graph in ${\cal H}$ by applying at most $k$ vertex identifications. We determine the classical and parameterized complexity of this problem for various subclasses ${\cal H}$ of chordal graphs, obtaining an almost complete picture for two parameters: $k$ and $n-k$. We also consider the {\sc Identification} problem, which is to test for two given graphs $G$ and $H$ if $G$ can be identified to $H$. We determine the parameterized complexity of this problem when $H$ is a graph from one of our testbed classes, taking the number of simplicial vertices of $H$ as the parameter.

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arXiv2026

Polynomial Kernels for Spanning Tree with Diversity Requirements

Given a connected undirected graph $G$, a spanning tree is a subgraph $T$ of $G$ such that $V(T) = V(G)$ and $T$ is a tree. A collection of $\ell$ spanning trees $T_1,\ldots,T_\ell$ is pairwise $k$-diverse if for every $i \neq j$, $|E(T_i) \triangle E(T_j)| \geq k$. Given a connected undirected graph $G$ and integers $p, q, k, \ell$, Leaf & Internal-Constrained Diverse Spanning Trees asks whether there are $\ell$ distinct spanning trees $T_1,\ldots,T_{\ell}$ of $G$ that are pairwise $k$-diverse such that each tree has at least $p$ leaves and at least $q$ internal vertices. Similarly, Leaf & Non-terminal-Constrained Diverse Spanning Trees takes a connected undirected graph $G$, $V_{NT}\subseteq V(G)$, and three integers $p, k, \ell$, and asks if $G$ has $\ell$ spanning trees that are pairwise $k$-diverse, and each has at least $p$ leaves and conains the vertices of $V_{NT}$ as internal. We consider these two problems from the kernelization perspective and provide polynomial kernels for Leaf & Internal-Constrained Diverse Spanning Trees and Leaf & Non-terminal-Constrained Diverse Spanning Trees, when parameterized by $p + q + k + \ell$ and $p + |V_{\rm NT}| + k + \ell$, respectively.

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arXiv2026

A Framework for Parameterized Subexponential-Subcubic-Time Algorithms for Weighted Problems in Planar Graphs

Many problems are known to be solvable in subexponential parameterized time when the input graph is planar. The bidimensionality framework of Demaine, Fomin, Hajiaghay, and Thilikos [JACM'05] and the treewidth-pattern-covering approach by Fomin, Lokshtanov, Marx, Pilipczuk, Pilipczuk, and Saurabh [SICOMP'22] give robust tools for designing such algorithms. However, there are still many problems for which we do not know whether subexponential parameterized algorithms exist. The bidimensionality framework is not able to handle weights or directed graphs and the treewidth-pattern-covering approach only works for finding connected solutions. Building on a result by Nederlof [STOC'20], we provide a framework that is able to solve a variety of problems in planar graphs in subexponential parameterized time for which this was previously not known (where the polynomial part of the running time is usually $O(n^{2.49})$). Our framework can handle weights, does not require solutions to contain only few connected components, and applies to cases where the number of potential patterns of a solution is exponential in the parameter. We then use the framework to show that various weighted problems like Weighted Partial Vertex Cover, Maximum-Weight Induced Forest, Minimum-Weight Rooted Simple Minor, and Maximum-Weight Rooted Parallel Induced Minor allow for subexponential parameterized algorithms. This was previously not known for any of them. Moreover, we present a very easy-to-use fragment of our framework. This fragment allows for significantly simpler proofs in the case of Maximum-Weight Independent Set and Maximum $(k, n-k)$-Cut and is able to show a subexponential parameterized algorithm for weighted versions of Densest $k$-Subgraph. Even the unweighted version was not known before and is stated as an open problem in the existing literature.

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arXiv2026

Line Cover and Related Problems

We study extensions of the classic \emph{Line Cover} problem, which asks whether a set of $n$ points in the plane can be covered using $k$ lines. Line Cover is known to be NP-hard, and we focus on two natural generalizations. The first is \textbf{Line Clustering}, where the goal is to find $k$ lines minimizing the sum of squared distances from the input points to their nearest line. The second is \textbf{Hyperplane Cover}, which asks whether $n$ points in $\mathbb{R}^d$ can be covered by $k$ hyperplanes. We also study the more general \textbf{Projective Clustering} problem, which unifies both settings and has applications in machine learning, data analysis, and computational geometry. In this problem, one seeks $k$ affine subspaces of dimension $r$ that minimize the sum of squared distances from the given points in $\mathbb{R}^d$ to the nearest subspace. Our results reveal notable differences in the parameterized complexity of these problems. While Line Cover is fixed-parameter tractable when parameterized by $k$, we show that Line Clustering is W[1]-hard with respect to $k$ and does not admit an algorithm with running time $n^{o(k)}$ unless the Exponential Time Hypothesis fails. Hyperplane Cover has been known to be NP-hard since the 1980s, following work of Megiddo and Tamir, even for $d=2$, we show that it remains NP-hard even when $k=2$. Finally, we present an algorithm for Projective Clustering running in $n^{O(dk(r+1))}$ time. This bound matches our lower bound for Line Clustering and generalizes the classic algorithm for $k$-Means Clustering ($r=0$) by Inaba, Katoh, and Imai [SoCG 1994].

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arXiv2026

Algorithms for Euclidean Distance Matrix Completion: Exploiting Proximity to Triviality

In the d-Euclidean Distance Matrix Completion (d-EDMC) problem, one aims to determine whether a given partial matrix of pairwise distances can be extended to a full Euclidean distance matrix in d dimensions. This problem is a cornerstone of computational geometry with numerous applications. While classical work on this problem often focuses on exploiting connections to semidefinite programming typically leading to approximation algorithms, we focus on exact algorithms and propose a novel distance-from-triviality parameterization framework to obtain tractability results for d-EDMC. We identify key structural patterns in the input that capture entry density, including chordal substructures and coverability of specified entries by fully specified principal submatrices. We obtain: (1) The first fixed-parameter algorithm (FPT algorithm) for d-EDMC parameterized by d and the maximum number of unspecified entries per row/column. This is achieved through a novel compression algorithm that reduces a given instance to a submatrix on O(1) rows (for fixed values of the parameters). (2) The first FPT algorithm for d-EDMC parameterized by d and the minimum number of fully specified principal submatrices whose entries cover all specified entries of the given matrix. This result is also achieved through a compression algorithm. (3) A polynomial-time algorithm for d-EDMC when both d and the minimum fill-in of a natural graph representing the specified entries are fixed constants. This result is achieved by combining tools from distance geometry and algorithms from real algebraic geometry. Our work identifies interesting parallels between EDM completion and graph problems, with our algorithms exploiting techniques from both domains.

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arXiv2026

Path Cover, Hamiltonicity, and Independence Number: An FPT Perspective

The classic theorem of Gallai and Milgram (1960) generalizes several fundamental results in Graph Theory, such as Dilworth's theorem on posets and Kőnig's theorem on matchings in bipartite graphs. The theorem asserts that for every graph G, the vertex set of G can be partitioned into at most α(G) vertex-disjoint paths, where α(G) is the maximum size of an independent set in G. The proof of the Gallai-Milgram theorem is constructive and yields a polynomial-time algorithm that computes a covering of G by at most α(G) vertex-disjoint paths. While the Gallai-Milgram theorem is tight, it was not known prior to our work whether deciding if a graph G could be covered by fewer than α(G) vertex-disjoint paths can be done in polynomial time. We resolve this question by proving the following algorithmic extension of the Gallai-Milgram theorem for undirected graphs: There is an algorithm that, for an n-vertex graph G and an integer parameter k \ge 1, runs in time 2^{2^{O(k^4\log{k})}} \cdot n^{O(1)} and outputs a path cover P of G together with - a correct conclusion that P is a minimum-size path cover, or - an independent set of size |P|+k, certifying that P contains at most α(G) - k paths. The proof of our algorithmic extension of the Gallai-Milgram theorem is non-trivial and builds on several novel algorithmic ideas. One of the key subroutines in our algorithm is an FPT algorithm, parameterized by α(G), for deciding whether G contains a Hamiltonian path. This result is of independent interest - prior to our work no polynomial-time algorithm for deciding Hamiltonicity was known even for graphs with independence number at most three. Moreover, the algorithmic techniques we develop apply to a wide array of problems in undirected graphs, including Hamiltonian Cycle, Path Cover, Largest Linkage, and Topological Minor Containment.

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arXiv2026

What Can Be Computed Locally Revisited: First-Order Logic on Sparse Graphs in Distributed Computing

The question of 'what can be computed locally?' lies at the heart of distributed computing in networks. As established in Naor and Stockmeyer's seminal paper (STOC 1993), this question is undecidable, even for graph problems whose solutions can be checked locally. In this paper, we adopt a novel perspective on the question, by asking for which classes $Π$ of problems, and for which classes $\mathcal{G}$ of graphs, all problems in $Π$ can be solved efficiently in a distributed manner in all graphs of $\mathcal{G}$. This paper focuses on two natural candidates for such an approach, namely the class of problems expressible in first-order logic (FO), because of their intrinsic form of locality thanks to Gaifman's theorem, and the class of graphs with bounded expansion, because they form a large class of graphs encompassing, e.g., planar, bounded-treewidth, and bounded-degree graphs. The starting point of our work is the decade-old open question of Nešetřil and Ossona de Mendez (Distributed Computing 2016) on the distributed complexity of local FO formula on graphs of bounded expansion, in the standard CONGEST model of distributed computing. A formula $φ(x)$ is local if the satisfaction of $φ(x)$ depends only on the $r$-neighborhood of its free variable $x$, for some fixed $r$. For instance, the formula '$x$ belongs to a triangle' is local. We resolve the open problem positively by showing that, for every local FO formula $φ(x)$, and for every graph class $\mathcal{G}$ of bounded expansion, there exists a deterministic algorithm that identifies, for every $n$-vertex graph $G\in \mathcal{G}$, all vertices $v$ of $G$ such that $G\models φ(v)$, in $O(\log n)$ rounds. When dropping the locality condition, we show that $O(D+\log n)$ rounds are sufficient for deciding any FO formula $φ$ on graphs of bounded expansion.

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arXiv2025

H-Planarity and Parametric Extensions: when Modulators Act Globally

We introduce a series of graph decompositions based on the modulator/target scheme of modification problems that enable several algorithmic applications that parametrically extend the algorithmic potential of planarity. In the core of our approach is a polynomial time algorithm for computing planar H-modulators. Given a graph class H, a planar H-modulator of a graph G is a set X \subseteq V(G) such that the ``torso'' of X is planar and all connected components of G - X belong to H. Here, the torso of X is obtained from G[X] if, for every connected component of G-X, we form a clique out of its neighborhood on G[X]. We introduce H-Planarity as the problem of deciding whether a graph G has a planar H-modulator. We prove that, if H is hereditary, CMSO-definable, and decidable in polynomial time, then H-Planarity is solvable in polynomial time. Further, we introduce two parametric extensions of H-Planarity by defining the notions of H-planar treedepth and H-planar treewidth, which generalize the concepts of elimination distance and tree decompositions to the class H. Combining this result with existing FPT algorithms for various H-modulator problems, we thereby obtain FPT algorithms parameterized by H-planar treedepth and H-planar treewidth for numerous graph classes H. By combining the well-known algorithmic properties of planar graphs and graphs of bounded treewidth, our methods for computing H-planar treedepth and H-planar treewidth lead to a variety of algorithmic applications. For instance, once we know that a given graph has bounded H-planar treedepth or bounded H-planar treewidth, we can derive additive approximation algorithms for graph coloring and polynomial-time algorithms for counting (weighted) perfect matchings. Furthermore, we design Efficient Polynomial-Time Approximation Schemes (EPTAS-es) for several problems, including Maximum Independent Set.

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arXiv2025

Tight Parameterized (In)tractability of Layered Crossing Minimization: Subexponential Algorithms and Kernelization

The starting point of our work is a decade-old open question concerning the subexponential parameterized complexity of \textsc{2-Layer Crossing Minimization}. In this problem, the input is an $n$-vertex graph $G$ whose vertices are partitioned into two independent sets $V_1$ and $V_2$, and a non-negative integer $k$. The question is whether $G$ admits a 2-layered drawing with at most $k$ crossings, where each $V_i$ lies on a distinct line parallel to the $x$-axis, and all edges are straight lines. We resolve this open question by giving the first subexponential fixed-parameter algorithm for this problem, running in time $2^{O(\sqrt{k}\log k)} + n \cdot k^{O(1)}$. We then ask whether the subexponential phenomenon extends beyond two layers. In the general $h$-Layer Crossing Minimization problem, the vertex set is partitioned into $h$ independent sets $V_1, \ldots, V_h$, and the goal is to decide whether an $h$-layered drawing with at most $k$ crossings exists. We present a subexponential FPT algorithm for three layers with running time $2^{O(k^{2/3}\log k)} + n \cdot k^{O(1)}$ for $h = 3$ layers. In contrast, we show that for all $h \ge 5$, no algorithm with running time $2^{o(k/\log k)} \cdot n^{O(1)}$ exists unless the Exponential-Time Hypothesis fails. Finally, we address polynomial kernelization. While a polynomial kernel was already known for $h=2$, we design a new polynomial kernel for $h=3$. These kernels are essential ingredients in our subexponential algorithms. Finally, we rule out polynomial kernels for all $h \ge 4$ unless the polynomial hierarchy collapses.

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arXiv2025

Computing Tree Decompositions with Small Independence Number

The independence number of a tree decomposition is the maximum of the independence numbers of the subgraphs induced by its bags. The tree-independence number of a graph is the minimum independence number of a tree decomposition of it. Several NP-hard graph problems, like maximum weight independent set, can be solved in time n^{O(k)} if the input n-vertex graph is given together with a tree decomposition of independence number k. Yolov, in [SODA 2018], gave an algorithm that, given an n-vertex graph G and an integer k, in time n^{O(k^3)} either constructs a tree decomposition of G whose independence number is O(k^3) or correctly reports that the tree-independence number of G is larger than k. In this paper, we first give an algorithm for computing the tree-independence number with a better approximation ratio and running time and then prove that our algorithm is, in some sense, the best one can hope for. More precisely, our algorithm runs in time 2^{O(k^2)} n^{O(k)} and either outputs a tree decomposition of G with independence number at most $8k$, or determines that the tree-independence number of G is larger than k. This implies 2^{O(k^2)} n^{O(k)}-time algorithms for various problems, like maximum weight independent set, parameterized by the tree-independence number k without needing the decomposition as an input. Assuming Gap-ETH, an n^{Ω(k)} factor in the running time is unavoidable for any approximation algorithm for the tree-independence number. Our second result is that the exact computation of the tree-independence number is para-NP-hard: We show that for every constant k \ge 4 it is NP-hard to decide if a given graph has the tree-independence number at most k.

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arXiv2025

Fault-Tolerant Matroid Bases

We investigate the problem of constructing fault-tolerant bases in matroids. Given a matroid M and a redundancy parameter k, a k-fault-tolerant basis is a minimum-size set of elements such that, even after the removal of any k elements, the remaining subset still spans the entire ground set. Since matroids generalize linear independence across structures such as vector spaces, graphs, and set systems, this problem unifies and extends several fault-tolerant concepts appearing in prior research. Our main contribution is a fixed-parameter tractable (FPT) algorithm for the k-fault-tolerant basis problem, parameterized by both k and the rank r of the matroid. This two-variable parameterization by k + r is shown to be tight in the following sense. On the one hand, the problem is already NP-hard for k=1. On the other hand, it is Para-NP-hard for r \geq 3 and polynomial-time solvable for r \leq 2.

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arXiv2025

Edge Clique Partition and Cover Beyond Independence

Covering and partitioning the edges of a graph into cliques are classical problems at the intersection of combinatorial optimization and graph theory, having been studied through a range of algorithmic and complexity-theoretic lenses. Despite the well-known fixed-parameter tractability of these problems when parameterized by the total number of cliques, such a parameterization often fails to be meaningful for sparse graphs. In many real-world instances, on the other hand, the minimum number of cliques in an edge cover or partition can be very close to the size of a maximum independent set α(G). Motivated by this observation, we investigate above αparameterizations of the edge clique cover and partition problems. Concretely, we introduce and study Edge Clique Cover Above Independent Set (ECC/α) and Edge Clique Partition Above Independent Set (ECP/α), where the goal is to cover or partition all edges of a graph using at most α(G) + k cliques, and k is the parameter. Our main results reveal a distinct complexity landscape for the two variants. We show that ECP/αis fixed-parameter tractable, whereas ECC/αis NP-complete for all k \geq 2, yet can be solved in polynomial time for k \in {0,1}. These findings highlight intriguing differences between the two problems when viewed through the lens of parameterization above a natural lower bound. Finally, we demonstrate that ECC/αbecomes fixed-parameter tractable when parameterized by k + ω(G), where ω(G) is the size of a maximum clique of the graph G. This result is particularly relevant for sparse graphs, in which ωis typically small. For H-minor free graphs, we design a subexponential algorithm of running time f(H)^{\sqrt{k}}n^{O(1)}.

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