arXiv subjects

Jakob Greilhuber

Publications and source records attributed to Jakob Greilhuber.

5 recordsLinked to original sources

arXiv2026

Faster Exponential Algorithms for Multi-Machine Scheduling Problems

Minimizing the weighted completion times ($P \mid \mid \Sigma w_j C_j$) and weighted number of tardy jobs ($P \mid \mid \Sigma w_j U_j$) on multiple identical machines are two classical NP-hard scheduling problems. As shown by Lent\'e et al. (2014), both problems can be solved in time ${O}^{\star}(3^n)$. In this paper, we improve these bounds to ${O}(2.755^n)$ and ${O}^{\star}(2^n)$, respectively. Our algorithm for $P \mid \mid \Sigma w_j C_j$ exploits the meet-in-the-middle paradigm and an efficient data structure answering linear programming queries. Additionally, when the number of machines is at most $6$, we show that the running time for $P \mid \mid \Sigma w_j C_j$ can further be improved. Both scheduling problems are generalizations of the classical Bin Packing problem, which can be solved in ${O}^{\star}(2^n)$ time. Improving this running time is an important open question. We show that, when assuming the Asymptotic Rank Conjecture (ARC), Bin Packing can be solved in time ${O}((2-\varepsilon)^n)$ for some $\varepsilon >0$. Our algorithm makes use of two main ingredients: the recent ${O}((2-\varepsilon)^n)$-time algorithm of Nederlof et al. [SICOMP'23] for Bin Packing when the number of bins is a fixed constant, and the ${O}((2-\varepsilon)^n)$-time algorithm of Bj\"orklund et al. [SODA'25] for special instances of the $3$-way Partitioning problem when assuming ARC.

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arXiv2026

Where Treewidth and Pathwidth Diverge: Towards a Uniform Kernel for Pathwidth-$\eta$ Deletion

For a constant $\eta \geq 0$, Pathwidth-$\eta$ Deletion is the problem of deciding whether, for a given graph $G$ and integer $k$, there is a set $S \subseteq V(G)$ of size at most $k$ such that the pathwidth of $G - S$ is at most $\eta$. The problems Treewidth-$\eta$ Deletion and Treedepth-$\eta$ Deletion are defined similarly for the parameters treewidth and treedepth, respectively. A landmark result of Fomin et al. [FOCS, 2012] shows that, for any constant $\eta$, all three problems admit a kernel on $O(k^{c(\eta)})$ vertices, where $c(\eta)$ is a constant depending on $\eta$. Giannopoulou et al. [ACM TALG, 2017] show that, in some sense, this result is optimal for Treewidth-$\eta$ Deletion: for $\eta \geq 2$ and even when parameterizing by the size of a vertex cover $M$ of the input graph, there is no kernel of size $O(|M|^{\frac{\eta+1}{2}-\varepsilon})$, for any $\varepsilon > 0$. Contrasting this result, they prove that Treedepth-$\eta$ Deletion admits a uniform polynomial kernel, that is, a kernel of size $O(k^c)$ for a constant $c$ that is independent of $\eta$. In comparison, the question whether Pathwidth-$\eta$ Deletion admits a uniform polynomial kernel has been neglected in the literature. As treewidth and pathwidth tend to behave similarly, it is natural to expect that no uniform kernel exists when parameterizing by the size of a vertex cover. Surprisingly, we show this not to be the case. More concretely, we prove the existence of a uniform polynomial kernel for Pathwidth-$\eta$ Deletion when parameterizing by (1) the solution size $k$ plus the size of a set $M$ such that $G - M$ has bounded treedepth; (2) the (vertex-deletion) distance to pathwidth-$1$ graphs; (3) the distance to the class of graphs with treedepth at most $\eta + 1$. This leads us to conjecture that Pathwidth-$\eta$ Deletion admits a uniform kernel when parameterizing by the solution size $k$.

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arXiv2026

A Dividing Line for Structural Kernelization of Component Order Connectivity via Distance to Bounded Pathwidth

In this work we study a classic generalization of the Vertex Cover (VC) problem, called the Component Order Connectivity (COC) problem. In COC, given an undirected graph $G$, integers $d \geq 1$ and $k$, the goal is to determine if there is a set of at most $k$ vertices whose deletion results in a graph where each connected component has at most $d$ vertices. When $d=1$, this is exactly VC. This work is inspired by polynomial kernelization results with respect to structural parameters for VC. On one hand, Jansen & Bodlaender [TOCS 2013] show that VC admits a polynomial kernel when the parameter is the distance to treewidth-$1$ graphs, on the other hand Cygan, Lokshtanov, Pilipczuk, Pilipczuk & Saurabh [TOCS 2014] showed that VC does not admit a polynomial kernel when the parameter is distance to treewidth-$2$ graphs. Greilhuber & Sharma [IPEC 2024] showed that, for any $d \geq 2$, $d$-COC cannot admit a polynomial kernel when the parameter is distance to a forest of pathwidth $2$. Here, $d$-COC is the same as COC only that $d$ is a fixed constant not part of the input. We complement this result and show that like for the VC problem where distance to treewidth-$1$ graphs versus distance to treewidth-$2$ graphs is the dividing line between structural parameterizations that allow and respectively disallow polynomial kernelization, for COC this dividing line happens between distance to pathwidth-$1$ graphs and distance to pathwidth-$2$ graphs. The main technical result of this work is that COC admits a polynomial kernel parameterized by distance to pathwidth-$1$ graphs plus $d$.

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arXiv2026

The Price of Being Partial: Complexity of Partial Generalized Dominating Set on Bounded-Treewidth Graphs

For fixed sets $σ, ρ$ of non-negative integers, the $(σ, ρ)$-domination framework introduced by Telle [Nord. J. Comput. 1994] captures many classical graph problems. For a graph $G$, a $(σ,ρ)$-set is a set $S$ of vertices such that for every $v\in V(G)$, we have (1) if $v \in S$, then $|N(v) \cap S| \in σ$, and (2) if $v \notin S$, then $|N(v) \cap S| \in ρ$. We initiate the study of a natural partial variant $(σ,ρ)$-MinParDomSet of the problem, in which the constraints given by $σ, ρ$ need not be fulfilled for all vertices, but we want to find a set of size at most $k$ that maximizes the number of vertices that are satisfied in the sense that they satisfy (1) or (2) above. Our goal is to understand whether $(σ,ρ)$-MinParDomSet can be solved in the same running time as the nonpartial version, or whether it is strictly harder. Formally, we consider nonempty finite or simple cofinite sets $σ$ and $ρ$ (simple cofinite sets are of the form $\mathbb{Z}_{\geq c}$), and we try to determine the smallest constant $c_{σ,ρ}$ such that there is a $c_{σ,ρ}^{tw}\cdot n^{O(1)}$ time algorithm for the problem if a tree decomposition of width $tw$ is given. We obtain matching upper and lower bounds on $c_{σ,ρ}$ for every such fixed $σ$ and $ρ$ under the Primal Pathwidth Strong Exponential Time Hypothesis, and establish whether the partial problem is harder than the nonpartial variant. For some sets $σ$ and $ρ$, the more general $(σ,ρ)$-MinParDomSet has the same complexity as the nonpartial special case (e.g., for Dominating Set), while for other choices, the partial version is significantly harder (e.g., for Perfect Code).

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arXiv2025

Residue Domination in Bounded-Treewidth Graphs

For the vertex selection problem $(σ,ρ)$-DomSet one is given two fixed sets $σ$ and $ρ$ of integers and the task is to decide whether we can select vertices of the input graph such that, for every selected vertex, the number of selected neighbors is in $σ$ and, for every unselected vertex, the number of selected neighbors is in $ρ$ [Telle, Nord. J. Comp. 1994]. This framework covers many fundamental graph problems such as Independent Set and Dominating Set. We significantly extend the recent result by Focke et al. [SODA 2023] to investigate the case when $σ$ and $ρ$ are two (potentially different) residue classes modulo $m\ge 2$. We study the problem parameterized by treewidth and present an algorithm that solves in time $m^{tw} \cdot n^{O(1)}$ the decision, minimization and maximization version of the problem. This significantly improves upon the known algorithms where for the case $m \ge 3$ not even an explicit running time is known. We complement our algorithm by providing matching lower bounds which state that there is no $(m-ε)^{pw} \cdot n^{O(1)}$-time algorithm parameterized by pathwidth $pw$, unless SETH fails. For $m = 2$, we extend these bounds to the minimization version as the decision version is efficiently solvable.

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