Minimizing the weighted completion times ($P \mid \mid Σw_j C_j$) and weighted number of tardy jobs ($P \mid \mid Σw_j U_j$) on multiple identical machines are two classical NP-hard scheduling problems. As shown by Lenté 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 $P \mid \mid Σw_j C_j$ 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 $P \mid \mid Σw_j C_j$ 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 Björklund et al. [SODA'25] for special instances of the $3$-way Partitioning problem when assuming ARC.

For a constant $η\geq 0$, Pathwidth-$η$ 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 $η$. The problems Treewidth-$η$ Deletion and Treedepth-$η$ 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 $η$, all three problems admit a kernel on $O(k^{c(η)})$ vertices, where $c(η)$ is a constant depending on $η$. Giannopoulou et al. [ACM TALG, 2017] show that, in some sense, this result is optimal for Treewidth-$η$ Deletion: for $η\geq 2$ and even when parameterizing by the size of a vertex cover $M$ of the input graph, there is no kernel of size $O(|M|^{\frac{η+1}{2}-\varepsilon})$, for any $\varepsilon > 0$. Contrasting this result, they prove that Treedepth-$η$ Deletion admits a uniform polynomial kernel, that is, a kernel of size $O(k^c)$ for a constant $c$ that is independent of $η$. In comparison, the question whether Pathwidth-$η$ 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-$η$ 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 $η+ 1$. This leads us to conjecture that Pathwidth-$η$ Deletion admits a uniform kernel when parameterizing by the solution size $k$.

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).

In this work we study a classic generalization of the ubiquitous Vertex Cover (VC) problem, called the Component Order Connectivity (COC) problem. In COC, given an undirected graph G, integers d ≥ 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 ≥ 2, d-COC cannot admit a polynomial kernel when the parameter is distance to a forest of pathwidth 2. Here, d-COC is the variant of COC where d is a fixed constant rather than part of the input. We complement this result and show that, analogously to the VC setting, where distance to treewidth-1 graphs versus distance to treewidth-2 graphs is the dividing line between structural parameterizations that admit and respectively do not admit polynomial kernelization, for COC this dividing line lies 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. The problem d-COC can also be expressed as an ℱ-MinorDeletion problem for an appropriate graph family ℱ. One of the central questions around ℱ-MinorDeletion is for which families ℱ and minor-closed graph classes 𝒢 the problem admits a polynomial kernel when parameterized by the distance to 𝒢. For some families ℱ complete dichotomies answering this question are known [Bougeret et al., SIDMA 2022][Bougeret et al., STACS 2026][Bougeret et al., arXiv 2026]. But, these results do not capture the 2-COC problem. We show that, when d ≥ 2, the line of tractability for polynomial kernelization of d-COC parameterized by the distance to 𝒢 is different from the tractability line of the ℱ-MinorDeletion problems for which the currently known dichotomies apply. Thus, with our result, d-COC serves as an outlier in the class of ℱ-MinorDeletion problems when it comes to understanding the dichotomies for polynomial kernelization when parameterizing by the distance to some minor-closed graph class.

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.