arXiv subjects

Hans L. Bodlaender

Publications and source records attributed to Hans L. Bodlaender.

At least 19 recordsLinked to original sources

arXiv2026

Trade-off between spread and width for tree decompositions

The spread of a vertex $v$ in a tree decomposition is the number of bags that contain $v$. We study the trade-off between spread and width in tree decompositions, answering every open question from Wood [arXiv:2509.01140]. First, Wood asked for the infimum of $c > 0$ such that there exists $c'$ such that each graph $G$ has a tree decomposition of width $c \cdot tw(G)$ in which each vertex $v$ has spread at most $c'(d(v)+1)$. We show that the answer is $3$. Second, we prove a conjecture of Wood, stating that every tree-decomposition of the $(n \times n)$-grid with width $n$ has a vertex with spread $Ω(n)$. Finally, we answer the last question of Wood by showing that near-optimal average spread can be achieved simultaneously with width $O(tw(G))$.

math.CO↗

arXiv2026

On Equivalent Characterizations of the Polynomial Hierarchy in Abstract Models of Computation

We investigate machine models similar to Turing machines that are augmented with the operations of a first-order structure $\mathcal{R}$, and we show that under weak conditions on $\mathcal{R}$, the complexity class $Σ_k \mathcal{R}$ may be characterized in four equivalent ways: (1) by polynomial-time algorithms implemented on $\mathcal{R}$-machines together with witness strings, (2) by the $Σ_k\mathcal{R}$-complete problem $Σ_k\text{SAT}(\mathcal{R})$, (3) by the $k$th existential fragment of second-order metafinite logic over $\mathcal{R}$ via descriptive complexity, and (4) via oracles. By characterizing $Σ_k\mathcal{R}$ in these four ways, we extend previous work and embed it in one coherent framework. In addition, we derive similar results for $\exists_k \mathcal{R}$, the constant-free Boolean part of $Σ_k\mathcal{R}$, by showing that $\exists_k\mathcal{R}$ may be characterized in four analogous ways. Some conditions on $\mathcal{R}$ must be assumed in order to achieve the above quaternity because there are infinite-vocabulary structures for which $\text{NP}(\mathcal{R}) = Σ_1 \mathcal{R}$ does not have a complete problem. Surprisingly, even in these cases, we show that $\text{NP}(\mathcal{R})$ does have a characterization in terms of existential second-order metafinite logic, suggesting that descriptive complexity theory is well suited to working with infinite-vocabulary structures, such as real vector spaces.

cs.LO↗

arXiv2026

The Parameterized Complexity of Problems on Outer k-Planar Graphs

A graph is outer k-planar if it admits a straight-line drawing in which all vertices lie on a circle and every edge is crossed by at most k other edges. We study the parameterized complexity of a broad collection of graph problems on outer k-planar graphs, with k as the parameter. Many graph problems are known to be XALP-hard when parameterized by treewidth or outerplanarity, and XNLP-hard when parameterized by pathwidth. We show that only a few such problems, including Binary CSP and Scattered Set, remain intractable on outer k-planar graphs, whereas a large class of the others become fixed-parameter tractable in this setting, assuming that an outer k-planar drawing of the input graph is given. These include List Coloring, Capacitated Dominating Set, Capacitated Vertex Cover, Target Outdegree Orientation, and Target Set Selection, among others. In addition to the algorithmic and complexity results, we establish several structural results. We show that outer k-planar graphs have mim-width at most k+2, that graphs of cut-width at most k are outer 2k-planar, and that graphs of feedback edge set number at most k are outer 6k-planar. We also show that many graph parameters are incomparable with outer k-planarity, thereby clarifying its position within the graph parameter hierarchy.

cs.DS↗

arXiv2026

The Parameterized Complexity of Scheduling with Precedence Delays: Shuffle Product and Directed Bandwidth

In this paper, we study the parameterized complexity of several variants of scheduling with precedence constraints between jobs. Namely, we consider the single machine setting with delay values on top of the precedence constraints. Such scheduling problems are related to several decades-old problems with open parameterized complexity status, notably Shuffle Product and Directed Bandwidth. We obtain XNLP-completeness results for both problems, and derive implications to scheduling with minimum (resp. maximum) delays parameterized by the width of the directed acyclic graph giving the precedence constraints, and/or by the maximum delay value in the input. Regarding Directed Bandwidth, we also settle the case of trees by showing XNLP-completeness parameterized by the target value. Beyond these results, we believe that Shuffle Product is an unusual and promising addition to the list of XNLP-complete problems.

cs.DS↗

arXiv2025

On Stable Cutsets in General and Minimum Degree Constrained Graphs

A stable cutset is a set of vertices $S$ of a connected graph, that is pairwise non-adjacent and when deleting $S$, the graph becomes disconnected. Determining the existence of a stable cutset in a graph is known to be NP-complete. In this paper, we introduce a new exact algorithm for Stable Cutset. By branching on graph configurations and using the $O^*(1.3645)$ algorithm for the (3,2)-Constraint Satisfaction Problem presented by Beigel and Eppstein, we achieve an improved running time of $O^*(1.2972^n)$. In addition, we investigate the Stable Cutset problem for graphs with a bound on the minimum degree $δ$. First, we show that if the minimum degree of a graph $G$ is at least $\frac{2}{3}(n-1)$, then $G$ does not contain a stable cutset. Furthermore, we provide a polynomial-time algorithm for graphs where $δ\geq \tfrac{1}{2}n$, and a similar kernelisation algorithm for graphs where $δ= \tfrac{1}{2}n - k$. Finally, we prove that Stable Cutset remains NP-complete for graphs with minimum degree $c$, where $c > 1$. We design an exact algorithm for this problem that runs in $O^*(λ^n)$ time, where $λ$ is the positive root of $x^{δ+ 2} - x^{δ+ 1} + 6$. This algorithm can also be applied to the \textsc{3-Colouring} problem with the same minimum degree constraint, leading to an improved exact algorithm as well.

cs.DS↗

arXiv2025

Hedonic Seat Arrangement Problems

In this paper, we study a variant of hedonic games, called \textsc{Seat Arrangement}. The model is defined by a bijection from agents with preferences for each other to vertices in a graph $G$. The utility of an agent depends on the neighbors assigned in the graph. More precisely, it is the sum over all neighbors of the preferences that the agent has towards the agent assigned to the neighbor. We first consider the price of stability and fairness for different classes of preferences. In particular, we show that there is an instance such that the price of fairness ({\sf PoF}) is unbounded in general. Moreover, we show an upper bound $\tilde{d}(G)$ and an almost tight lower bound $\tilde{d}(G)-1/4$ of {\sf PoF}, where $\tilde{d}(G)$ is the average degree of an input graph. Then we investigate the computational complexity of problems to find certain ``good'' seat arrangements, say \textsc{Utilitarian Arrangement}, \textsc{Egalitarian Arrangement}, \textsc{Stable Arrangement}, and \textsc{Envy-free Arrangement}. We give dichotomies of computational complexity of four \textsc{Seat Arrangement} problems from the perspective of the maximum order of connected components in an input graph. For the parameterized complexity, \textsc{Utilitarian Arrangement} can be solved in time $n^{O(γ)}$, while it cannot be solved in time $f(γ)n^{o(γ)}$ under ETH, where $n$ is the number of agents and $γ$ is the vertex cover number of an input graph. Moreover, we show that \textsc{Egalitarian Arrangement} and \textsc{Envy-free Arrangement} are weakly NP-hard even on graphs of bounded vertex cover number. Finally, we prove that determining whether a stable arrangement can be obtained from a given arrangement by $k$ swaps is W[1]-hard when parameterized by $k+γ$, whereas it can be solved in time $n^{O(k)}$.

cs.GT↗

arXiv2025

Finding sparse induced subgraphs on graphs of bounded induced matching treewidth

The induced matching width of a tree decomposition of a graph $G$ is the cardinality of a largest induced matching $M$ of $G$, such that there exists a bag that intersects every edge in $M$. The induced matching treewidth of a graph $G$, denoted by $\mathsf{tree-}μ(G)$, is the minimum induced matching width of a tree decomposition of $G$. The parameter $\mathsf{tree-}μ$ was introduced by Yolov [SODA '18], who showed that, for example, Maximum-Weight Independent Set can be solved in polynomial-time on graphs of bounded $\mathsf{tree-}μ$. Lima, Milanič, Muršič, Okrasa, Rzążewski, and Štorgel [ESA '24] conjectured that this algorithm can be generalized to a meta-problem called Maximum-Weight Induced Subgraph of Bounded Treewidth, where we are given a vertex-weighted graph $G$, an integer $w$, and a $\mathsf{CMSO}_2$-sentence $Φ$, and are asked to find a maximum-weight set $X \subseteq V(G)$ so that $G[X]$ has treewidth at most $w$ and satisfies $Φ$. They proved the conjecture for some special cases, such as for the problem Maximum-Weight Induced Forest. In this paper, we prove the general case of the conjecture. In particular, we show that Maximum-Weight Induced Subgraph of Bounded Treewidth is polynomial-time solvable when $\mathsf{tree-}μ(G)$, $w$, and $|Φ|$ are bounded. The running time of our algorithm for $n$-vertex graphs $G$ with $\mathsf{tree} - μ(G) \le k$ is $f(k, w, |Φ|) \cdot n^{O(k w^2)}$ for a computable function $f$.

cs.DS↗

arXiv2025

Concurrency Constrained Scheduling with Tree-Like Constraints

This paper investigates concurrency-constrained scheduling problems, where the objective is to construct a schedule for a set of jobs subject to concurrency restrictions. Formally, we are given a conflict graph $G$ defined over a set of $n$ jobs, where an edge between two jobs in $G$ indicates that these jobs cannot be executed concurrently. Each job may have distinct attributes, such as processing time, due date, weight, and release time. The goal is to determine a schedule that optimizes a specified scheduling criterion while adhering to all concurrency constraints. This framework offers a versatile model for analyzing resource allocation problems where processes compete for shared resources, such as access to shared memory. From a theoretical perspective, it encompasses several classical graph coloring problems, including Chromatic Number, Sum Coloring, and Interval Chromatic Number. Given that even the simplest concurrency-constrained scheduling problems are NP-hard for general conflict graphs, this study focuses on conflict graphs with bounded treewidth. Our results establish a dichotomy: Some problems in this setting can be solved in FPT time, while others are shown to be XALP-complete for treewidth as parameter. Along the way, we generalize several previously known results on coloring problems for bounded treewidth graphs. Several of the FPT algorithms are based on the insight that completion times are bounded by the Grundy number of the conflict graph - the fact that this number is bounded by the product of treewidth and the logarithm of the number of vertices then leads to the FPT time bound.

cs.DM↗

arXiv2025

On the parameterized complexity of computing tree-partitions

We study the parameterized complexity of computing the tree-partition-width, a graph parameter equivalent to treewidth on graphs of bounded maximum degree. On one hand, we can obtain approximations of the tree-partition-width efficiently: we show that there is an algorithm that, given an $n$-vertex graph $G$ and an integer $k$, constructs a tree-partition of width $O(k^7)$ for $G$ or reports that $G$ has tree-partition-width more than $k$, in time $k^{O(1)}n^2$. We can improve slightly on the approximation factor by sacrificing the dependence on $k$, or on $n$. On the other hand, we show the problem of computing tree-partition-width exactly is XALP-complete, which implies that it is $W[t]$-hard for all $t$. We deduce XALP-completeness of the problem of computing the domino treewidth. Next, we adapt some known results on the parameter tree-partition-width and the topological minor relation, and use them to compare tree-partition-width to tree-cut width. Finally, for the related parameter weighted tree-partition-width, we give a similar approximation algorithm (with ratio now $O(k^{15})$) and show XALP-completeness for the special case where vertices and edges have weight 1.

cs.DM↗

arXiv2025

XALP-completeness of Parameterized Problems on Planar Graphs

The class XNLP consists of (parameterized) problems that can be solved nondeterministically in $f(k)n^{O(1)}$ time and $f(k)\log n$ space, where $n$ is the size of the input instance and $k$ the parameter. The class XALP consists of problems that can be solved in the above time and space with access to an additional stack. These two classes are a "natural home" for many standard graph problems and their generalizations. In this paper, we show the hardness of several problems on planar graphs, parameterized by outerplanarity, treewidth and pathwidth, thus strengthening several existing results. In particular, we show the XALP-completeness of the following problems parameterized by outerplanarity: All-or-Nothing Flow, Target Outdegree Orientation, Capacitated (Red-Blue) Dominating Set, Target Set Selections etc. We also show the XNLP-completeness of Scattered Set parameterized by pathwidth and XALP-completeness parameterized by treewidth and outerplanarity.

cs.CC↗

arXiv2024

On the Complexity of Problems on Tree-structured Graphs

In this paper, we introduce a new class of parameterized problems, which we call XALP: the class of all parameterized problems that can be solved in $f(k)n^{O(1)}$ time and $f(k)\log n$ space on a non-deterministic Turing Machine with access to an auxiliary stack (with only top element lookup allowed). Various natural problems on `tree-structured graphs' are complete for this class: we show that List Colouring and All-or-Nothing Flow parameterized by treewidth are XALP-complete. Moreover, Independent Set and Dominating Set parameterized by treewidth divided by $\log n$, and Max Cut parameterized by cliquewidth are also XALP-complete. Besides finding a `natural home' for these problems, we also pave the road for future reductions. We give a number of equivalent characterisations of the class XALP, e.g., XALP is the class of problems solvable by an Alternating Turing Machine whose runs have tree size at most $f(k)n^{O(1)}$ and use $f(k)\log n$ space. Moreover, we introduce `tree-shaped' variants of Weighted CNF-Satisfiability and Multicolour Clique that are XALP-complete.

cs.CC↗

arXiv2023

Parameterized Problems Complete for Nondeterministic FPT time and Logarithmic Space

Let XNLP be the class of parameterized problems such that an instance of size n with parameter k can be solved nondeterministically in time $f(k)n^{O(1)}$ and space $f(k)\log(n)$ (for some computable function f). We give a wide variety of XNLP-complete problems, such as List Coloring and Precoloring Extension with pathwidth as parameter, Scheduling of Jobs with Precedence Constraints, with both number of machines and partial order width as parameter, Bandwidth and variants of Weighted CNF-Satisfiability. In particular, this implies that all these problems are W[t]-hard for all t. This also answers a long standing question on the parameterized complexity of the Bandwidth problem.

cs.CC↗

arXiv2023

Parameterized Complexities of Dominating and Independent Set Reconfiguration

We settle the parameterized complexities of several variants of independent set reconfiguration and dominating set reconfiguration, parameterized by the number of tokens. We show that both problems are XL-complete when there is no limit on the number of moves, XNL-complete when a maximum length $\ell$ for the sequence is given in binary in the input, and XNLP-complete when $\ell$ is given in unary. The problems were known to be $\mathrm{W}[1]$- and $\mathrm{W}[2]$-hard respectively when $\ell$ is also a parameter. We complete the picture by showing membership in those classes. Moreover, we show that for all the variants that we consider, token sliding and token jumping are equivalent under pl-reductions. We introduce partitioned variants of token jumping and token sliding, and give pl-reductions between the four variants that have precise control over the number of tokens and the length of the reconfiguration sequence.

cs.CC↗

arXiv2023

The Parameterised Complexity of Integer Multicommodity Flow

The Integer Multicommodity Flow problem has been studied extensively in the literature. However, from a parameterised perspective, mostly special cases, such as the Disjoint Paths problem, have been considered. Therefore, we investigate the parameterised complexity of the general Integer Multicommodity Flow problem. We show that the decision version of this problem on directed graphs for a constant number of commodities, when the capacities are given in unary, is XNLP-complete with pathwidth as parameter and XALP-complete with treewidth as parameter. When the capacities are given in binary, the problem is NP-complete even for graphs of pathwidth at most 13. We give related results for undirected graphs. These results imply that the problem is unlikely to be fixed-parameter tractable by these parameters. In contrast, we show that the problem does become fixed-parameter tractable when weighted tree partition width (a variant of tree partition width for edge weighted graphs) is used as parameter.

cs.DM↗

arXiv2023

Complexity Framework for Forbidden Subgraphs IV: The Steiner Forest Problem

We study Steiner Forest on $H$-subgraph-free graphs, that is, graphs that do not contain some fixed graph $H$ as a (not necessarily induced) subgraph. We are motivated by a recent framework that completely characterizes the complexity of many problems on $H$-subgraph-free graphs. However, in contrast to e.g. the related Steiner Tree problem, Steiner Forest falls outside this framework. Hence, the complexity of Steiner Forest on $H$-subgraph-free graphs remained tantalizingly open. In this paper, we make significant progress towards determining the complexity of Steiner Forest on $H$-subgraph-free graphs. Our main results are four novel polynomial-time algorithms for different excluded graphs $H$ that are central to further understand its complexity. Along the way, we study the complexity of Steiner Forest for graphs with a small $c$-deletion set, that is, a small set $S$ of vertices such that each component of $G-S$ has size at most $c$. Using this parameter, we give two noteworthy algorithms that we later employ as subroutines. First, we prove Steiner Forest is FPT parameterized by $|S|$ when $c=1$ (i.e. the vertex cover number). Second, we prove Steiner Forest is polynomial-time solvable for graphs with a 2-deletion set of size at most 2. The latter result is tight, as the problem is NP-complete for graphs with a 3-deletion set of size 2.

math.CO↗

arXiv2023

Parameterized Complexity of Binary CSP: Vertex Cover, Treedepth, and Related Parameters

We investigate the parameterized complexity of Binary CSP parameterized by the vertex cover number and the treedepth of the constraint graph, as well as by a selection of related modulator-based parameters. The main findings are as follows: i) Binary CSP parameterized by the vertex cover number is $\mathrm{W}[3]$-complete. More generally, for every positive integer $d$, Binary CSP parameterized by the size of a modulator to a treedepth-d graph is $\mathrm{W}[2d+1]$-complete. This provides a new family of natural problems that are complete for odd levels of the W-hierarchy. ii) We introduce a new complexity class XSLP, defined so that Binary CSP parameterized by treedepth is complete for this class. We provide two equivalent characterizations of XSLP: the first one relates XSLP to a model of an alternating Turing machine with certain restrictions on conondeterminism and space complexity, while the second one links XSLP to the problem of model-checking first-order logic with suitably restricted universal quantification. Interestingly, the proof of the machine characterization of XSLP uses the concept of universal trees, which are prominently featured in the recent work on parity games iii) We describe a new complexity hierarchy sandwiched between the W-hierarchy and the A-hierarchy: For every odd $t$, we introduce a parameterized complexity class $\mathrm{S}[t]$ with $\mathrm{W}[t]\subseteq \mathrm{S}[t]\subseteq \mathrm{A}[t]$, defined using a parameter that interpolates between the vertex cover number and the treedepth. We expect that many of the studied classes will be useful in the future for pinpointing the complexity of various structural parameterizations of graph problems.

cs.DM↗

arXiv2023

Treewidth is NP-Complete on Cubic Graphs (and related results)

In this paper, we give a very simple proof that Treewidth is NP-complete; this proof also shows NP-completeness on the class of co-bipartite graphs. We then improve the result by Bodlaender and Thilikos from 1997 that Treewidth is NP-complete on graphs with maximum degree at most 9, by showing that Treewidth is NP-complete on cubic graphs.

cs.CC↗

arXiv2023

An ETH-Tight Exact Algorithm for Euclidean TSP

We study exact algorithms for Euclidean TSP in $\mathbb{R}^d$. In the early 1990s algorithms with $n^{O(\sqrt{n})}$ running time were presented for the planar case, and some years later an algorithm with $n^{O(n^{1-1/d})}$ running time was presented for any $d\geq 2$. Despite significant interest in subexponential exact algorithms over the past decade, there has been no progress on Euclidean TSP, except for a lower bound stating that the problem admits no $2^{O(n^{1-1/d-ε})}$ algorithm unless ETH fails. Up to constant factors in the exponent, we settle the complexity of Euclidean TSP by giving a $2^{O(n^{1-1/d})}$ algorithm and by showing that a $2^{o(n^{1-1/d})}$ algorithm does not exist unless ETH fails.

cs.CG↗