arXiv subjects

Christian Komusiewicz

Publications and source records attributed to Christian Komusiewicz.

At least 19 recordsLinked to original sources

arXiv2026

The Influence of Agent Models on the Complexity of Bus Routing

In bus routing, the task is to plan a bus route in a network with several agents, each of whom wants to travel from a starting point to a destination. A bus route should account for several factors, including agents' cost for reaching the bus stops, their travel time, or the energy consumption of the buses. We study the complexity of several variants of this problem, focusing on how the objective function and the models for agents' walking costs influence the problem complexity. After observing that even the simplest agent cost model leads to hardness on general networks, we consider networks with tree structure. Our main findings are as follows. First, allowing agent-specific cost models leads to hardness even on extremely limited trees such as stars. Second, consistent agent models (where agents differ only in their starting points and destinations) make the problem easier in some cases. Finally, allowing agents to choose between using the bus and walking directly can make the problem considerably harder. Most of our hardness results show not only classical NP-hardness but also parameterized intractability for the natural parameter $k$, the number of bus stops.

cs.CC↗

arXiv2026

Clustering with Locally Bounded Ignorance

In Correlation Clustering, the input is a graph $G=(V,E)$ with weight function $\omega: {V \choose 2}\to Z$ and the task is to partition the vertex set into clusters such that the total weight of edges between clusters and missing edges inside clusters is minimized. Due to close connections between Correlation Clustering and Edge Multicut, deciding whether there is a partition with total cost at most $k$ is FPT with respect to $k$ but a polynomial kernel is presumably impossible. We study the influence of the structure of the fuzzy edge graph, that is, the graph induced by the weight-0 edges, on the problem complexity. We show in particular that Correlation Clustering admits a polynomial problem kernel when parameterized by $k+d$, where $d$ is the degeneracy of the fuzzy edge graph, and when parameterized by $k+c$, where $c$ is the closure of the fuzzy edge graph. We complement these positive results by showing hardness for several settings where the graph induced by the edges and nonedges has very restricted structure.

cs.DS↗

arXiv2026

Parameterized Local Search for Vertex Cover: When only the Search Radius is Crucial

A vertex set $W$ in a graph $G$ is a valid $k$-swap for a vertex cover $S$ of $G$ if $W$ has size at most $k$ and $S'=(S \setminus W) \cup (W \setminus S)$, the symmetric difference of $S$ and $W$, is a vertex cover of $G$. If $|S'| < |S|$, then $W$ is improving. In LS Vertex Cover, one is given a vertex cover $S$ of a graph $G$ and wants to know if there is a valid improving $k$-swap for $S$ in $G$. In applications of LS Vertex Cover, $k$ is a very small parameter that can be set by a user to determine the trade-off between running time and solution quality. Consequently, $k$ can be considered to be a constant. Motivated by this and the fact that LS Vertex Cover is W[1]-hard with respect to $k$, we aim for algorithms with running time $\ell^{f(k)}\cdot n^{\mathcal{O}(1)}$ where $\ell$ is a structural graph parameter upper-bounded by $n$. We say that such a running time grows mildly with respect to $\ell$ and strongly with respect to $k$. We obtain algorithms with such a running time for $\ell$ being the $h$-index of $G$, the treewidth of $G$, or the modular-width of $G$. In addition, we consider a novel parameter, the maximum degree over all quotient graphs in a modular decomposition of $G$. Moreover, we adapt these algorithms to the more general problem where each vertex is assigned a weight and where we want to find a valid $d$-improving $k$-swap, that is, a valid $k$-swap which decreases the weight of the vertex cover by at least $d$.

cs.DS↗

arXiv2026

On the Hardness of Finding Temporally Connected Subgraphs of Any Size

Temporal graphs are graphs whose edges are present only at certain points in time. Reachability in these graphs is defined via temporal paths, in which edges are traversed in chronological order. A temporal graph is temporally connected (or TC) if every ordered pair of vertices is connected by a temporal path. When the graph itself is not TC, a natural question is whether it admits a TC subgraph (a.k.a.~closed temporal component) of a given size $k$. This question was one of the earliest and most studied in the field, shown to be NP-hard by Bhadra and Ferreira in 2003. We strengthen this result dramatically, showing that deciding if a temporal graph admits a TC subgraph \emph{of any size} (beyond the trivial case of a single vertex in the directed and a single edge in the undirected case) is already NP-hard. Our result holds for all standard temporal graph settings, answering a series of open questions in the field and strengthening several existing results. This sharply separates closed components from open ones (where temporal paths can travel outside the component), for which the analogous problem is trivially solvable in polynomial time. More precisely, our reductions imply that the size of the largest TC subgraph cannot even be approximated within a factor of~$(1-\epsilon)n$ in directed graphs, and within a factor of~$(1-\epsilon)\frac{n}{2}$ in undirected graphs. They also complete the complexity landscape for TC subgraphs of size exactly $k$ when parameterized by $k$ (answering the missing ``non-strict'' case). Our results also have structural implications. In particular, they imply that there exist arbitrarily large TC graphs without nontrivial TC subgraphs, and that there exist TC graphs of arbitrary girth, both facts being of independent interest.

cs.CC↗

arXiv2026

The Descriptive Complexity of Relation Modification Problems

A relation modification problem gets a logical structure and a natural number k as input and asks whether k modifications of the structure suffice to make it satisfy a predefined property. We provide a complete classification of the classical and parameterized complexity of relation modification problems - the latter w. r. t. the modification budget k - based on the descriptive complexity of the respective target property. We consider different types of logical structures on which modifications are performed: Whereas monadic structures and undirected graphs without self-loops each yield their own complexity landscapes, we find that modifying undirected graphs with self-loops, directed graphs, or arbitrary logical structures is equally hard w. r. t. quantifier patterns. Moreover, we observe that all classes of problems considered in this paper are subject to a strong dichotomy in the sense that they are either very easy to solve (that is, they lie in paraAC^{0\uparrow} or TC^0) or intractable (that is, they contain W[2]-hard or NP-hard problems).

cs.LO↗

arXiv2026

A Parameterized-Complexity Framework for Finding Local Optima

Local search is a fundamental optimization technique that is both widely used in practice and deeply studied in theory, yet its computational complexity remains poorly understood. The traditional frameworks, PLS and the standard algorithm problem, introduced by Johnson, Papadimitriou, and Yannakakis (1988) fail to capture the methodology of local search algorithms: PLS is concerned with finding a local optimum and not with using local search, while the standard algorithm problem restricts each improvement step to follow a fixed pivoting rule. In this work, we introduce a novel formulation of local search which provides a middle ground between these models. In particular, the task is to output not only a local optimum but also a chain of local improvements leading to it. With this framework, we aim to capture the challenge in designing a good pivoting rule. Especially, when combined with the parameterized complexity paradigm, it enables both strong lower bounds and meaningful tractability results. Unlike previous works that combined parameterized complexity with local search, our framework targets the whole task of finding a local optimum and not only a single improvement step. Focusing on two representative meta-problems -- Subset Weight Optimization Problem with the $c$-swap neighborhood and Weighted Circuit with the flip neighborhood -- we establish fixed-parameter tractability results related to the number of distinct weights, while ruling out an analogous result when parameterized by the distance to the nearest optimum via a new type of reduction.

cs.CC↗

arXiv2025

A Complexity Analysis of the c-Closed Vertex Deletion Problem

A graph is $c$-closed when every pair of nonadjacent vertices has at most $c-1$ common neighbors. In $c$-Closed Vertex Deletion, the input is a graph $G$ and an integer $k$ and we ask whether $G$ can be transformed into a $c$-closed graph by deleting at most $k$ vertices. We study the classic and parameterized complexity of $c$-Closed Vertex Deletion. We obtain, for example, NP-hardness for the case that $G$ is bipartite with bounded maximum degree. We also show upper and lower bounds on the size of problem kernels for the parameter $k$ and introduce a new parameter, the number $x$ of vertices in bad pairs, for which we show a problem kernel of size $O(x^3 + x^2\cdot c)$. Here, a pair of nonadjacent vertices is bad if they have at least $c$ common neighbors. Finally, we show that $c$-Closed Vertex Deletion can be solved in polynomial time on unit interval graphs with depth at most $c+1$ and that it is fixed-parameter tractable with respect to the neighborhood diversity of $G$.

cs.DS↗

arXiv2025

Timeline Problems in Temporal Graphs: Vertex Cover vs. Dominating Set

A temporal graph is a finite sequence of graphs, called snapshots, over the same vertex set. Many temporal graph problems turn out to be much more difficult than their static counterparts. One such problem is \textsc{Timeline Vertex Cover} (also known as \textsc{MinTimeline$_\infty$}), a temporal analogue to the classical \textsc{Vertex Cover} problem. In this problem, one is given a temporal graph $\mathcal{G}$ and two integers $k$ and $\ell$, and the goal is to cover each edge of each snapshot by selecting for each vertex at most $k$ activity intervals of length at most $\ell$ each. Here, an edge $uv$ in the $i$th snapshot is covered, if an activity interval of $u$ or $v$ is active at time $i$. In this work, we continue the algorithmic study of \textsc{Timeline Vertex Cover} and introduce the \textsc{Timeline Dominating Set} problem where we want to dominate all vertices in each snapshot by the selected activity intervals. We analyze both problems from a classical and parameterized point of view and also consider partial problem versions, where the goal is to cover (dominate) at least $t$ edges (vertices) of the snapshots. With respect to the parameterized complexity, we consider the temporal graph parameters vertex-interval-membership-width $(vimw)$ and interval-membership-width $(imw)$. We show that all considered problems admit FPT-algorithms when parameterized by $vimw + k+\ell$. This provides a smaller parameter combination than the ones used for previously known FPT-algorithms for \textsc{Timeline Vertex Cover}. Surprisingly, for $imw+ k+\ell$, \textsc{Timeline Dominating Set} turns out to be easier than \textsc{Timeline Vertex Cover}, by also admitting an FPT-algorithm, whereas the vertex cover version is NP-hard even if $imw+\, k+\ell$ is constant. We also consider parameterization by combinations of $n$, the vertex set size, with $k$ or $\ell$ and parameterization by $t$.

cs.DS↗

arXiv2025

The Parameter Report: An Orientation Guide for Data-Driven Parameterization

A strength of parameterized algorithmics is that each problem can be parameterized by an essentially inexhaustible set of parameters. Usually, the choice of the considered parameter is informed by the theoretical relations between parameters with the general goal of achieving FPT-algorithms for smaller and smaller parameters. However, the FPT-algorithms for smaller parameters usually have higher running times and it is unclear whether the decrease in the parameter value or the increase in the running time bound dominates in real-world data. This question cannot be answered from purely theoretical considerations and any answer requires knowledge on typical parameter values. To provide a data-driven guideline for parameterized complexity studies of graph problems, we present the first comprehensive comparison of parameter values for a set of benchmark graphs originating from real-world applications. Our study covers degree-related parameters, such as maximum degree or degeneracy, neighborhood-based parameters such as neighborhood diversity and modular-width, modulator-based parameters such as vertex cover number and feedback vertex set number, and the treewidth of the graphs. Our results may help assess the significance of FPT-running time bounds on the solvability of real-world instances. For example, the vertex cover number $vc$ of $n$-vertex graphs is often only slightly below $n/2$. Thus, a running time bound of $O(2^{vc})$ is only slightly better than a running time bound of $O(1.4^{n})$. In contrast, the treewidth $tw$ is almost always below $n/3$ and often close to $n/10$, making a running time of $O(2^{tw})$ much more practical on real-world instances. We make our implementation and full experimental data openly available. In particular, this provides the first implementations for several graph parameters such as 4-path vertex cover number and vertex integrity.

cs.CC↗

arXiv2025

Polynomial-Size Enumeration Kernelizations for Long Path Enumeration

Enumeration kernelization for parameterized enumeration problems was defined by Creignou et al. [Theory Comput. Syst. 2017] and was later refined by Golovach et al. [J. Comput. Syst. Sci. 2022, STACS 2021] to polynomial-delay enumeration kernelization. We consider ENUM LONG-PATH, the enumeration variant of the Long-Path problem, from the perspective of enumeration kernelization. Formally, given an undirected graph G and an integer k, the objective of ENUM LONG-PATH is to enumerate all paths of G having exactly k vertices. We consider the structural parameters vertex cover number, dissociation number, and distance to clique and provide polynomial-delay enumeration kernels of polynomial size for each of these parameters.

cs.DM↗

arXiv2024

Witty: An Efficient Solver for Computing Minimum-Size Decision Trees

Decision trees are a classic model for summarizing and classifying data. To enhance interpretability and generalization properties, it has been proposed to favor small decision trees. Accordingly, in the minimum-size decision tree training problem (MSDT), the input is a set of training examples in $\mathbb{R}^d$ with class labels and we aim to find a decision tree that classifies all training examples correctly and has a minimum number of nodes. MSDT is NP-hard and therefore presumably not solvable in polynomial time. Nevertheless, Komusiewicz et al. [ICML '23] developed a promising algorithmic paradigm called witness trees which solves MSDT efficiently if the solution tree is small. In this work, we test this paradigm empirically. We provide an implementation, augment it with extensive heuristic improvements, and scrutinize it on standard benchmark instances. The augmentations achieve a mean 324-fold (median 84-fold) speedup over the naive implementation. Compared to the state of the art they achieve a mean 32-fold (median 7-fold) speedup over the dynamic programming based MurTree solver [Demirovi\'c et al., J. Mach. Learn. Res. '22] and a mean 61-fold (median 25-fold) speedup over SAT-based implementations [Janota and Morgado, SAT '20]. As a theoretical result we obtain an improved worst-case running-time bound for MSDT.

cs.DS↗

arXiv2024

Parameterized Local Search for Max $c$-Cut

In the NP-hard Max $c$-Cut problem, one is given an undirected edge-weighted graph $G$ and aims to color the vertices of $G$ with $c$ colors such that the total weight of edges with distinctly colored endpoints is maximal. The case with $c=2$ is the famous Max Cut problem. To deal with the NP-hardness of this problem, we study parameterized local search algorithms. More precisely, we study LS Max $c$-Cut where we are also given a vertex coloring and an integer $k$ and the task is to find a better coloring that changes the color of at most $k$ vertices, if such a coloring exists; otherwise, the given coloring is $k$-optimal. We show that, for all $c\ge 2$, LS Max $c$-Cut presumably cannot be solved in $f(k)\cdot n^{\mathcal{O}(1)}$ time even on bipartite graphs. We then present an algorithm for LS Max $c$-Cut with running time $\mathcal{O}((3eΔ)^k\cdot c\cdot k^3\cdotΔ\cdot n)$, where $Δ$ is the maximum degree of the input graph. Finally, we evaluate the practical performance of this algorithm in a hill-climbing approach as a post-processing for a state-of-the-art heuristic for Max $c$-Cut. We show that using parameterized local search, the results of this state-of-the-art heuristic can be further improved on a set of standard benchmark instances.

cs.CC↗

arXiv2024

Maximizing Phylogenetic Diversity under Ecological Constraints: A Parameterized Complexity Study

In the NP-hard Optimizing PD with Dependencies (PDD) problem, the input consists of a phylogenetic tree $T$ over a set of taxa $X$, a food-web that describes the prey-predator relationships in $X$, and integers $k$ and $D$. The task is to find a set $S$ of $k$ species that is viable in the food-web such that the subtree of $T$ obtained by retaining only the vertices of $S$ has total edge weight at least $D$. Herein, viable means that for every predator taxon of $S$, the set $S$ contains at least one prey taxon. We provide the first systematic analysis of PDD and its special case s-PDD from a parameterized complexity perspective. For solution-size related parameters, we show that PDD is FPT with respect to $D$ and with respect to $k$ plus the height of the phylogenetic tree. Moreover, we consider structural parameterizations of the food-web. For example, we show an FPT-algorithm for the parameter that measures the vertex deletion distance to graphs where every connected component is a complete graph. Finally, we show that s-PDD admits an FPT-algorithm for the treewidth of the food-web. This disproves a conjecture of Faller et al. [Annals of Combinatorics, 2011] who conjectured that s-PDD is NP-hard even when the food-web is a tree.

cs.CC↗

arXiv2024

On the Complexity of Community-aware Network Sparsification

Network sparsification is the task of reducing the number of edges of a given graph while preserving some crucial graph property. In community-aware network sparsification, the preserved property concerns the subgraphs that are induced by the communities of the graph which are given as vertex subsets. This is formalized in the $Π$-Network Sparsification problem: given an edge-weighted graph $G$, a collection $Z$ of $c$ subsets of $V(G)$ (communities), and two numbers $\ell, b$, the question is whether there exists a spanning subgraph $G'$ of $G$ with at most $\ell$ edges of total weight at most $b$ such that $G'[C]$ fulfills $Π$ for each community $C$. Here, we consider two graph properties $Π$: the connectivity property (Connectivity NWS) and the property of having a spanning star (Stars NWS). Since both problems are NP-hard, we study their parameterized and fine-grained complexity. We provide a tight $2^{Ω(n^2+c)} poly(n+|Z|)$-time running time lower bound based on the ETH for both problems, where $n$ is the number of vertices in $G$. The lower bound holds even in the restricted case when all communities have size at most 4, $G$ is a clique, and every edge has unit weight. For the connectivity property, the unit weight case with $G$ being a clique is the well-studied problem of computing a hypergraph support with a minimum number of edges. We then study the complexity of both problems parameterized by the feedback edge number $t$ of the solution graph $G'$. For Stars NWS, we present an XP-algorithm for $t$. This answers an open question by Korach and Stern [Disc. Appl. Math. '08] who asked for the existence of polynomial-time algorithms for $t=0$. In contrast, we show for Connectivity NWS that known polynomial-time algorithms for $t=0$ [Korach and Stern, Math. Program. '03; Klemz et al., SWAT '14] cannot be extended by showing that Connectivity NWS is NP-hard for $t=1$.

cs.DS↗

arXiv2023

Enumeration Kernels of Polynomial Size for Cuts of Bounded Degree

Enumeration kernelization was first proposed by Creignou et al. [TOCS 2017] and was later refined by Golovach et al. [JCSS 2022] into two different variants: fully-polynomial enumeration kernelization and polynomial-delay enumeration kernelization. In this paper, we consider the d-CUT problem from the perspective of (polynomial-delay) enumeration kenrelization. Given an undirected graph G = (V, E), a cut F = (A, B) is a d-cut of G if every $u \in A$ has at most d neighbors in B and every $v \in B$ has at most d neighbors in A. Checking the existence of a d-cut in a graph is a well-known NP-hard problem and is well-studied in parameterized complexity [Algorithmica 2021, IWOCA 2021]. This problem also generalizes a well-studied problem MATCHING CUT (set d = 1) that has been a central problem in the literature of polynomial-delay enumeration kernelization. In this paper, we study three different enumeration variants of this problem, ENUM d-CUT, ENUM MIN-d-CUT and ENUM MAX-d-CUT that intends to enumerate all the d-cuts, all the minimal d-cuts and all the maximal d-cuts respectively. We consider various structural parameters of the input, e.g. vertex cover number, neighborhood diversity, and clique partition number. When vertex cover number and neighborhood diversity are considered as parameters, we provide polynomial-delay enumeration kernelizations of polynomial size for ENUM d-CUT and ENUM MAX-d-CUT and fully-polynomial enumeration kernels of polynomial size for ENUM MIN-d-CUT. When clique partition number is considered as the parameter, we provide bijective enumeration kernels for each of these three problems.

cs.DS↗

arXiv2023

A Multivariate Complexity Analysis of the Generalized Noah's Ark Problem

In the Generalized Noah's Ark Problem, one is given a phylogenetic tree on a set of species X and a set of conservation projects for each species. Each project comes with a cost and raises the survival probability of the corresponding species. The aim is to select a conservation project for each species such that the total cost of the selected projects does not exceed some given threshold and the expected phylogenetic diversity is as large as possible. We study the complexity of Generalized Noah's Ark Problem and some of its special cases with respect to several parameters related to the input structure, such as the number of different costs, the number of different survival probabilities, or the number of species, |X|.

cs.DS↗

arXiv2023

On Computing Optimal Tree Ensembles

Random forests and, more generally, (decision\nobreakdash-)tree ensembles are widely used methods for classification and regression. Recent algorithmic advances allow to compute decision trees that are optimal for various measures such as their size or depth. We are not aware of such research for tree ensembles and aim to contribute to this area. Mainly, we provide two novel algorithms and corresponding lower bounds. First, we are able to carry over and substantially improve on tractability results for decision trees: We obtain an algorithm that, given a training-data set and an size bound $S \in \mathbb{R}$, computes a tree ensemble of size at most $S$ that classifies the data correctly. The algorithm runs in $(4\delta D S)^S \cdot poly$-time, where $D$ the largest domain size, $\delta$ is the largest number of features in which two examples differ, $n$ the number of input examples, and $poly$ a polynomial of the input size. For decision trees, that is, ensembles of size 1, we obtain a running time of $(\delta D s)^s \cdot poly$, where $s$ is the size of the tree. To obtain these algorithms, we introduce the witness-tree technique, which seems promising for practical implementations. Secondly, we show that dynamic programming, which has been applied successfully to computing decision trees, may also be viable for tree ensembles, providing an $\ell^n \cdot poly$-time algorithm, where $\ell$ is the number of trees. Finally, we compare the number of cuts necessary to classify training data sets for decision trees and tree ensembles, showing that ensembles may need exponentially fewer cuts for increasing number of trees.

cs.LG↗

arXiv2023

Exact and Heuristic Approaches to Speeding Up the MSM Time Series Distance Computation

The computation of the distance of two time series is time-consuming for any elastic distance function that accounts for misalignments. Among those functions, DTW is the most prominent. However, a recent extensive evaluation has shown that the move-split merge (MSM) metric is superior to DTW regarding the analytical accuracy of the 1-NN classifier. Unfortunately, the running time of the standard dynamic programming algorithm for MSM distance computation is $Ω(n^2)$, where $n$ is the length of the longest time series. In this paper, we provide approaches to reducing the cost of MSM distance computations by using lower and upper bounds for early pruning paths in the underlying dynamic programming table. For the case of one time series being a constant, we present a linear-time algorithm. In addition, we propose new linear-time heuristics and adapt heuristics known from DTW to computing the MSM distance. One heuristic employs the metric property of MSM and the previously introduced linear-time algorithm. Our experimental studies demonstrate substantial speed-ups in our approaches compared to previous MSM algorithms. In particular, the running time for MSM is faster than a state-of-the-art DTW distance computation for a majority of the popular UCR data sets.

cs.DS↗