arXiv subjects

Christian Komusiewicz

Publications and source records attributed to Christian Komusiewicz.

At least 37 records · Page 2Linked to original sources

arXiv2023

Destroying Multicolored Paths and Cycles in Edge-Colored Graphs

We study the computational complexity of $c$-Colored $P_\ell$ Deletion and $c$-Colored $C_\ell$ Deletion. In these problems, one is given a $c$-edge-colored graph and wants to destroy all induced $c$-colored paths or cycles, respectively, on $\ell$ vertices by deleting at most $k$ edges. Herein, a path or cycle is $c$-colored if it contains edges of $c$ distinct colors. We show that $c$-Colored $P_\ell$ Deletion and $c$-Colored $C_\ell$ Deletion are NP-hard for each non-trivial combination of $c$ and $\ell$. We then analyze the parameterized complexity of these problems. We extend the notion of neighborhood diversity to edge-colored graphs and show that both problems are fixed-parameter tractable with respect to the colored neighborhood diversity of the input graph. We also provide hardness results to outline the limits of parameterization by the standard parameter solution size $k$. Finally, we consider bicolored input graphs and show a special case of $2$-Colored $P_4$ Deletion that can be solved in polynomial time.

cs.DS↗

arXiv2022

Efficient Branch-and-Bound Algorithms for Finding Triangle-Constrained 2-Clubs

In the Vertex Triangle 2-Club problem, we are given an undirected graph $G$ and aim to find a maximum-vertex subgraph of $G$ that has diameter at most 2 and in which every vertex is contained in at least $\ell$ triangles in the subgraph. So far, the only algorithm for solving Vertex Triangle 2-Club relies on an ILP formulation [Almeida and Brás, Comput. Oper. Res. 2019]. In this work, we develop a combinatorial branch-and-bound algorithm that, coupled with a set of data reduction rules, outperforms the existing implementation and is able to find optimal solutions on sparse real-world graphs with more than 100,000 vertices in a few minutes. We also extend our algorithm to the Edge Triangle 2-Club problem where the triangle constraint is imposed on all edges of the subgraph.

cs.DS↗

arXiv2022

Computing Dense and Sparse Subgraphs of Weakly Closed Graphs

A graph $G$ is weakly $γ$-closed if every induced subgraph of $G$ contains one vertex $v$ such that for each non-neighbor $u$ of $v$ it holds that $|N(u)\cap N(v)|<γ$. The weak closure $γ(G)$ of a graph, recently introduced by Fox et al. [SIAM J. Comp. 2020], is the smallest number such that $G$ is weakly $γ$-closed. This graph parameter is never larger than the degeneracy (plus one) and can be significantly smaller. Extending the work of Fox et al. [SIAM J. Comp. 2020] on clique enumeration, we show that several problems related to finding dense subgraphs, such as the enumeration of bicliques and $s$-plexes, are fixed-parameter tractable with respect to $γ(G)$. Moreover, we show that the problem of determining whether a weakly $γ$-closed graph $G$ has a subgraph on at least $k$ vertices that belongs to a graph class $\mathcal{G}$ which is closed under taking subgraphs admits a kernel with at most $γk^2$ vertices. Finally, we provide fixed-parameter algorithms for Independent Dominating Set and Dominating Clique when parameterized by $γ+k$ where $k$ is the solution size.

cs.DM↗

arXiv2022

Covering Many (or Few) Edges with k Vertices in Sparse Graphs

We study the following two fixed-cardinality optimization problems (a maximization and a minimization variant). For a fixed $α$ between zero and one we are given a graph and two numbers $k \in \mathbb{N}$ and $t \in \mathbb{Q}$. The task is to find a vertex subset $S$ of exactly $k$ vertices that has value at least (resp. at most for minimization) $t$. Here, the value of a vertex set computes as $α$ times the number of edges with exactly one endpoint in $S$ plus $1-α$ times the number of edges with both endpoints in $S$. These two problems generalize many prominent graph problems, such as Densest $k$-Subgraph, Sparsest $k$-Subgraph, Partial Vertex Cover, and Max ($k$,$n-k$)-Cut. In this work, we complete the picture of their parameterized complexity on several types of sparse graphs that are described by structural parameters. In particular, we provide kernelization algorithms and kernel lower bounds for these problems. A somewhat surprising consequence of our kernelizations is that Partial Vertex Cover and Max $(k,n-k)$-Cut not only behave in the same way but that the kernels for both problems can be obtained by the same algorithms.

cs.DS↗

arXiv2022

On Computing Exact Means of Time Series Using the Move-Split-Merge Metric

Computing an accurate mean of a set of time series is a critical task in applications like nearest-neighbor classification and clustering of time series. While there are many distance functions for time series, the most popular distance function used for the computation of time series means is the non-metric dynamic time warping (DTW) distance. A recent algorithm for the exact computation of a DTW-Mean has a running time of $\mathcal{O}(n^{2k+1}2^kk)$, where $k$ denotes the number of time series and $n$ their maximum length. In this paper, we study the mean problem for the move-split-merge (MSM) metric that not only offers high practical accuracy for time series classification but also carries of the advantages of the metric properties that enable further diverse applications. The main contribution of this paper is an exact and efficient algorithm for the MSM-Mean problem of time series. The running time of our algorithm is $\mathcal{O}(n^{k+3}2^k k^3 )$, and thus better than the previous DTW-based algorithm. The results of an experimental comparison confirm the running time superiority of our algorithm in comparison to the DTW-Mean competitor. Moreover, we introduce a heuristic to improve the running time significantly without sacrificing much accuracy.

cs.DS↗

arXiv2022

The role of twins in computing planar supports of hypergraphs

A support or realization of a hypergraph $H$ is a graph $G$ on the same vertex as $H$ such that for each hyperedge of $H$ it holds that its vertices induce a connected subgraph of $G$. The NP-hard problem of finding a planar support has applications in hypergraph drawing and network design. Previous algorithms for the problem assume that twins -- pairs of vertices that are in precisely the same hyperedges -- can safely be removed from the input hypergraph. We prove that this assumption is generally wrong, yet that the number of twins necessary for a hypergraph to have a planar support only depends on its number of hyperedges. We give an explicit upper bound on the number of twins necessary for a hypergraph with $m$ hyperedges to have an $r$-outerplanar support, which depends only on $r$ and $m$. Since all additional twins can be safely removed, we obtain a linear-time algorithm for computing $r$-outerplanar supports for hypergraphs with $m$ hyperedges if $m$ and $r$ are constant; in other words, the problem is fixed-parameter linear-time solvable with respect to the parameters $m$ and $r$.

cs.DM↗

arXiv2022

Exploiting $\mathbf{c}$-Closure in Kernelization Algorithms for Graph Problems

A graph is c-closed if every pair of vertices with at least c common neighbors is adjacent. The c-closure of a graph G is the smallest number such that G is c-closed. Fox et al. [ICALP '18] defined c-closure and investigated it in the context of clique enumeration. We show that c-closure can be applied in kernelization algorithms for several classic graph problems. We show that Dominating Set admits a kernel of size k^O(c), that Induced Matching admits a kernel with O(c^7*k^8) vertices, and that Irredundant Set admits a kernel with O(c^(5/2)*k^3) vertices. Our kernelization exploits the fact that c-closed graphs have polynomially-bounded Ramsey numbers, as we show.

cs.DM↗

arXiv2022

The Parameterized Complexity of s-Club with Triangle and Seed Constraints

The s-Club problem asks, for a given undirected graph $G$, whether $G$ contains a vertex set $S$ of size at least $k$ such that $G[S]$, the subgraph of $G$ induced by $S$, has diameter at most $s$. We consider variants of $s$-Club where one additionally demands that each vertex of $G[S]$ is contained in at least $\ell$ triangles in $G[S]$, that each edge of $G[S]$ is contained in at least $\ell$~triangles in $G[S]$, or that $S$ contains a given set $W$ of seed vertices. We show that in general these variants are W[1]-hard when parameterized by the solution size $k$, making them significantly harder than the unconstrained $s$-Club problem. On the positive side, we obtain some FPT algorithms for the case when $\ell=1$ and for the case when $G[W]$, the graph induced by the set of seed vertices, is a clique.

cs.DS↗

arXiv2022

Efficient Bayesian Network Structure Learning via Parameterized Local Search on Topological Orderings

In Bayesian Network Structure Learning (BNSL), one is given a variable set and parent scores for each variable and aims to compute a DAG, called Bayesian network, that maximizes the sum of parent scores, possibly under some structural constraints. Even very restricted special cases of BNSL are computationally hard, and, thus, in practice heuristics such as local search are used. A natural approach for a local search algorithm is a hill climbing strategy, where one replaces a given BNSL solution by a better solution within some pre-defined neighborhood as long as this is possible. We study ordering-based local search, where a solution is described via a topological ordering of the variables. We show that given such a topological ordering, one can compute an optimal DAG whose ordering is within inversion distance $r$ in subexponential FPT time; the parameter $r$ allows to balance between solution quality and running time of the local search algorithm. This running time bound can be achieved for BNSL without structural constraints and for all structural constraints that can be expressed via a sum of weights that are associated with each parent set. We also introduce a related distance called `window inversions distance' and show that the corresponding local search problem can also be solved in subexponential FPT time for the parameter $r$. For two further natural modification operations on the variable orderings, we show that algorithms with an FPT time for $r$ are unlikely. We also outline the limits of ordering-based local search by showing that it cannot be used for common structural constraints on the moralized graph of the network.

cs.DS↗

arXiv2021

Learning Bayesian Networks Under Sparsity Constraints: A Parameterized Complexity Analysis

We study the problem of learning the structure of an optimal Bayesian network when additional constraints are posed on the network or on its moralized graph. More precisely, we consider the constraint that the network or its moralized graph are close, in terms of vertex or edge deletions, to a sparse graph class $Π$. For example, we show that learning an optimal network whose moralized graph has vertex deletion distance at most $k$ from a graph with maximum degree 1 can be computed in polynomial time when $k$ is constant. This extends previous work that gave an algorithm with such a running time for the vertex deletion distance to edgeless graphs [Korhonen & Parviainen, NIPS 2015]. We then show that further extensions or improvements are presumably impossible. For example, we show that learning optimal networks where the network or its moralized graph have maximum degree $2$ or connected components of size at most $c$, $c\ge 3$, is NP-hard. Finally, we show that learning an optimal network with at most $k$ edges in the moralized graph presumably has no $f(k)\cdot |I|^{O(1)}$-time algorithm and that, in contrast, an optimal network with at most $k$ arcs can be computed in $2^{O(k)}\cdot |I|^{O(1)}$ time where $|I|$ is the total input size.

cs.DS↗

arXiv2021

Preventing Small $\mathbf{(s,t)}$-Cuts by Protecting Edges

We introduce and study Weighted Min $(s,t)$-Cut Prevention, where we are given a graph $G=(V,E)$ with vertices $s$ and $t$ and an edge cost function and the aim is to choose an edge set $D$ of total cost at most $d$ such that $G$ has no $(s,t)$-edge cut of capacity at most $a$ that is disjoint from $D$. We show that Weighted Min $(s,t)$-Cut Prevention is NP-hard even on subcubcic graphs when all edges have capacity and cost one and provide a comprehensive study of the parameterized complexity of the problem. We show, for example W[1]-hardness with respect to $d$ and an FPT algorithm for $a$.

cs.DS↗

arXiv2021

Destroying Bicolored $P_3$s by Deleting Few Edges

We introduce and study the Bicolored $P_3$ Deletion problem defined as follows. The input is a graph $G=(V,E)$ where the edge set $E$ is partitioned into a set $E_r$ of red edges and a set $E_b$ of blue edges. The question is whether we can delete at most $k$ edges such that $G$ does not contain a bicolored $P_3$ as an induced subgraph. Here, a bicolored $P_3$ is a path on three vertices with one blue and one red edge. We show that Bicolored $P_3$ Deletion is NP-hard and cannot be solved in $2^{o(|V|+|E|)}$ time on bounded-degree graphs if the ETH is true. Then, we show that Bicolored $P_3$ Deletion is polynomial-time solvable when $G$ does not contain a bicolored $K_3$, that is, a triangle with edges of both colors. Moreover, we provide a polynomial-time algorithm for the case that $G$ contains no blue $P_3$, red $P_3$, blue $K_3$, and red $K_3$. Finally, we show that Bicolored $P_3$ Deletion can be solved in $ O(1.84^k\cdot |V| \cdot |E|)$ time and that it admits a kernel with $ O(kΔ\min(k,Δ))$ vertices, where $Δ$ is the maximum degree of $G$.

cs.DS↗

arXiv2021

On the Parameterized Complexity of Polytree Learning

A Bayesian network is a directed acyclic graph that represents statistical dependencies between variables of a joint probability distribution. A fundamental task in data science is to learn a Bayesian network from observed data. \textsc{Polytree Learning} is the problem of learning an optimal Bayesian network that fulfills the additional property that its underlying undirected graph is a forest. In this work, we revisit the complexity of \textsc{Polytree Learning}. We show that \textsc{Polytree Learning} can be solved in $3^n \cdot |I|^{\mathcal{O}(1)}$ time where $n$ is the number of variables and $|I|$ is the total instance size. Moreover, we consider the influence of the number of variables $d$ that might receive a nonempty parent set in the final DAG on the complexity of \textsc{Polytree Learning}. We show that \textsc{Polytree Learning} has no $f(d)\cdot |I|^{\mathcal{O}(1)}$-time algorithm, unlike Bayesian network learning which can be solved in $2^d \cdot |I|^{\mathcal{O}(1)}$ time. We show that, in contrast, if $d$ and the maximum parent set size are bounded, then we can obtain efficient algorithms.

cs.DS↗

arXiv2021

Parameterized String Equations

We study systems of String Equations where block variables need to be assigned strings so that their concatenation gives a specified target string. We investigate this problem under a multivariate complexity framework, searching for tractable special cases such as systems of equations with few block variables or few equations. Our main results include a polynomial-time algorithm for size-2 equations, and hardness for size-3 equations, as well as hardness for systems of two equations, even with tight constraints on the block variables. We also study a variant where few deletions are allowed in the target string, and give XP algorithms in this setting when the number of block variables is constant.

cs.CC↗

arXiv2021

Your Rugby Mates Don't Need to Know your Colleagues: Triadic Closure with Edge Colors

Given an undirected graph $G=(V,E)$ the NP-hard Strong Triadic Closure (STC) problem asks for a labeling of the edges as \emph{weak} and \emph{strong} such that at most $k$ edges are weak and for each induced $P_3$ in $G$ at least one edge is weak. In this work, we study the following generalizations of STC with $c$ different strong edge colors. In Multi-STC an induced $P_3$ may receive two strong labels as long as they are different. In Edge-List Multi-STC and Vertex-List Multi-STC we may additionally restrict the set of permitted colors for each edge of $G$. We show that, under the Exponential Time Hypothesis (ETH), Edge-List Multi-STC and Vertex-List Multi-STC cannot be solved in time $2^{o(|V|^2)}$. We then proceed with a parameterized complexity analysis in which we extend previous fixed-parameter tractability results and kernelizations for STC [Golovach et al., Algorithmica '20, Grüttemeier and Komusiewicz, Algorithmica '20] to the three variants with multiple edge colors or outline the limits of such an extension.

cs.DS↗

arXiv2021

Essentially Tight Kernels for (Weakly) Closed Graphs

We study kernelization of classic hard graph problems when the input graphs fulfill triadic closure properties. More precisely, we consider the recently introduced parameters closure number $c$ and the weak closure number $γ$ [Fox et al., SICOMP 2020] in addition to the standard parameter solution size $k$. For Capacitated Vertex Cover, Connected Vertex Cover, and Induced Matching we obtain the first kernels of size $k^{\mathcal{O}(γ)}$ and $(γk)^{\mathcal{O}(γ)}$, respectively, thus extending previous kernelization results on degenerate graphs. The kernels are essentially tight, since these problems are unlikely to admit kernels of size $k^{o(γ)}$ by previous results on their kernelization complexity in degenerate graphs [Cygan et al., ACM TALG 2017]. In addition, we provide lower bounds for the kernelization of Independent Set on graphs with constant closure number~$c$ and kernels for Dominating Set on weakly closed split graphs and weakly closed bipartite graphs.

cs.DS↗

arXiv2021

Refined Notions of Parameterized Enumeration Kernels with Applications to Matching Cut Enumeration

An enumeration kernel as defined by Creignou et al. [Theory Comput. Syst. 2017] for a parameterized enumeration problem consists of an algorithm that transforms each instance into one whose size is bounded by the parameter plus a solution-lifting algorithm that efficiently enumerates all solutions from the set of the solutions of the kernel. We propose to consider two new versions of enumeration kernels by asking that the solutions of the original instance can be enumerated in polynomial time or with polynomial delay from the kernel solutions. Using the NP-hard Matching Cut problem parameterized by structural parameters such as the vertex cover number or the cyclomatic number of the input graph, we show that the new enumeration kernels present a useful notion of data reduction for enumeration problems which allows to compactly represent the set of feasible solutions.

cs.DS↗

arXiv2020

Maximum Edge-Colorable Subgraph and Strong Triadic Closure Parameterized by Distance to Low-Degree Graphs

Given an undirected graph $G$ and integers $c$ and $k$, the Maximum Edge-Colorable Subgraph problem asks whether we can delete at most $k$ edges in $G$ to obtain a graph that has a proper edge coloring with at most $c$ colors. We show that Maximum Edge-Colorable Subgraph admits, for every fixed $c$, a linear-size problem kernel when parameterized by the edge deletion distance of $G$ to a graph with maximum degree $c-1$. This parameterization measures the distance to instances that, due to Vizing's famous theorem, are trivial yes-instances. For $c\le 4$, we also provide a linear-size kernel for the same parameterization for Multi Strong Triadic Closure, a related edge coloring problem with applications in social network analysis. We provide further results for Maximum Edge-Colorable Subgraph parameterized by the vertex deletion distance to graphs where every component has order at most $c$ and for the list-colored versions of both problems.

cs.DS↗