arXiv subjects

Henning Fernau

Publications and source records attributed to Henning Fernau.

At least 55 records · Page 3Linked to original sources

arXiv2018

Extension of vertex cover and independent set in some classes of graphs and generalizations

We consider extension variants of the classical graph problems Vertex Cover and Independent Set. Given a graph $G=(V,E)$ and a vertex set $U \subseteq V$, it is asked if there exists a minimal vertex cover (resp.\ maximal independent set) $S$ with $U\subseteq S$ (resp.\ $U \supseteq S$). Possibly contradicting intuition, these problems tend to be NP-hard, even in graph classes where the classical problem can be solved in polynomial time. Yet, we exhibit some graph classes where the extension variant remains polynomial-time solvable. We also study the parameterized complexity of these problems, with parameter $|U|$, as well as the optimality of simple exact algorithms under the Exponential-Time Hypothesis. All these complexity considerations are also carried out in very restricted scenarios, be it degree or topological restrictions (bipartite, planar or chordal graphs). This also motivates presenting some explicit branching algorithms for degree-bounded instances. We further discuss the price of extension, measuring the distance of $U$ to the closest set that can be extended, which results in natural optimization problems related to extension problems for which we discuss polynomial-time approximability.

cs.CC↗

arXiv2018

On the Complexity of Solution Extension of Optimization Problems

The question if a given partial solution to a problem can be extended reasonably occurs in many algorithmic approaches for optimization problems. For instance, when enumerating minimal dominating sets of a graph $G=(V,E)$, one usually arrives at the problem to decide for a vertex set $U \subseteq V$, if there exists a \textit{minimal} dominating set $S$ with $U\subseteq S$. We propose a general, partial-order based formulation of such extension problems and study a number of specific problems which can be expressed in this framework. Possibly contradicting intuition, these problems tend to be NP-hard, even for problems where the underlying optimisation problem can be solved in polynomial time. This raises the question of how fixing a partial solution causes this increase in difficulty. In this regard, we study the parameterised complexity of extension problems with respect to parameters related to the partial solution, as well as the optimality of simple exact algorithms under the Exponential-Time Hypothesis. All complexity considerations are also carried out in very restricted scenarios, be it degree restrictions or topological restrictions (planarity) for graph problems or the size of the given partition for the considered extension variant of Bin Packing.

cs.CC↗

arXiv2017

Combinatorial Properties and Recognition of Unit Square Visibility Graphs

Unit square (grid) visibility graphs (USV and USGV, resp.) are described by axis-parallel visibility between unit squares placed (on integer grid coordinates) in the plane. We investigate combinatorial properties of these graph classes and the hardness of variants of the recognition problem, i.e., the problem of representing USGV with fixed visibilities within small area and, for USV, the general recognition problem.

cs.CG↗

arXiv2017

Diminishable Parameterized Problems and Strict Polynomial Kernelization

Kernelization---a mathematical key concept for provably effective polynomial-time preprocessing of NP-hard problems---plays a central role in parameterized complexity and has triggered an extensive line of research. This is in part due to a lower bounds framework that allows to exclude polynomial-size kernels under the assumption of NP $\nsubseteq$ coNP$/$poly. In this paper we consider a restricted yet natural variant of kernelization, namely strict kernelization, where one is not allowed to increase the parameter of the reduced instance (the kernel) by more than an additive constant. Building on earlier work of Chen, Flum, and Müller [Theory Comput. Syst. 2011] and developing a general and remarkably simple framework, we show that a variety of FPT problems does not admit strict polynomial kernels under the weaker assumption of P $\neq$ NP. In particular, we show that various (multicolored) graph problems and Turing machine computation problems do not admit strict polynomial kernels unless P $=$ NP. To this end, a key concept we use are diminishable problems; these are parameterized problems that allow to decrease the parameter of the input instance by at least one in polynomial time, thereby outputting an equivalent problem instance. Finally, we study a relaxation of the notion of strict kernels and reveal its limitations.

cs.CC↗

arXiv2017

Two-Dimensional Pattern Languages

We introduce several classes of array languages obtained by generalising Angluin's pattern languages to the two-dimensional case. These classes of two-dimensional pattern languages are compared with respect to their expressive power and their closure properties are investigated.

cs.FL↗

arXiv2015

Characterization and Complexity Results on Jumping Finite Automata

In a jumping finite automaton, the input head can jump to an arbitrary position within the remaining input after reading and consuming a symbol. We characterize the corresponding class of languages in terms of special shuffle expressions and survey other equivalent notions from the existing literature. Moreover, we present several results concerning computational hardness and algorithms for parsing and other basic tasks concerning jumping finite automata.

cs.FL↗

arXiv2015

Algorithmic Aspects of Upper Domination

In this paper we study combinatorial and algorithmic resp. complexity questions of upper domination, i.e., the maximum cardinality of a minimal dominating set in a graph. We give a full classification of the related maximisation and minimisation problems, as well as the related parameterised problems, on general graphs and on graphs of bounded degree, and we also study planar graphs.

cs.CC↗

arXiv2015

Kernelization Algorithms for Packing Problems Allowing Overlaps (Extended Version)

We consider the problem of discovering overlapping communities in networks which we model as generalizations of Graph Packing problems with overlap. We seek a collection $\mathcal{S}' \subseteq \mathcal{S}$ consisting of at least $k$ sets subject to certain disjointness restrictions. In the $r$-Set Packing with $t$-Membership, each element of $\mathcal{U}$ belongs to at most $t$ sets of $\mathcal{S'}$ while in $t$-Overlap each pair of sets in $\mathcal{S'}$ overlaps in at most $t$ elements. Each set of $\mathcal{S}$ has at most $r$ elements. Similarly, both of our graph packing problems seek a collection $\mathcal{K}$ of at least $k$ subgraphs in a graph $G$ each isomorphic to a graph $H \in \mathcal{H}$. In $\mathcal{H}$-Packing with $t$-Membership, each vertex of $G$ belongs to at most $t$ subgraphs of $\mathcal{K}$ while in $t$-Overlap each pair of subgraphs in $\mathcal{K}$ overlaps in at most $t$ vertices. Each member of $\mathcal{H}$ has at most $r$ vertices and $m$ edges. We show NP-Completeness results for all of our packing problems and we give a dichotomy result for the $\mathcal{H}$-Packing with $t$-Membership problem analogous to the Kirkpatrick and Hell \cite{Kirk78}. We reduce the $r$-Set Packing with $t$-Membership to a problem kernel with $O((r+1)^r k^{r})$ elements while we achieve a kernel with $O(r^r k^{r-t-1})$ elements for the $r$-Set Packing with $t$-Overlap. In addition, we reduce the $\mathcal{H}$-Packing with $t$-Membership and its edge version to problem kernels with $O((r+1)^r k^{r})$ and $O((m+1)^{m} k^{m})$ vertices, respectively. On the other hand, we achieve kernels with $O(r^r k^{r-t-1})$ and $O(m^{m} k^{m-t-1})$ vertices for the $\mathcal{H}$-Packing with $t$-Overlap and its edge version, respectively. In all cases, $k$ is the input parameter while $t$, $r$, and $m$ are constants.

cs.DS↗

arXiv2014

Data Reductions and Combinatorial Bounds for Improved Approximation Algorithms

Kernelization algorithms in the context of Parameterized Complexity are often based on a combination of reduction rules and combinatorial insights. We will expose in this paper a similar strategy for obtaining polynomial-time approximation algorithms. Our method features the use of approximation-preserving reductions, akin to the notion of parameterized reductions. We exemplify this method to obtain the currently best approximation algorithms for \textsc{Harmless Set}, \textsc{Differential} and \textsc{Multiple Nonblocker}, all of them can be considered in the context of securing networks or information propagation.

cs.DS↗

arXiv2014

Cooperating Distributed Grammar Systems of Finite Index Working in Hybrid Modes

We study cooperating distributed grammar systems working in hybrid modes in connection with the finite index restriction in two different ways: firstly, we investigate cooperating distributed grammar systems working in hybrid modes which characterize programmed grammars with the finite index restriction; looking at the number of components of such systems, we obtain surprisingly rich lattice structures for the inclusion relations between the corresponding language families. Secondly, we impose the finite index restriction on cooperating distributed grammar systems working in hybrid modes themselves, which leads us to new characterizations of programmed grammars of finite index.

cs.FL↗

arXiv2013

Alliances and related parameters in graphs

In this paper, we show that several graph parameters are known in different areas under completely different names. More specifically, our observations connect signed domination, monopolies, $α$-domination, $α$-independence, positive influence domination, and a parameter associated to fast information propagation in networks to parameters related to various notions of global $r$-alliances in graphs. We also propose a new framework, called (global) $(D,O)$-alliances, not only in order to characterize various known variants of alliance and domination parameters, but also to suggest a unifying framework for the study of alliances and domination. Finally, we also give a survey on the mentioned graph parameters, indicating how results transfer due to our observations.

math.CO↗

arXiv2013

On the (adjacency) metric dimension of corona and strong product graphs and their local variants: combinatorial and computational results

The metric dimension is quite a well-studied graph parameter. Recently, the adjacency metric dimension and the local metric dimension have been introduced. We combine these variants and introduce the local adjacency metric dimension. We show that the (local) metric dimension of the corona product of a graph of order $n$ and some non-trivial graph $H$ equals $n$ times the (local) adjacency metric dimension of $H$. This strong relation also enables us to infer computational hardness results for computing the (local) metric dimension, based on according hardness results for (local) adjacency metric dimension that we also provide. We also study combinatorial properties of the strong product of graphs and emphasize the role different types of twins play in determining in particular the adjacency metric dimension of a graph.

math.CO↗

arXiv2013

On the partition dimension of unicyclic graphs

Given an ordered partition $Π=\{P_1,P_2, ...,P_t\}$ of the vertex set $V$ of a connected graph $G=(V,E)$, the \emph{partition representation} of a vertex $v\in V$ with respect to the partition $Π$ is the vector $r(v|Π)=(d(v,P_1),d(v,P_2),...,d(v,P_t))$, where $d(v,P_i)$ represents the distance between the vertex $v$ and the set $P_i$. A partition $Π$ of $V$ is a \emph{resolving partition} if different vertices of $G$ have different partition representations, i.e., for every pair of vertices $u,v\in V$, $r(u|Π)\ne r(v|Π)$. The \emph{partition dimension} of $G$ is the minimum number of sets in any resolving partition for $G$. In this paper we obtain several tight bounds on the partition dimension of unicyclic graphs.

math.CO↗

arXiv2011

The Complexity of Probabilistic Lobbying

We propose models for lobbying in a probabilistic environment, in which an actor (called "The Lobby") seeks to influence voters' preferences of voting for or against multiple issues when the voters' preferences are represented in terms of probabilities. In particular, we provide two evaluation criteria and two bribery methods to formally describe these models, and we consider the resulting forms of lobbying with and without issue weighting. We provide a formal analysis for these problems of lobbying in a stochastic environment, and determine their classical and parameterized complexity depending on the given bribery/evaluation criteria and on various natural parameterizations. Specifically, we show that some of these problems can be solved in polynomial time, some are NP-complete but fixed-parameter tractable, and some are W[2]-complete. Finally, we provide approximability and inapproximability results for these problems and several variants.

cs.CC↗

arXiv2009

A Faster Exact Algorithm for the Directed Maximum Leaf Spanning Tree Problem

Given a directed graph $G=(V,A)$, the Directed Maximum Leaf Spanning Tree problem asks to compute a directed spanning tree (i.e., an out-branching) with as many leaves as possible. By designing a Branch-and-Reduced algorithm combined with the Measure & Conquer technique for running time analysis, we show that the problem can be solved in time $\Oh^*(1.9043^n)$ using polynomial space. Hitherto, there have been only few examples. Provided exponential space this run time upper bound can be lowered to $\Oh^*(1.8139^n)$.

cs.DS↗

arXiv2009

Breaking the 2^n-Barrier for Irredundance: A Parameterized Route to Solving Exact Puzzles

The lower and the upper irredundance numbers of a graph $G$, denoted $ir(G)$ and $IR(G)$ respectively, are conceptually linked to domination and independence numbers and have numerous relations to other graph parameters. It is a long-standing open question whether determining these numbers for a graph $G$ on $n$ vertices admits exact algorithms running in time less than the trivial $Ω(2^n)$ enumeration barrier. We solve these open problems by devising parameterized algorithms for the dual of the natural parameterizations of the problems with running times faster than $O^*(4^{k})$. For example, we present an algorithm running in time $O^*(3.069^{k})$ for determining whether $IR(G)$ is at least $n-k$. Although the corresponding problem has been known to be in FPT by kernelization techniques, this paper offers the first parameterized algorithms with an exponential dependency on the parameter in the running time. Additionally, our work also appears to be the first example of a parameterized approach leading to a solution to a problem in exponential time algorithmics where the natural interpretation as an exact exponential-time algorithm fails.

cs.DS↗

arXiv2009

Exact Exponential Time Algorithms for Max Internal Spanning Tree

We consider the NP-hard problem of finding a spanning tree with a maximum number of internal vertices. This problem is a generalization of the famous Hamiltonian Path problem. Our dynamic-programming algorithms for general and degree-bounded graphs have running times of the form O*(c^n) (c <= 3). The main result, however, is a branching algorithm for graphs with maximum degree three. It only needs polynomial space and has a running time of O*(1.8669^n) when analyzed with respect to the number of vertices. We also show that its running time is 2.1364^k n^O(1) when the goal is to find a spanning tree with at least k internal vertices. Both running time bounds are obtained via a Measure & Conquer analysis, the latter one being a novel use of this kind of analyses for parameterized algorithms.

cs.DS↗

arXiv2008

Kernel(s) for Problems With no Kernel: On Out-Trees With Many Leaves

The {\sc $k$-Leaf Out-Branching} problem is to find an out-branching (i.e. a rooted oriented spanning tree) with at least $k$ leaves in a given digraph. The problem has recently received much attention from the viewpoint of parameterized algorithms {alonLNCS4596,AlonFGKS07fsttcs,BoDo2,KnLaRo}. In this paper we step aside and take a kernelization based approach to the {\sc $k$-Leaf-Out-Branching} problem. We give the first polynomial kernel for {\sc Rooted $k$-Leaf-Out-Branching}, a variant of {\sc $k$-Leaf-Out-Branching} where the root of the tree searched for is also a part of the input. Our kernel has cubic size and is obtained using extremal combinatorics. For the {\sc $k$-Leaf-Out-Branching} problem we show that no polynomial kernel is possible unless polynomial hierarchy collapses to third level %$PH=Σ_p^3$ by applying a recent breakthrough result by Bodlaender et al. {BDFH08} in a non-trivial fashion. However our positive results for {\sc Rooted $k$-Leaf-Out-Branching} immediately imply that the seemingly intractable the {\sc $k$-Leaf-Out-Branching} problem admits a data reduction to $n$ independent $O(k^3)$ kernels. These two results, tractability and intractability side by side, are the first separating {\it many-to-one kernelization} from {\it Turing kernelization}. This answers affirmatively an open problem regarding "cheat kernelization" raised in {IWPECOPEN08}.

cs.DS↗