Graph separation is a well-known tool to make (hard) graph problems accessible to a divide-and-co... more Graph separation is a well-known tool to make (hard) graph problems accessible to a divide-and-conquer approach. We show how to use graph separator theorems in combination with (linear) problem kernels in order to develop fixed parameter algorithms for many wellknown NP-hard (planar) graph problems. We coin the key notion of glueable select&verify graph problems and derive from that a prospective way to easily check whether a planar graph problem will allow for a fixed parameter algorithm of running time c √ k • n O(1) for constant c. One of the main contributions of the paper is to exactly compute the base c of the exponential term and its dependence on the various parameters specified by the employed separator theorem and the underlying graph problem. We discuss several strategies to improve on the involved constant c.
Kernelization-a mathematical key concept for provably effective polynomial-time preprocessing of ... more 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 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 = 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.
We establish refined search tree techniques for the parameterized dominating set problem on plana... more We establish refined search tree techniques for the parameterized dominating set problem on planar graphs. We derive a fixed parameter algorithm with running time O(8 k n), where k is the size of the dominating set and n is the number of vertices in the graph. For our search tree, we firstly provide a set of reduction rules. Secondly, we prove an intricate branching theorem based on the Euler formula. In addition, we give an example graph showing that the bound of the branching theorem is optimal with respect to our reduction rules. Our final algorithm is very easy (to implement); its analysis, however, is involved.
Utrecht University: Information and Computing Sciences eBooks, 2000
We present an algorithm that constructively produces a solution to the k-dominating set problem f... more We present an algorithm that constructively produces a solution to the k-dominating set problem for planar graphs in time O(c √ k n), where c = 3 6 √ 34. To obtain this result, we show that the treewidth of a planar graph with domination number γ(G) is O(γ(G)), and that such a tree decomposition can be found in O(γ(G)n) time. The same technique can be used to show that the k-face cover problem (find a size k set of faces that cover all vertices of a given plane graph) can be solved in O(c √ k 1 n + n 2) time, where c 1 = 2 36 √ 34 and k is the size of the face cover set. Similar results can be obtained in the planar case for some variants of k-dominating set, e.g., k-independent dominating set and k-weighted dominating set.
We propose models for lobbying in a probabilistic environment, in which an actor (called "The Lob... more 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.
When we consider {ε, a, aa} and {a, aa} as being "the same" language, we can take {a, aaa} as exa... more When we consider {ε, a, aa} and {a, aa} as being "the same" language, we can take {a, aaa} as example.
A vertex $v\in V(G)$ is said to {distinguish} two vertices $x,y\in V(G)$ of a graph $G$ if the d... more A vertex $v\in V(G)$ is said to {distinguish} two vertices $x,y\in V(G)$ of a graph $G$ if the distance from $v$ to $x$ is different from the distance from $v$ to $y$. A set $W\subseteq V(G)$ is a \textit{total resolving set} for a graph $G$ if for every pair of vertices $x,y\in V(G),$ there exists some vertex $w\in W-\{x,y\}$ which distinguishes $x$ and $y$, while $W$ is a \textit{weak total resolving set} if for every $x\in V(G)-W$ and $y\in W$, there exists some $w\in W-\{y\}$ which distinguishes $x$ and $y$. A weak total resolving set of minimum cardinality is called \textit{weak total metric basis} of $G$ and its cardinality, the {\it weak total metric dimension} of $G$. Our main contributions are the following ones: (a) Graphs with small and large weak total metric bases are characterised. (b) We explore the (tight) relation to independent 2-domination. (c) We introduce a new graph parameter, called \textit{weak total adjacency dimension} and present results that are analogous to those presented for weak total dimension. (d) For trees, we derive a characterisation of the weak total (adjacency) metric dimension. Also, exact figures for our parameters are presented for (generalised) fans and wheels. (e) We show that for Cartesian product graphs, the weak total (adjacency) metric dimension is usually pretty small. (f) The weak total (adjacency) dimension is studied for lexicographic products of graphs.
Given a directed graph G = (V, A), the Directed Maximum Leaf Spanning Tree problem asks to comput... more 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 O * (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 O * (1.8139 n ).
We present an algorithm that constructively produces a solution to the k-dominating set problem f... more We present an algorithm that constructively produces a solution to the k-dominating set problem for planar graphs in time O(c √ k n), where c = 3 6 √ 34 . To obtain this result, we show that the treewidth of a planar graph with domination number γ(G) is O( γ(G)), and that such a tree decomposition can be found in O( γ(G)n) time. The same technique can be used to show that the k-face cover problem (find a size k set of faces that cover all vertices of a given plane graph) can be solved in O(c √ k 1 n + n 2 ) time, where c 1 = 2 36 √ 34 and k is the size of the face cover set. Similar results can be obtained in the planar case for some variants of k-dominating set, e.g., k-independent dominating set and k-weighted dominating set.
. Valences are a very simple and yet powerful method of regulatedrewriting. In this paper we give... more . Valences are a very simple and yet powerful method of regulatedrewriting. In this paper we give an overview on different aspectsof this subject. We discuss closure properties of valence languages. It isshown that matrix grammars can be simulated by valence grammars overfinite monoids. A Chomsky normal form theorem is proved for multiplicativevalence grammars, thereby solving the open question of the existenceof normal forms for unordered vector grammars. This also gives an alternative...
Valence grammars were introduced by Gh. Păun in [8] as a grammatical model of chemical processes.... more Valence grammars were introduced by Gh. Păun in [8] as a grammatical model of chemical processes. Here, we focus on discussing a simpler variant which we call valuated grammars.We give some algebraic characterizations of the corresponding language classes. Similarly,we obtain an algebraic characterization of the linear languages. We also give some Nivat-like representations of valence transductions.
We show how appropriately chosen functions f which we call distinguishing can be used to make det... more We show how appropriately chosen functions f which we call distinguishing can be used to make deterministic finite automata back- ward deterministic. These ideas have been exploited to design regular lan- guage classes called f -distinguishable which are identifiable in the limit from positive samples. Special cases of this approach are the k -reversible and terminal distinguishable languages as discussed in [1, 3, 5, 15, 16].Here, we give new characterizations of these language classes. Moreover, we show that all regular languages can be approximated in the setting introduced by Kobayashi and Yokomori [12, [13]. Finally, we prove that the class of all function-distinguishable languages is equal to the class of regular languages.
This paper continues our research on the use of graph separator theorems for designing fixed para... more This paper continues our research on the use of graph separator theorems for designing fixed parameter algorithms started with the COCOON’01 contribution [2], showing how a more elaborated use of these theorems can bring down the algorithmically relevant constants. More precisely, if a cÖ</font >k c^{\sqrt k } -algorithm is obtainable with the help of applying the well-known Lipton/Tarjan planar separator theorem, our new approach will lead to a c2 \mathord/ \vphantom 2 3 3Ö</font >k c^{{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-\nulldelimiterspace} 3}\sqrt k } -algorithm, this way also improving on the direct use of the “best” known planar separator theorem. For several problems, the constants can be even improved more by analyzing other separator theorems.
We sketch possible applications of grammatical inference techniques to problems arising in the co... more We sketch possible applications of grammatical inference techniques to problems arising in the context of XML. The idea is to infer document type defnitions (DTDs) of XML documents in situations when either the original DTD is missing or should be (re)designed or should be restricted to a more user-oriented view on a subset of the (given) DTD. The usefulness of
. We consider conditional context-free grammars that generatelanguages of finite index. Thereby, ... more . We consider conditional context-free grammars that generatelanguages of finite index. Thereby, we solve an open problem statedin Dassow and Paun's monograph on regulated rewriting. Moreover, weshow that conditional context-free languages with context-free conditionsof finite index are more powerful than conditional context-free languageswith regular conditions of finite index. Furthermore, we study thecomplexity of membership and non-emptiness for conditional and programmedlanguages...
... We will now exhibit this relation by investigating multiplex select gates which were introduc... more ... We will now exhibit this relation by investigating multiplex select gates which were introduced by Reinhardt [17]. They were inspired by the select gates of Nie-pel and Rossmanith [16,19]. ... from gate g ~ "~" from gate . from gate d V ~ from gate dk Fig. 3. Select and deselect gates ...
Graph separation is a well-known tool to make (hard) graph problems accessible to a divide-and-co... more Graph separation is a well-known tool to make (hard) graph problems accessible to a divide-and-conquer approach. We show how to use graph separator theorems in combination with (linear) problem kernels in order to develop fixed parameter algorithms for many wellknown NP-hard (planar) graph problems. We coin the key notion of glueable select&verify graph problems and derive from that a prospective way to easily check whether a planar graph problem will allow for a fixed parameter algorithm of running time c √ k • n O(1) for constant c. One of the main contributions of the paper is to exactly compute the base c of the exponential term and its dependence on the various parameters specified by the employed separator theorem and the underlying graph problem. We discuss several strategies to improve on the involved constant c.
Kernelization-a mathematical key concept for provably effective polynomial-time preprocessing of ... more 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 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 = 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.
We establish refined search tree techniques for the parameterized dominating set problem on plana... more We establish refined search tree techniques for the parameterized dominating set problem on planar graphs. We derive a fixed parameter algorithm with running time O(8 k n), where k is the size of the dominating set and n is the number of vertices in the graph. For our search tree, we firstly provide a set of reduction rules. Secondly, we prove an intricate branching theorem based on the Euler formula. In addition, we give an example graph showing that the bound of the branching theorem is optimal with respect to our reduction rules. Our final algorithm is very easy (to implement); its analysis, however, is involved.
Utrecht University: Information and Computing Sciences eBooks, 2000
We present an algorithm that constructively produces a solution to the k-dominating set problem f... more We present an algorithm that constructively produces a solution to the k-dominating set problem for planar graphs in time O(c √ k n), where c = 3 6 √ 34. To obtain this result, we show that the treewidth of a planar graph with domination number γ(G) is O(γ(G)), and that such a tree decomposition can be found in O(γ(G)n) time. The same technique can be used to show that the k-face cover problem (find a size k set of faces that cover all vertices of a given plane graph) can be solved in O(c √ k 1 n + n 2) time, where c 1 = 2 36 √ 34 and k is the size of the face cover set. Similar results can be obtained in the planar case for some variants of k-dominating set, e.g., k-independent dominating set and k-weighted dominating set.
We propose models for lobbying in a probabilistic environment, in which an actor (called "The Lob... more 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.
When we consider {ε, a, aa} and {a, aa} as being "the same" language, we can take {a, aaa} as exa... more When we consider {ε, a, aa} and {a, aa} as being "the same" language, we can take {a, aaa} as example.
A vertex $v\in V(G)$ is said to {distinguish} two vertices $x,y\in V(G)$ of a graph $G$ if the d... more A vertex $v\in V(G)$ is said to {distinguish} two vertices $x,y\in V(G)$ of a graph $G$ if the distance from $v$ to $x$ is different from the distance from $v$ to $y$. A set $W\subseteq V(G)$ is a \textit{total resolving set} for a graph $G$ if for every pair of vertices $x,y\in V(G),$ there exists some vertex $w\in W-\{x,y\}$ which distinguishes $x$ and $y$, while $W$ is a \textit{weak total resolving set} if for every $x\in V(G)-W$ and $y\in W$, there exists some $w\in W-\{y\}$ which distinguishes $x$ and $y$. A weak total resolving set of minimum cardinality is called \textit{weak total metric basis} of $G$ and its cardinality, the {\it weak total metric dimension} of $G$. Our main contributions are the following ones: (a) Graphs with small and large weak total metric bases are characterised. (b) We explore the (tight) relation to independent 2-domination. (c) We introduce a new graph parameter, called \textit{weak total adjacency dimension} and present results that are analogous to those presented for weak total dimension. (d) For trees, we derive a characterisation of the weak total (adjacency) metric dimension. Also, exact figures for our parameters are presented for (generalised) fans and wheels. (e) We show that for Cartesian product graphs, the weak total (adjacency) metric dimension is usually pretty small. (f) The weak total (adjacency) dimension is studied for lexicographic products of graphs.
Given a directed graph G = (V, A), the Directed Maximum Leaf Spanning Tree problem asks to comput... more 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 O * (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 O * (1.8139 n ).
We present an algorithm that constructively produces a solution to the k-dominating set problem f... more We present an algorithm that constructively produces a solution to the k-dominating set problem for planar graphs in time O(c √ k n), where c = 3 6 √ 34 . To obtain this result, we show that the treewidth of a planar graph with domination number γ(G) is O( γ(G)), and that such a tree decomposition can be found in O( γ(G)n) time. The same technique can be used to show that the k-face cover problem (find a size k set of faces that cover all vertices of a given plane graph) can be solved in O(c √ k 1 n + n 2 ) time, where c 1 = 2 36 √ 34 and k is the size of the face cover set. Similar results can be obtained in the planar case for some variants of k-dominating set, e.g., k-independent dominating set and k-weighted dominating set.
. Valences are a very simple and yet powerful method of regulatedrewriting. In this paper we give... more . Valences are a very simple and yet powerful method of regulatedrewriting. In this paper we give an overview on different aspectsof this subject. We discuss closure properties of valence languages. It isshown that matrix grammars can be simulated by valence grammars overfinite monoids. A Chomsky normal form theorem is proved for multiplicativevalence grammars, thereby solving the open question of the existenceof normal forms for unordered vector grammars. This also gives an alternative...
Valence grammars were introduced by Gh. Păun in [8] as a grammatical model of chemical processes.... more Valence grammars were introduced by Gh. Păun in [8] as a grammatical model of chemical processes. Here, we focus on discussing a simpler variant which we call valuated grammars.We give some algebraic characterizations of the corresponding language classes. Similarly,we obtain an algebraic characterization of the linear languages. We also give some Nivat-like representations of valence transductions.
We show how appropriately chosen functions f which we call distinguishing can be used to make det... more We show how appropriately chosen functions f which we call distinguishing can be used to make deterministic finite automata back- ward deterministic. These ideas have been exploited to design regular lan- guage classes called f -distinguishable which are identifiable in the limit from positive samples. Special cases of this approach are the k -reversible and terminal distinguishable languages as discussed in [1, 3, 5, 15, 16].Here, we give new characterizations of these language classes. Moreover, we show that all regular languages can be approximated in the setting introduced by Kobayashi and Yokomori [12, [13]. Finally, we prove that the class of all function-distinguishable languages is equal to the class of regular languages.
This paper continues our research on the use of graph separator theorems for designing fixed para... more This paper continues our research on the use of graph separator theorems for designing fixed parameter algorithms started with the COCOON’01 contribution [2], showing how a more elaborated use of these theorems can bring down the algorithmically relevant constants. More precisely, if a cÖ</font >k c^{\sqrt k } -algorithm is obtainable with the help of applying the well-known Lipton/Tarjan planar separator theorem, our new approach will lead to a c2 \mathord/ \vphantom 2 3 3Ö</font >k c^{{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-\nulldelimiterspace} 3}\sqrt k } -algorithm, this way also improving on the direct use of the “best” known planar separator theorem. For several problems, the constants can be even improved more by analyzing other separator theorems.
We sketch possible applications of grammatical inference techniques to problems arising in the co... more We sketch possible applications of grammatical inference techniques to problems arising in the context of XML. The idea is to infer document type defnitions (DTDs) of XML documents in situations when either the original DTD is missing or should be (re)designed or should be restricted to a more user-oriented view on a subset of the (given) DTD. The usefulness of
. We consider conditional context-free grammars that generatelanguages of finite index. Thereby, ... more . We consider conditional context-free grammars that generatelanguages of finite index. Thereby, we solve an open problem statedin Dassow and Paun's monograph on regulated rewriting. Moreover, weshow that conditional context-free languages with context-free conditionsof finite index are more powerful than conditional context-free languageswith regular conditions of finite index. Furthermore, we study thecomplexity of membership and non-emptiness for conditional and programmedlanguages...
... We will now exhibit this relation by investigating multiplex select gates which were introduc... more ... We will now exhibit this relation by investigating multiplex select gates which were introduced by Reinhardt [17]. They were inspired by the select gates of Nie-pel and Rossmanith [16,19]. ... from gate g ~ "~" from gate . from gate d V ~ from gate dk Fig. 3. Select and deselect gates ...
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Papers by Henning Fernau
A set $W\subseteq V(G)$ is a \textit{total resolving set} for a graph $G$ if for every pair of vertices $x,y\in V(G),$ there exists some vertex $w\in W-\{x,y\}$ which distinguishes $x$ and $y$,
while $W$ is a \textit{weak total resolving set} if for every $x\in V(G)-W$ and $y\in W$, there exists some $w\in W-\{y\}$ which distinguishes $x$ and $y$.
A weak total resolving set of minimum cardinality is called \textit{weak total metric basis} of $G$ and its cardinality, the {\it weak total metric dimension} of $G$.
Our main contributions are the following ones:
(a) Graphs with small and large weak total metric bases are characterised. (b) We explore the (tight) relation to independent 2-domination.
(c) We introduce a new graph parameter, called \textit{weak total adjacency dimension} and present results that are analogous to those presented for weak total dimension.
(d) For trees, we derive a characterisation of the weak total (adjacency) metric dimension. Also, exact figures for our parameters are presented for (generalised) fans and wheels.
(e) We show that for Cartesian product graphs, the weak total (adjacency) metric dimension is usually pretty small.
(f) The weak total (adjacency) dimension is studied for lexicographic products of graphs.
A set $W\subseteq V(G)$ is a \textit{total resolving set} for a graph $G$ if for every pair of vertices $x,y\in V(G),$ there exists some vertex $w\in W-\{x,y\}$ which distinguishes $x$ and $y$,
while $W$ is a \textit{weak total resolving set} if for every $x\in V(G)-W$ and $y\in W$, there exists some $w\in W-\{y\}$ which distinguishes $x$ and $y$.
A weak total resolving set of minimum cardinality is called \textit{weak total metric basis} of $G$ and its cardinality, the {\it weak total metric dimension} of $G$.
Our main contributions are the following ones:
(a) Graphs with small and large weak total metric bases are characterised. (b) We explore the (tight) relation to independent 2-domination.
(c) We introduce a new graph parameter, called \textit{weak total adjacency dimension} and present results that are analogous to those presented for weak total dimension.
(d) For trees, we derive a characterisation of the weak total (adjacency) metric dimension. Also, exact figures for our parameters are presented for (generalised) fans and wheels.
(e) We show that for Cartesian product graphs, the weak total (adjacency) metric dimension is usually pretty small.
(f) The weak total (adjacency) dimension is studied for lexicographic products of graphs.