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Computational complexity in graph theory
Doucha, Martin ; Kratochvíl, Jan (advisor) ; Dvořák, Zdeněk (referee)
This work introduces two new parameterizations of graph problems generalizing vertex cover which fill part of the space between vertex cover and clique width in the hierarchy of graf parameterizations. We also study parameterized complexity of Hamiltonian path and cycle, vertex coloring, precoloring extension and equitable coloring parameterized by these two parameterizations. With the exception of precoloring extension which is W[1]-hard in one case, all the other problems listed above are tractable for both parameterizations. The boundary between tractability and intractability of these problems can therefore be moved closer to parameterization by clique width.
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Computational complexity in graph theory
Melka, Jakub ; Kratochvíl, Jan (advisor) ; Fiala, Jiří (referee)
In the present work we study the problem of reconstructing a graph from its closed neighbourhood list. We will explore this problem, formulated by V. Sós, from the point of view of the fixed parameter complexity. We study the graph reconstruction problem in a more general setting, when the reconstructed graph is required to belong to some special graph class. In the present work we prove that this general problem lies in the complexity class FPT, when parametrized by the treewidth and maximum degree of the reconstructed graph, or by the number of certain special induced subgraphs if the reconstructed graph is 2-degenerate. Also, we prove that the graph reconstruction problem lies in the complexity class XP when parametrized by the vertex cover number. Finally, we prove mutual independence of the results
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Parameterized Complexity
Suchý, Ondřej ; Kratochvíl, Jan (advisor) ; Telle, Jan Arne (referee) ; Obdržálek, Jan (referee)
Title: Parameterized Complexity Author: Ondřej Suchý Department: Department of Applied Mathematics Advisor: Prof. RNDr. Jan Kratochvíl, CSc. Advisor's e-mail address: honza@kam.mff.cuni.cz Abstract: This thesis deals with the parameterized complexity of NP-hard graph problems. We explore the complexity of the problems in various scenarios, with respect to miscellaneous parameters and their combina- tions. Our aim is rather to classify in this multivariate manner whether the particular parameters make the problem fixed-parameter tractable or intractable than to present the algorithm achieving the best running time. In the questions we study typically the first-choice parameter is unsuccessful, in which case we propose to use less standard ones. The first family of problems investigated provides a common general- ization of many well known and studied domination and independence problems. Here we suggest using the dual parameterization and show that, in contrast to the standard solution-size, it can confine the in- evitable combinatorial explosion. Further studied problems are ana- logues of the Steiner problem in directed graphs. Here the parameter- ization by the number of terminals to be connected seems to be previ- ously unexplored in the directed setting. Unfortunately, the problems are shown to be...
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Structural properties of graphs and eficient algorithms: Problems Between Parameters
Knop, Dušan ; Fiala, Jiří (advisor)
Structural Properties of Graphs and Eficient Algorithms: Problems Between Parameters Dušan Knop Parameterized complexity became over last two decades one of the most impor- tant subfield of computational complexity. Structural graph parameters (widths) play important role both in graph theory and (parameterized) algoritmh design. By studying some concrete problems we exhibit the connection between struc- tural graph parameters and parameterized tractability. We do this by examining tractability and hardness results for the Target Set Selection, Minimum Length Bounded Cut, and other problems. In the Minimum Length Bounded Cut problem we are given a graph, source, sink, and a positive integer L and the task is to remove edges from the graph such that the distance between the source and the sink exceeds L in the resulting graph. We show that an optimal solution to the Minimum Length Bounded Cut problem can be computed in time f(k)n, where f is a computable function and k denotes the tree-depth of the input graph. On the other hand we prove that (under assumption that FPT ̸= W[1]) no such algorithm can exist if the parameter k is the tree-width of the input graph. Currently only few such problems are known. The Target Set Selection problem exibits the same phenomenon for the vertex cover number and...
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Variants of graph labeling problems
Masařík, Tomáš ; Fiala, Jiří (advisor) ; Fellows, Michael R. (referee) ; Hell, Pavol (referee)
This thesis consists of three parts devoted to graph labeling, hereditary graph classes, and parameterized complexity. Packing coloring, originally Broadcasting Chromatic number, assigns natural numbers to vertices such that vertices with the same label are in distance at least the value of the label. This problem is motivated by the assignment of frequencies to the transmitters. We improve hardness on chordal graphs. We proof that packing coloring on chordal graphs with diameter 3 is very hard to approximate. Moreover, we discuss several positive results on interval graphs and on related structural graph parameters. Hereditary graph classes are preserved under vertex deletion. We study graphs that do not contain an induced subgraph H. We prove that 3-coloring is polynomial-time solvable for (P3 + P4)-free and (P2 + P5)-free graphs and thus we have solved the last open cases for the problem on H-free graphs where H has up to 7 vertices. Fair problems are a modification of graph deletion problems, where, instead of minimizing the size of the solution, the aim is to minimize the maximum number of neighbors in the deleted set. We show that those problems can be solved in FPT time for an MSO1 formula parameterized by the size of the formula and the twin cover of the graph. Moreover, we define a basic...
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Structural properties of graphs and eficient algorithms: Problems Between Parameters
Knop, Dušan ; Fiala, Jiří (advisor) ; Niedermeier, Rolf (referee) ; Feldmann, Andreas Emil (referee)
Structural Properties of Graphs and Eficient Algorithms: Problems Between Parameters Dušan Knop Parameterized complexity became over last two decades one of the most impor- tant subfield of computational complexity. Structural graph parameters (widths) play important role both in graph theory and (parameterized) algoritmh design. By studying some concrete problems we exhibit the connection between struc- tural graph parameters and parameterized tractability. We do this by examining tractability and hardness results for the Target Set Selection, Minimum Length Bounded Cut, and other problems. In the Minimum Length Bounded Cut problem we are given a graph, source, sink, and a positive integer L and the task is to remove edges from the graph such that the distance between the source and the sink exceeds L in the resulting graph. We show that an optimal solution to the Minimum Length Bounded Cut problem can be computed in time f(k)n, where f is a computable function and k denotes the tree-depth of the input graph. On the other hand we prove that (under assumption that FPT ̸= W[1]) no such algorithm can exist if the parameter k is the tree-width of the input graph. Currently only few such problems are known. The Target Set Selection problem exibits the same phenomenon for the vertex cover number and...
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Computational complexity in graph theory
Melka, Jakub ; Kratochvíl, Jan (advisor) ; Fiala, Jiří (referee)
In the present work we study the problem of reconstructing a graph from its closed neighbourhood list. We will explore this problem, formulated by V. Sós, from the point of view of the fixed parameter complexity. We study the graph reconstruction problem in a more general setting, when the reconstructed graph is required to belong to some special graph class. In the present work we prove that this general problem lies in the complexity class FPT, when parametrized by the treewidth and maximum degree of the reconstructed graph, or by the number of certain special induced subgraphs if the reconstructed graph is 2-degenerate. Also, we prove that the graph reconstruction problem lies in the complexity class XP when parametrized by the vertex cover number. Finally, we prove mutual independence of the results
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Structural properties of graphs and eficient algorithms: Problems Between Parameters
Knop, Dušan ; Fiala, Jiří (advisor)
Structural Properties of Graphs and Eficient Algorithms: Problems Between Parameters Dušan Knop Parameterized complexity became over last two decades one of the most impor- tant subfield of computational complexity. Structural graph parameters (widths) play important role both in graph theory and (parameterized) algoritmh design. By studying some concrete problems we exhibit the connection between struc- tural graph parameters and parameterized tractability. We do this by examining tractability and hardness results for the Target Set Selection, Minimum Length Bounded Cut, and other problems. In the Minimum Length Bounded Cut problem we are given a graph, source, sink, and a positive integer L and the task is to remove edges from the graph such that the distance between the source and the sink exceeds L in the resulting graph. We show that an optimal solution to the Minimum Length Bounded Cut problem can be computed in time f(k)n, where f is a computable function and k denotes the tree-depth of the input graph. On the other hand we prove that (under assumption that FPT ̸= W[1]) no such algorithm can exist if the parameter k is the tree-width of the input graph. Currently only few such problems are known. The Target Set Selection problem exibits the same phenomenon for the vertex cover number and...
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Obtížné problémy vzhledem k parametru různorodost sousedství
Koutecký, Martin ; Kolman, Petr (advisor) ; Fiala, Jiří (referee)
Parameterized complexity is a part of computer science dealing with the computational complexity of problems measured not only by the length of their input but also some parameter of the input. Nei- ghborhood diversity is a recently introduced parameter describing a certain structure of a graph. is parameter is aractive for resear especially because some problems whi are hard with respect to other parameters that are incomparable with neighborhood diversity become fixed-parameter tractable with respect to neighborhood diversity. In this thesis we show fixed-parameter tractability for three problems that are hard with respect to treewidth. is constitutes the main part of this thesis and it is our original work. Next it contains an overview of other interesting problems and also a survey of the state of the art in the area of parameters for sparse and dense graphs. 1
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Computational complexity in graph theory
Doucha, Martin ; Kratochvíl, Jan (advisor) ; Dvořák, Zdeněk (referee)
This work introduces two new parameterizations of graph problems generalizing vertex cover which fill part of the space between vertex cover and clique width in the hierarchy of graf parameterizations. We also study parameterized complexity of Hamiltonian path and cycle, vertex coloring, precoloring extension and equitable coloring parameterized by these two parameterizations. With the exception of precoloring extension which is W[1]-hard in one case, all the other problems listed above are tractable for both parameterizations. The boundary between tractability and intractability of these problems can therefore be moved closer to parameterization by clique width.
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