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Algebraic Equations Comparisons
Nečasová, Gabriela ; Kunovský, Jiří (referee) ; Šátek, Václav (advisor)
The thesis deals with the topic of comparative calculation of algebraic equations. First it describes the comparison of the overall number of operations at direct and iteration met\-hods, as well as gives concrete examples of the methods and explains solutions of direct and iteration methods. Another part focuses on possible methods of converting systems of linear algebraic equations to the system of differential equations. The end of the thesis describes method of working with TKSL/C, Matlab and Maple. In this thesis, there was designed graphical user interface serving for comfortable communication with TKSL/C programme. Graphical user interface was tested on concrete tasks demonstrating the conversion of system of linear algebraic equations to the system of differential equations.
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Mathematical models in logistics
Nevrlý, Vlastimír ; Holešovský, Jan (referee) ; Popela, Pavel (advisor)
The aim of the bachelor thesis is to model and generate test transportation networks, which are similar to real-world waste management networks. Several software tools (Mathematica, GAMS, Excel, VBA) are used to develop suitable procedures. The next task is to analyze dependence of computational complexity on the size of network by using statistical test computations. The existing original optimization model called NERUDA that supports decision-making in the eld of waste management is utilized. The obtained results are processed, analysed and interpreted in detail.
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Statistical analysis of interval data
Troshkov, Kirill ; Antoch, Jaromír (advisor) ; Branda, Martin (referee)
Traditional statistical analysis starts with computing the basic statisti- cal characteristics such as the population mean E, population variance V , cova- riance and correlation. In computing these characteristics, it is usually assumed that the corresponding data values are known exactly. In real life there are many situations in which a more complete information can be achieved by describing a set of statistical units in terms of interval data. For example, daily tempera- tures registered as minimum and maximum values offer a more realistic view on the weather conditions variations with respect to the simple average values. In environmental analysis, we observe a pollution level x(t) in a lake at different mo- ments of time t, and we would like to estimate standard statistical characteristics such as mean, variance and correlation with other measurements. Another exam- ple can be given by financial series. The minimum and the maximum transaction prices recorded daily for a set of stocks represent a more relevant information for experts in order to evaluate the stocks tendency and volatility in the same day. We must therefore modify the existing statistical algorithms to process such interval data. In this work we will analyze algorithms and their modifications for computing various statistics under...
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Computational Complexity in Graph Theory
Jelínková, Eva ; Kratochvíl, Jan (advisor) ; Manlove, David (referee) ; Fiala, Jiří (referee)
Title: Computational Complexity in Graph Theory Author: Eva Jelínková Department: Department of Applied Mathematics Supervisor: Prof. RNDr. Jan Kratochvíl, CSc., Department of Applied Mathematics Abstract: We address problems from graph theory, especially from the computational complexity point of view. In the first part of the thesis we address the computational complexity of problems related to Seidel's switch- ing of graphs. We prove that the problem to decide if a given graph can be switched to contain at most a given number of edges is NP-complete, even for graphs with bounded density. We thus partially answer a question of Matoušek and Wagner [Discrete Comput. Geom. 52, no. 1, 2014]. We also describe infinitely many graphs H such that it is NP-complete to decide if a given graph is switching-equivalent to a graph that does not contain H as an induced subgraph. We thus close an open problem of Kratochvíl, Nešetřil and Zýka [Annals of Discrete Math. 51, 1992]. In the second part of the thesis we address the topic of matchings under preferences. We focus on the housing market problem, in particular, on the model with duplicate houses. We present a 2-approximation algorithm for the maximum number of satisfied agents when the preference lists of agents are trichotomic. On the other hand, we...
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NP vyhledávací problémy
Jirotka, Tomáš ; Krajíček, Jan (advisor) ; Pudlák, Pavel (referee)
Title: NP search problems Author: Tomáš Jirotka Department: Department of Algebra Supervisor: Prof. RNDr. Jan Krajíček, DrSc. Abstract: The thesis summarizes known results in the field of NP search pro- blems. We discuss the complexity of integer factoring in detail, and we propose new results which place the problem in known classes and aim to separate it from PLS in some sense. Furthermore, we define several new search problems. Keywords: Computational complexity, TFNP, integer factorization. 1
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Computational Homotopy Theory
Krčál, Marek ; Matoušek, Jiří (advisor) ; Pultr, Aleš (referee) ; Romero Ibáñez, Ana (referee)
of doctoral thesis "Computational Homotopy Theory": We consider several basic problems of algebraic topology, with connections to combinatorial and geometric questions, from the point of view of compu- tational complexity. The extension problem asks, given topological spaces X, Y , a subspace A ⊆ X, and a (continuous) map f : A → Y , whether f can be extended to a map X → Y . For computational purposes, we assume that A, X, Y are represented as finite simplicial complexes and f as a simplicial map. We study the problem under the assumption that, for some d ≥ 1, Y is d- connected, otherwise the problem is undecidable by uncomputability of the fundamental group; We prove that, this problem is still undecidable for dim X = 2d + 2. On the other hand, for every fixed dim X ≤ 2d + 1, we obtain an algorithm that solves the extension problem in polynomial time. We obtain analogous complexity results for the problem of determining the set of homotopy classes of maps X → Y . We also consider the computation of the homotopy groups πk(Y ), k ≥ 2, for a 1-connected Y . Their computability was established by Brown in 1957; we show that πk(Y ) can be computed in polynomial time for every fixed k ≥ 2. On the other hand, we prove that computing πk(Y ) is #P-hard if k is a part of input. It is a strengthening of...
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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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Computational complexity of combinatorial problems in specific graph classes
Masařík, Tomáš ; Fiala, Jiří (advisor)
The topic of this diploma thesis is the edge distance labeling problem with specified parametres p, q and λ. We found a dychotomy for p = 2 and q = 1. So the problem is polynomial if λ ≤ 4 and it is NP-complete for λ > 4. The boundary is shifted by one prior to the vertex distance labeling problem, which has already been solved. Polynomial cases are characterized as some special paths and cycles with a few additional vertices. To show NP-completeness we use a well-known NP-complete problem of Monotone not all equal 3-SAT. That section has four parts: One for odd λ, one for even λ and two more reductions for λ = 5 and λ = 6. 1
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Logic circuits as models of computation
Naumenko, Mykhailo ; Kazda, Alexandr (advisor) ; Kompatscher, Michael (referee)
This work focuses on the study of logic circuits. We investigated the basics of the theory of logic circuits following the textbook "Models of Computation" by John E. Savage and we used this knowledge to solve some of the examples and problems suggested in the textbook. In this work, you can find key concepts related to logical circuits. Our main topic is the estimation of the lower bounds of the circuit size and formula size of general Boolean function. We constructed simple examples of some known circuits and showed how the circuit designs may be offered. 1
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