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CSP over oriented trees
Tatarko, William ; Bulín, Jakub (advisor) ; Barto, Libor (referee)
In this thesis we present an oriented tree with only 26 vertices, whose CSP is NP-complete. This serves as a counterexample for a conjecture that any oriented tree with this property has at least 39 vertices. The work itself is divided into three chapters. In the first one, basic definitions and tools of this topic are introduced. Then, tractability of trees of special shapes is shown. Finally, NP-completeness of a certain oriented tree is proven. 1
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Squares in integer sequences
Hudcová, Barbora ; Holub, Štěpán (advisor) ; Bulín, Jakub (referee)
This work is based on an article which provides a construction of the first infinite word over a finite alphabet avoiding additive cubes. We construct other words with the same properties and we choose one of them to show the main idea of the proof that this word avoids additive cubes. 1
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Finitely generated clones
Draganov, Ondřej ; Barto, Libor (advisor) ; Bulín, Jakub (referee)
A clone is a set of finitary operations closed under composition and contain- ing all projections. We say it is finitely generated if there exist a finite subset {f1, . . . , fn} such that all the other operations can be expressed as compositions of f1, . . . , fn. We present examples of finitely and non-finitely genreated clones on finite sets. First, we demonstrate an explicit construction of operations in finitely generated clones. Secondly, we define relations such that the clones of compatible operations have restricted essential arity, and discuss several modifi- cations. Lastly, for every binary operation f which cannot be composed to yield an essentially ternary operation, we find a maximal clone of essentially at most binary operations containing f. 1
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Constraint satisfaction, graphs and algebras
Bulín, Jakub ; Barto, Libor (advisor) ; Růžička, Pavel (referee) ; Bulatov, Andrei (referee)
Constraint satisfaction, graphs and algebras Jakub Bulín Abstract This thesis consists of three papers in the area of algebraic approach to the constraint satisfaction problem. In the first paper, a joint work with Deli'c, Jackson and Niven, we study the reduction of fixed template CSPs to digraphs. We construct, for every relational structure A, a digraph D(A) such that CSP(A) and CSP(D(A)) are logspace equivalent and most rele- vant properties carry over from A to D(A). As a consequence, the algebraic conjectures characterizing CSPs solvable in P, NL and L are equivalent to their restrictions to digraphs. In the second paper we prove that, given a core relational structure A of bounded width and B ⊆ A, it is decidable whether B is an absorbing subuniverse of the algebra of polymorphisms of A. As a by-product, we show that Jónsson absorption coincides with the usual absorption in this case. In the third paper we establish, using modern algebraic tools (e.g. absorption theory and pointing operations), the CSP dichotomy conjecture for so-called special oriented trees and in particular prove that all tractable core special trees have bounded width. Keywords: constraint satisfaction problem, algebra of polymorphisms, absorbing subuniverse, bounded width, oriented tree.
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Algebraický přístup k CSP
Bulín, Jakub
For a finite relational structure A, the Constraint Satisfaction Problem with template A, or CSP(A), is the problem of deciding whether an input relational structure X admits a homomorphism to A. The CSP dichotomy conjecture of Feder and Vardi states that for any A, CSP(A) is either in P or NP-complete. In the first part we present the algebraic approach to CSP and summarize known results about CSP for digraphs, also known as the H-coloring problem. In the second part we study a class of oriented trees called special polyads. Using the algebraic approach we confirm the dichotomy conjecture for special polyads. We provide a finer description of the tractable cases and give a construction of a special polyad T such that CSP(T) is tractable, but T does not have width 1 and admits no near-unanimity polymorphisms.
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An elementary proof of the existence of primitive elements
Majerčík, Miroslav ; Kepka, Tomáš (advisor) ; Bulín, Jakub (referee)
Vedúcí bakalárskej práce: prof. RNDr. Tomáš Kepka, DrSc. Abstrakt: Tento text je venovaný elementárnym dôkazom dvoch významných viet teórie čísel a to Gaussovmu kvadratickému zákonu reciprocity a vete o primitívnom prvku. Dôkazy týchto viet sú vo forme menších na seba nadväzujúcich lemmat a dôkazov. Úvod je venovaný historickému priblíženiu a metóde dôkazov viet. Prvá kapitola smeruje k dôkazu Gaussovmu kvadratickému zákonu reciprocity a druhá k dôkazu vety o primitívnom prvku a k určeniu prirodzených čísel n pre ktoré existuje primitívny prvok modulo n a pre ktoré nie. K dôkazu týchto viet bolo potrebné dokázat aj niekol'ko dalších viet, napríklad malú Fermatovu vetu alebo schému rozdielu mocnín. Klúčové slová: Kvadratický zbytok, Primitívny koreň, Rád prvku modulo n, Eulerova funkcia Title: An elementary proof of the existence of primitive elements Author: Miroslav Majerčík Department: Department of Algebra Supervisor: prof. RNDr. Tomáš Kepka, DrSc. Abstract: This text is about elementary proofs of two well known number theory statements, Gauss quadratic reciprocity law and proof of the existence of primitive elements. These proofs are in form of simpler interlinked lemmas and proofs. Introduction is about historical background and about...
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Algebraický přístup k CSP
Bulín, Jakub ; Barto, Libor (advisor) ; Růžička, Pavel (referee)
For a finite relational structure A, the Constraint Satisfaction Problem with template A, or CSP(A), is the problem of deciding whether an input relational structure X admits a homomorphism to A. The CSP dichotomy conjecture of Feder and Vardi states that for any A, CSP(A) is either in P or NP-complete. In the first part we present the algebraic approach to CSP and summarize known results about CSP for digraphs, also known as the H-coloring problem. In the second part we study a class of oriented trees called special polyads. Using the algebraic approach we confirm the dichotomy conjecture for special polyads. We provide a finer description of the tractable cases and give a construction of a special polyad T such that CSP(T) is tractable, but T does not have width 1 and admits no near-unanimity polymorphisms.
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