113 research outputs found

    Complexity of Chess Domination Problems

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    We study different domination problems of attacking and non-attacking rooks and queens on polyominoes and polycubes of all dimensions. Our main result proves that maximal domination is NP-complete for non-attacking queens and for non-attacking rooks on polycubes of dimension three and higher. We also analyse these problems for polyominoes and convex polyominoes, conjecture the complexity classes and provide a computer tool for investigation. We have also computed new values for classical queen domination problems on chessboards (square polyominoes). For our computations, we have translated the problem into an integer linear programming instance. Finally, using this computational implementation and the game engine Godot, we have developed a video game of minimal domination of queens and rooks on randomly generated polyominoes.Comment: 19 pages, 20 figures, 4 tables. Theorem 1 now for d>2, added results on approximation, fixed typos, reorganised some proof

    DNA Tile Self-Assembly for 3D-Surfaces: Towards Genus Identification

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    We introduce a new DNA tile self-assembly model: the Surface Flexible Tile Assembly Model (SFTAM), where 2D tiles are placed on host 3D surfaces made of axis-parallel unit cubes glued together by their faces, called polycubes. The bonds are flexible, so that the assembly can bind on the edges of the polycube. We are interested in the study of SFTAM self-assemblies on 3D surfaces which are not always embeddable in the Euclidean plane, in order to compare their different behaviors and to compute the topological properties of the host surfaces. We focus on a family of polycubes called order-1 cuboids. Order-0 cuboids are polycubes that have six rectangular faces, and order-1 cuboids are made from two order-0 cuboids by substracting one from the other. Thus, order-1 cuboids can be of genus 0 or of genus 1 (then they contain a tunnel). We are interested in the genus of these structures, and we present a SFTAM tile assembly system that determines the genus of a given order-1 cuboid. The SFTAM tile assembly system which we design, contains a specific set Y of tile types with the following properties. If the assembly is made on a host order-1 cuboid C of genus 0, no tile of Y appears in any producible assembly, but if C has genus 1, every terminal assembly contains at least one tile of Y. Thus, for order-1 cuboids our system is able to distinguish the host surfaces according to their genus, by the tiles used in the assembly. This system is specific to order-1 cuboids but we can expect the techniques we use to be generalizable to other families of shapes

    Combinatorics of the Permutahedra, Associahedra, and Friends

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    I present an overview of the research I have conducted for the past ten years in algebraic, bijective, enumerative, and geometric combinatorics. The two main objects I have studied are the permutahedron and the associahedron as well as the two partial orders they are related to: the weak order on permutations and the Tamari lattice. This document contains a general introduction (Chapters 1 and 2) on those objects which requires very little previous knowledge and should be accessible to non-specialist such as master students. Chapters 3 to 8 present the research I have conducted and its general context. You will find: * a presentation of the current knowledge on Tamari interval and a precise description of the family of Tamari interval-posets which I have introduced along with the rise-contact involution to prove the symmetry of the rises and the contacts in Tamari intervals; * my most recent results concerning q, t-enumeration of Catalan objects and Tamari intervals in relation with triangular partitions; * the descriptions of the integer poset lattice and integer poset Hopf algebra and their relations to well known structures in algebraic combinatorics; * the construction of the permutree lattice, the permutree Hopf algebra and permutreehedron; * the construction of the s-weak order and s-permutahedron along with the s-Tamari lattice and s-associahedron. Chapter 9 is dedicated to the experimental method in combinatorics research especially related to the SageMath software. Chapter 10 describes the outreach efforts I have participated in and some of my approach towards mathematical knowledge and inclusion.Comment: 163 pages, m\'emoire d'Habilitation \`a diriger des Recherche

    Reconstruction of Convex Sets from One or Two X-rays

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    We consider a class of problems of Discrete Tomography which has been deeply investigated in the past: the reconstruction of convex lattice sets from their horizontal and/or vertical X-rays, i.e. from the number of points in a sequence of consecutive horizontal and vertical lines. The reconstruction of the HV-convex polyominoes works usually in two steps, first the filling step consisting in filling operations, second the convex aggregation of the switching components. We prove three results about the convex aggregation step: (1) The convex aggregation step used for the reconstruction of HV-convex polyominoes does not always provide a solution. The example yielding to this result is called \textit{the bad guy} and disproves a conjecture of the domain. (2) The reconstruction of a digital convex lattice set from only one X-ray can be performed in polynomial time. We prove it by encoding the convex aggregation problem in a Directed Acyclic Graph. (3) With the same strategy, we prove that the reconstruction of fat digital convex sets from their horizontal and vertical X-rays can be solved in polynomial time. Fatness is a property of the digital convex sets regarding the relative position of the left, right, top and bottom points of the set. The complexity of the reconstruction of the lattice sets which are not fat remains an open question.Comment: 31 pages, 24 figure

    LIPIcs, Volume 274, ESA 2023, Complete Volume

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    LIPIcs, Volume 274, ESA 2023, Complete Volum

    LIPIcs, Volume 258, SoCG 2023, Complete Volume

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    LIPIcs, Volume 258, SoCG 2023, Complete Volum

    Lattices of acyclic pipe dreams

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    We show that for any permutation ω\omega, the increasing flip graph on acyclic pipe dreams with exiting permutation ω\omega is a lattice quotient of the interval [e,ω][e,\omega] of the weak order. We then discuss conjectural generalizations of this result to acyclic facets of subword complexes on arbitrary finite Coxeter groups.Comment: 34 pages, 19 figures. Version 2: New section 4.5 on nu-Tamari lattice

    A counterexample to the periodic tiling conjecture

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    The periodic tiling conjecture asserts that any finite subset of a lattice Zd\mathbb{Z}^d which tiles that lattice by translations, in fact tiles periodically. In this work we disprove this conjecture for sufficiently large dd, which also implies a disproof of the corresponding conjecture for Euclidean spaces Rd\mathbb{R}^d. In fact, we also obtain a counterexample in a group of the form Z2×G0\mathbb{Z}^2 \times G_0 for some finite abelian 22-group G0G_0. Our methods rely on encoding a "Sudoku puzzle" whose rows and other non-horizontal lines are constrained to lie in a certain class of "22-adically structured functions", in terms of certain functional equations that can be encoded in turn as a single tiling equation, and then demonstrating that solutions to this Sudoku puzzle exist but are all non-periodic.Comment: 50 pages, 13 figures. Minor changes and additions of new reference

    Scalable multi-class sampling via filtered sliced optimal transport

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    We propose a multi-class point optimization formulation based on continuous Wasserstein barycenters. Our formulation is designed to handle hundreds to thousands of optimization objectives and comes with a practical optimization scheme. We demonstrate the effectiveness of our framework on various sampling applications like stippling, object placement, and Monte-Carlo integration. We a derive multi-class error bound for perceptual rendering error which can be minimized using our optimization. We provide source code at https://github.com/iribis/filtered-sliced-optimal-transport.Comment: 15 pages, 17 figures, ACM Trans. Graph., Vol. 41, No. 6, Article 261. Publication date: December 202

    Information Technology Outlines (Professional English Basics). Підручник з англійської мови. Видання друге, перероблене і доповнене

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    Підручник покликаний розвивати професійно орієнтовану англомовну комунікативну компетентність через запропонований у ньому студентоцентрований підхід до навчання іноземної мови, який становить основу методики індивідуалізації викладання іноземних мов. Автори пропонують принципово нову структуру посібника, у якому кожен урок побудовано з урахуванням індивідуальних особливостей студентів. Підручник забезпечує аудиторну та самостійну роботу студентів. Для студентів галузі знань "Інформаційні технології", викладачів іноземної мови у вищих технічних навчальних закладах, та всіх, хто цікавиться проблемами викладання професійно орієнтованої англійської мови.The textbook is designed to develop professionally oriented English-language communicative competence through student-centered approach to foreign language teaching, which is the basis of the methodology of individualization of foreign language teaching. The authors propose a fundamentally new structure of the textbook, in which each lesson is built taking into account the individual characteristics of students. The textbook proposes tasks for class work and self-work of students. This book will be useful for students majoring in IT and related specialties, foreign language teachers in higher technical educational institutions, and anyone interested in teaching professional English
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