113 research outputs found
Complexity of Chess Domination Problems
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
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
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
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
LIPIcs, Volume 274, ESA 2023, Complete Volum
LIPIcs, Volume 258, SoCG 2023, Complete Volume
LIPIcs, Volume 258, SoCG 2023, Complete Volum
Lattices of acyclic pipe dreams
We show that for any permutation , the increasing flip graph on
acyclic pipe dreams with exiting permutation is a lattice quotient of
the interval 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
The periodic tiling conjecture asserts that any finite subset of a lattice
which tiles that lattice by translations, in fact tiles
periodically. In this work we disprove this conjecture for sufficiently large
, which also implies a disproof of the corresponding conjecture for
Euclidean spaces . In fact, we also obtain a counterexample in a
group of the form for some finite abelian -group
. Our methods rely on encoding a "Sudoku puzzle" whose rows and other
non-horizontal lines are constrained to lie in a certain class of "-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
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
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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