Search

The topic of recurrence relations has recently been introduced in many discrete mathematics textbooks. Recurrence relations are efficient modelling and problem-solving techniques used in mathematic...

Several articles deal with tilings with squares and dominoes on 2-dimensional boards, but only a few on boards in 3-dimensional space. We examine a tiling problem with coloured cubes and bricks on ...

Submitted to LMCS. Full version of our LICS 2019 paper; Adding propositional quantification to the modal logics K, T or S4 is known to lead to undecidability but CTL with propositional quantification under the tree semantics (tQCTL) admits a non-elementary Tower-complete satisfiability problem. We investigate the complexity of strict fragments of tQCTL as well as of the modal logic K with propositional quantification under the tree semantics. More specifically, we show that tQCTL restricted to the temporal operator EX is already Tower-hard, which is unexpected as EX can only enforce local properties. When tQCTL restricted to EX is interpreted on N-bounded trees for some N >= 2, we prove that the satisfiability problem is AExpPol-complete; AExpPol-hardness is established by reduction from a recently introduced tiling problem, instrumental for studying the model-checking problem for interval temporal logics. As consequences of our proof method, we prove Tower-hardness of tQCTL restricted to EF or to EXEF and of the well-known modal logics such as K, KD, GL, K4 and S4 with propositional quantification under a semantics based on classes of trees.

Several articles deal with tilings with squares and dominoes of the well-known regular square mosaic in Euclidean plane, but not any with the hyperbolic regular square mosaics. In this article, we examine the tiling problem with colored squares and dominoes of one type of the possible hyperbolic generalization of $(2\times n)$-board., Comment: 10 pages, 8 figures

This paper defines a tiling problem related to the automorphism group of $$F_2$$ . Our main result is that the corresponding tilings admit a complete, concrete classification.

We study the computational hardness of the tiling puzzle with polyominoes, where a polyomino is a right-angled polygon (i.e., a polygon made by connecting unit squares along their edges). In the tiling problem, we are given a right-angled polygon P and a set S of polyominoes, and asked whether P can be covered without any overlap using translated copies of polyominoes in S. In this paper, we focus on trominoes and bars as polyominoes; a tromino is a polyomino consisting of three unit squares, and a bar is a rectangle of either height one or width one. Notice that there are essentially two shapes of trominoes, that is, I-shape (i.e., a bar) and L-shape. We consider the tiling problem when restricted to only L-shape trominoes, only I-shape trominoes, both L-shape and I-shape trominoes, or only two bars. In this paper, we prove that the tiling problem remains NP-complete even for such restricted sets of polyominoes. All reductions are carefully designed so that we can also prove the # P-completeness and ASP-completeness of the counting and the another-solution-problem variants, respectively. Our results answer two open questions proposed by Moore and Robson (Discrete Comput Geom 26:573–590, 2001) and Pak and Yang (J Comb Theory 120:1804–1816, 2013).

Adding propositional quantification to the modal logics $\mathsf{K, T}$ or S4 is known to lead to undecidability but CTL with propositional quantification under the tree semantics (QCTLt) admits a non-elementary Tower-complete satisfiability problem. We investigate the complexity of strict fragments of $\mathsf{QCTL}^{t}$ as well as of the modal logic $\mathsf{K}$ with propositional quantification under the tree semantics. More specifically, we show that $\mathsf{QCTL}^{t}$ restricted to the temporal operator EX is already Tower-hard, which is unexpected as EX can only enforce local properties. When QCTLt restricted to EX is interpreted on N-bounded trees for some $N\geq 2$ , we prove that the satisfiability problem is $\mathbf{AExp}_{\mathrm{pol}^{-}}$ complete; $\mathbf{AExp}_{\mathrm{pol}}$ -hardness is established by reduction from a recently introduced tiling problem, instrumental for studying the model-checking problem for interval temporal logics. As consequences of our proof method, we prove Tower-hardness of QCTLtrestricted to EF or to EXEF and of the well-known modal logics $\mathsf{K}$ , KD, GL, S4, K4 and D4, with propositional quantification under a semantics based on classes of trees.

In this paper, new integer programming models of the problem of irregular polyomino tiling are introduced. We consider tiling of finite, square, NxN-sized structure with L-shaped trominoes without any restriction on their number. Each polyomino can be rotated 90\(^{\circ }\), so there are four orientations of the L-tromino. Developed models are effective for small-size instances. For medium- and large-size instances we suggest dividing the initial structure into several equally sized parts and combine the solutions of optimized tilings. We tried to apply new models to the existing information-theoretic entropy-based approach. We conducted computational experiments using IBM ILOG CPLEX package. The problem of irregular polyomino tiling can be applied to the design of phased array antennas where polyomino-shaped subarrays are used to reduce the cost of the array antenna and to reduce the undesired sidelobes radiation. Computational results along with antenna simulation results are presented in the paper.

We consider a certain tiling problem of a planar region in which there are no long horizontal or vertical strips consisting of copies of the same tile. Intuitively speaking, we would like to create a dappled pattern with two or more kinds of tiles. We give an efficient algorithm to turn any tiling into one satisfying the condition, and discuss its applications in texturing.

The Square Tiling Problem was recently introduced as equivalent to the problem of reconstructing an image from patches and a possible general-purpose indexing tool. Unfortunately, the Square Tiling Problem was shown to be NP -hard. A 1/2-approximation is known.We show that if the tile alphabet is fixed and finite, there is a Polynomial Time Approximation Scheme (PTAS) for the Square Tiling Problem with approximation ratio of ( 1 - ? 2 log ? n ) for any given ? ? 1 .Another topic handled in this paper is the NP -hardness of the Tiling problem with an infinite alphabet. We show that when the alphabet is not bounded, even the decision version for rectangles of size 3n is NP -Complete.

Combinatorial optimization problems have become the target of many scientific researches for their importance in solving academic problems and real problems encountered in the field of engineering and industry. Solving these problems by exact methods is often intractable because of the exorbitant time processing that these methods would require to reach the optimal solution(s). In this thesis, we were interested in the algorithmic context of solving combinatorial problems, and the modeling context of these problems. At the algorithmic level, we have explored the hybrid methods which excel in their ability to cooperate exact methods and approximate methods in order to produce rapidly solutions of best quality. At the modeling level, we worked on the specification and the exact resolution of complex problems in pattern set mining, in particular, by studying scaling issues in large databases. On the one hand, we proposed a first parallelization of the DGVNS algorithm, called CPDGVNS, which explores in parallel the different clusters of the tree decomposition by sharing the best overall solution on a master-worker model. Two other strategies, called RADGVNS and RSDGVNS, have been proposed which improve the frequency of exchanging intermediate solutions between the different processes. Experiments carried out on difficult combinatorial problems show the effectiveness of our parallel methods. On the other hand, we proposed a hybrid approach combining techniques of both Integer Linear Programming (ILP) and pattern mining. Our approach is comprehensive and takes advantage of the general ILP framework (by providing a high level of flexibility and expressiveness) and specialized heuristics for data mining (to improve computing time). In addition to the general framework for the pattern set mining, two problems were studied: conceptual clustering and the tiling problem. The experiments carried out showed the contribution of our proposition in relation to constraint-based approaches and specialized heuristics., Les problèmes d’optimisation combinatoire sont devenus la cible de nombreuses recherches scientifiques pour leur importance dans la résolution de problèmes académiques et de problèmes réels rencontrés dans le domaine de l’ingénierie et dans l’industrie. La résolution de ces problèmes par des méthodes exactes ne peut être envisagée à cause des délais de traitement souvent exorbitants que nécessiteraient ces méthodes pour atteindre la (les) solution(s) optimale(s). Dans cette thèse, nous nous sommes intéressés au contexte algorithmique de résolution des problèmes combinatoires, et au contexte de modélisation de ces problèmes. Au niveau algorithmique, nous avons appréhendé les méthodes hybrides qui excellent par leur capacité à faire coopérer les méthodes exactes et les méthodes approchées afin de produire rapidement des solutions. Au niveau modélisation, nous avons travaillé sur la spécification et la résolution exacte des problématiques complexes de fouille des ensembles de motifs en étudiant tout particulièrement le passage à l’échelle sur des bases de données de grande taille. D'une part, nous avons proposé une première parallélisation de l'algorithme DGVNS, appelée CPDGVNS, qui explore en parallèle les différents clusters fournis par la décomposition arborescente en partageant la meilleure solution trouvée sur un modèle maître-travailleur. Deux autres stratégies, appelées RADGVNS et RSDGVNS, ont été proposées qui améliorent la fréquence d'échange des solutions intermédiaires entre les différents processus. Les expérimentations effectuées sur des problèmes combinatoires difficiles montrent l'adéquation et l'efficacité de nos méthodes parallèles. D'autre part, nous avons proposé une approche hybride combinant à la fois les techniques de programmation linéaire en nombres entiers (PLNE) et la fouille de motifs. Notre approche est complète et tire profit du cadre général de la PLNE (en procurant un haut niveau de flexibilité et d’expressivité) et des heuristiques spécialisées pour l’exploration et l’extraction de données (pour améliorer les temps de calcul). Outre le cadre général de l’extraction des ensembles de motifs, nous avons étudié plus particulièrement deux problèmes : le clustering conceptuel et le problème de tuilage (tiling). Les expérimentations menées ont montré l’apport de notre proposition par rapport aux approches à base de contraintes et aux heuristiques spécialisées.

A lattice L L in R n \mathbb {R}^n is said to be equivalent to an integral lattice if there exists a real number r r such that the dot product of any pair of vectors in r L rL is an integer. We show that if n ≥ 3 n \geq 3 and L L is equivalent to an integral lattice, then there is no measurable Steinhaus set for L L , a set which no matter how translated and rotated contains exactly one vector in L L .