OPTIMIZATION-BASED APPROACH TO TILING OF FINITE AREAS WITH ARBITRARY SETS OF WANG TILES

Wang tiles proved to be a convenient tool for the design of aperiodic tilings in computer graphics and in materials engineering. While there are several algorithms for generation of finite-sized tilings, they exploit the specific structure of individual tile sets, which prevents their general usage. In this contribution, we reformulate the NP-complete tiling generation problem as a binary linear program, together with its linear and semidefinite relaxations suitable for the branch and bound method. Finally, we assess the performance of the established formulations on generations of several aperiodic tilings reported in the literature, and conclude that the linear relaxation is better suited for the problem

On bounded Wang tilings

Wang tiles enable efficient pattern compression while avoiding the periodicity in tile distribution via programmable matching rules. However, most research in Wang tilings has considered tiling the infinite plane. Motivated by emerging applications in materials engineering, we consider the bounded version of the tiling problem and offer four integer programming formulations to construct valid or nearly-valid Wang tilings: a decision, maximum-rectangular tiling, maximum cover, and maximum adjacency constraint satisfaction formulations. To facilitate a finer control over the resulting tilings, we extend these programs with tile-based, color-based, packing, and variable-sized periodic constraints. Furthermore, we introduce an efficient heuristic algorithm for the maximum-cover variant based on the shortest path search in directed acyclic graphs and derive simple modifications to provide a $1/2$ approximation guarantee for arbitrary tile sets, and a $2/3$ guarantee for tile sets with cyclic transducers. Finally, we benchmark the performance of the integer programming formulations and of the heuristic algorithms showing that the heuristics provides very competitive outputs in a fraction of time. As a by-product, we reveal errors in two well-known aperiodic tile sets: the Knuth tile set contains a tile unusable in two-way infinite tilings, and the Lagae corner tile set is not aperiodic

Proceedings of JAC 2010. Journées Automates Cellulaires

The second Symposium on Cellular Automata “Journ´ees Automates Cellulaires” (JAC 2010) took place in Turku, Finland, on December 15-17, 2010. The first two conference days were held in the Educarium building of the University of Turku, while the talks of the third day were given onboard passenger ferry boats in the beautiful Turku archipelago, along the route Turku–Mariehamn–Turku. The conference was organized by FUNDIM, the Fundamentals of Computing and Discrete Mathematics research center at the mathematics department of the University of Turku. The program of the conference included 17 submitted papers that were selected by the international program committee, based on three peer reviews of each paper. These papers form the core of these proceedings. I want to thank the members of the program committee and the external referees for the excellent work that have done in choosing the papers to be presented in the conference. In addition to the submitted papers, the program of JAC 2010 included four distinguished invited speakers: Michel Coornaert (Universit´e de Strasbourg, France), Bruno Durand (Universit´e de Provence, Marseille, France), Dora Giammarresi (Universit` a di Roma Tor Vergata, Italy) and Martin Kutrib (Universit¨at Gie_en, Germany). I sincerely thank the invited speakers for accepting our invitation to come and give a plenary talk in the conference. The invited talk by Bruno Durand was eventually given by his co-author Alexander Shen, and I thank him for accepting to make the presentation with a short notice. Abstracts or extended abstracts of the invited presentations appear in the first part of this volume. The program also included several informal presentations describing very recent developments and ongoing research projects. I wish to thank all the speakers for their contribution to the success of the symposium. I also would like to thank the sponsors and our collaborators: the Finnish Academy of Science and Letters, the French National Research Agency project EMC (ANR-09-BLAN-0164), Turku Centre for Computer Science, the University of Turku, and Centro Hotel. Finally, I sincerely thank the members of the local organizing committee for making the conference possible. These proceedings are published both in an electronic format and in print. The electronic proceedings are available on the electronic repository HAL, managed by several French research agencies. The printed version is published in the general publications series of TUCS, Turku Centre for Computer Science. We thank both HAL and TUCS for accepting to publish the proceedings.Siirretty Doriast

Universality in algorithmic self-assembly

Tile-based self-assembly is a model of algorithmic crystal growth in which square tiles represent molecules that bind to each other via specific and variable-strength bonds on their four sides, driven by random mixing in solution but constrained by the local binding rules of the tile bonds. In the late 1990s, Erik Winfree introduced a discrete mathematical model of DNA tile assembly called the abstract Tile Assembly Mode. Winfree proved that the Tile Assembly Model is computationally universal, i.e., that any Turing machine can be encoded into a finite set of tile types whose self-assembly simulates that Turing machine. In this thesis, we investigate tile-based self-assembly systems that exhibit Turing universality, geometric universality and intrinsic universality. We first establish a novel characterization of the computably enumerable languages in terms of self-assembly--the proof of which is a novel proof of the Turing-universality of the Tile Assembly Model in which a particular Turing machine is simulated on all inputs in parallel in the two-dimensional discrete Euclidean plane. Then we prove that the multiple temperature tile assembly model (introduced by Aggarwal, Cheng, Goldwasser, Kao, and Schweller) exhibits a kind of geometric universality in the sense that there is a small (constant-size) universal tile set that can be programmed via deliberate changes in the system temperature to uniquely produce any finite shape. Just as other models of computation such as the Turing machine and cellular automaton are known to be intrinsically universal, (e.g., Turing machines can simulate other Turing machines, and cellular automata other cellular automata), we show that tile assembly systems satisfying a natural condition known as local consistency are able to simulate other locally consistent tile assembly systems. In other words, we exhibit a particular locally consistent tile assembly system that can simulate the behavior--as opposed to only the final result--of any other locally consistent tile assembly system. Finally, we consider the notion of universal fault-tolerance in algorithmic self-assembly with respect to the two-handed Tile Assembly Model, in which large aggregations of tiles may attach to each other, in contrast to the seeded Tile Assembly Model, in which tiles aggregate one at a time to a single specially-designated seed assembly. We introduce a new model of fault-tolerance in self-assembly: the fuzzy temperature model of faults having the following informal characterization: the system temperature is normally 2, but may drift down to 1, allowing unintended temperature-1 growth for an arbitrary period of time. Our main construction, which is a tile set capable of uniquely producing an $n \times n$ square with log n unique tile types in the fuzzy temperature model, is not universal but presents novel technique that we hope will ultimately pave the way for a universal fuzzy-fault-tolerant tile assembly system in the future

Proceedings of the 8th Cologne-Twente Workshop on Graphs and Combinatorial Optimization

International audienceThe Cologne-Twente Workshop (CTW) on Graphs and Combinatorial Optimization started off as a series of workshops organized bi-annually by either Köln University or Twente University. As its importance grew over time, it re-centered its geographical focus by including northern Italy (CTW04 in Menaggio, on the lake Como and CTW08 in Gargnano, on the Garda lake). This year, CTW (in its eighth edition) will be staged in France for the first time: more precisely in the heart of Paris, at the Conservatoire National d’Arts et Métiers (CNAM), between 2nd and 4th June 2009, by a mixed organizing committee with members from LIX, Ecole Polytechnique and CEDRIC, CNAM