Related papers: The Complexity of Tiling Problems
It is well-known that the question of whether a given finite region can be tiled with a given set of tiles is NP-complete. We show that the same is true for the right tromino and square tetromino on the square lattice, or for the right…
Weighted counting problems are a natural generalization of counting problems where a weight is associated with every computational path of polynomial-time non-deterministic Turing machines and the goal is to compute the sum of the weights…
To understand an aperiodic tiling (or a quasicrystal modeled on an aperiodic tiling), we construct a space of similar tilings, on which the group of translations acts naturally. This space is then an (abstract) dynamical system. Dynamical…
This chapter does not deal with specific tools and techniques for managing complex systems, but proposes some basic concepts that help us to think and speak about complexity. We review classical thinking and its intrinsic drawbacks when…
We study the tiling of a two-dimensional region of the plane by $K$-cell one-dimensional tiles, or $K$-mers. Unlike previous studies, which typically allowed for one single value of $K$ or sometimes a small assortment of fixed values, here…
We synthesize and unify notions of regularity, both of individual sets and of collections of sets, as they appear in the convergence theory of projection methods for consistent feasibility problems. Several new characterizations of…
We present here an elementary construction of an aperiodic tile set. Although there already exist dozens of examples of aperiodic tile sets we believe this construction introduces an approach that is different enough to be interesting and…
Classical complexity theory measures the cost of computing a function, but many computational tasks require committing to one valid output among several. We introduce determination depth -- the minimum number of sequential layers of…
The Heesch problem 'grades' polygons that fail to tile the plane in terms of the number of layers (or corollas) of copies of it that can be formed around a central unit. We study the different topology of ' walls', which we define to be…
Altenbernd, Thomas and W\"ohrle have considered acceptance of languages of infinite two-dimensional words (infinite pictures) by finite tiling systems, with usual acceptance conditions, such as the B\"uchi and Muller ones [1]. It was proved…
We examine various categorical structures that can and cannot be constructed. We show that total computable functions can be mimicked by constructible functors. More generally, whatever can be done by a Turing machine can be constructed by…
There is a rich history of domino tilings in two dimensions. Through a variety of techniques we can answer questions such as: how many tilings are there of a given region or what does the space of all tilings look like? These questions and…
In a recent paper, the author has shown how Interaction Graphs models for linear logic can be used to obtain implicit characterisations of non-deterministic complexity classes. In this paper, we show how this semantic approach to Implicit…
The computational complexity of tiling finite simply connected regions with a fixed set of tiles is studied in this paper. We show that the problem of tiling simply connected regions with a fixed set of $23$ Wang tiles is NP-complete. As a…
We consider training probabilistic classifiers in the case of a large number of classes. The number of classes is assumed too large to perform exact normalisation over all classes. To account for this we consider a simple approach that…
We study the computational complexity theory of smooth, finite-dimensional dynamical systems. Building off of previous work, we give definitions for what it means for a smooth dynamical system to simulate a Turing machine. We then show that…
Nature hosts a wealth of materials showcasing intricate structures intertwining order, disorder, and hierarchy, delivering resilient multifunctionality surpassing perfect crystals or simplistic disordered materials. The engineering of such…
While many classes of cutting-planes are at the disposal of integer programming solvers, our scientific understanding is far from complete with regards to cutting-plane selection, i.e., the task of selecting a portfolio of cutting-planes to…
A sliding puzzle is a combination puzzle where a player slide pieces along certain routes on a board to reach a certain end-configuration. In this paper, we propose a novel measurement of complexity of massive sliding puzzles with…
A variant of the well-known Set Covering Problem is studied in this paper, where subsets of a collection have to be selected, and pairwise conflicts among subsets of items exist. The selection of each subset has a cost, and the inclusion of…