Eric Rémila

A Near-Optimal Solution to a Two-Dimensional Cutting Stock Problem

We present an asymptotic fully polynomial approximation scheme for strip-packing, or packing rectangles into a rectangle of fixed width and minimum height, a classicalNP-hard cutting-stock problem. The algorithm, based on a new linear-programming relaxation, finds a packing ofn rectangles whose total height is within a factor of (1 +e) of optimal (up to an additive term), and has running time polynomial both in n and in 1/ e.

Self-assemblying classes of shapes with a minimum number of tiles, and in optimal time

In this paper we construct fixed finite tile systems that assemble into particular classes of shapes. Moreover, given an arbitrary n, we show how to calculate the tile concentrations in order to ensure that the expected size of the produced shape is n. For rectangles and squares our constructions are optimal (with respect to the size of the systems). We also introduce the notion of parallel time, which is a good approximation of the classical asynchronous time. We prove that our tile systems produce the rectangles and squares in linear parallel time (with respect to the diameter). Those results are optimal. Finally, we introduce the class of diamonds. For these shapes we construct a non trivial tile system having also a linear parallel time complexity.

Rooted-tree solutions for tree games

In this paper, we study cooperative games with limited cooperation possibilities, represented by a tree on the set of agents. Agents in the game can cooperate if they are connected in the tree. We introduce natural extensions of the average (rooted)-tree solution (see [Herings, P., van der Laan, G., Talman, D., 2008. The average tree solution for cycle free games. Games and Economic Behavior 62, 77-92]): the marginalist tree solutions and the random tree solutions. We provide an axiomatic characterization of each of these sets of solutions. By the way, we obtain a new characterization of the average tree solution.

Configurations induced by discrete rotations: periodicity and quasi-periodicity properties

A discretized rotation acts on a pixel grid: the edges of the neighborhood relation are affected in particular way. Two types of configurations (i.e. applications from Z^2 to a finite set of states) are introduced to code locally the transformations of the neighborhood. All the characteristics of discretized rotations are encoded within the configurations. We prove that their structure is linked to a subgroup of the bidimensional torus. Using this link, we obtain a characterization of periodical configurations and we prove their quasi-periodicity for any angle.

Perfect matchings in the triangular lattice

Abstract Let V be a finite subset of vertices of the infinite planar triangular lattice T , and G denote the subgraph induced by V . We assume that G is simply connected (i.e. G and the subgraph induced by T − V are both connected). Firstly, we prove that if the vertex-connectivity of G is at least 2, and V is even, then G has a perfect matching. Secondly, we devise a linear-time algorithm for finding a perfect matching of G when it exists. Thirdly, we prove that any perfect matching of G can be transformed into any other perfect matching of G by a linear number of local transformations involving at most 4 edges.

Tiling figures of the plane with two bars

Abstract Given two “bars”, a horizontal one, and a vertical one (both of length at least two), we are interested in the following decision problem: is a finite figure drawn on a plane grid tilable with these bars. It turns out that if one of the bars has length at least three, the problem is NP- complete . If bars are dominoes, the problem is in P, and even linear (in the size of the figure) for certain classes of figures. Given a general pair of bars, we give two results: (1) a necessary condition to have a unique tiling for finite figures without holes, (2) a linear algorithm (in the size of the figure) deciding whether a unique tiling exists, and computing this one if it does exist. Finally, given a tiling of a figure (not necessarily finite), this tiling is the unique one for the figure if and only if there exists no subtiling covering a “canonical” rectangle .

Tiling groups: New applications in the triangular lattice

We first give a new presentation of an algorithm, from W. Thurston, to tile polygons with “calissons” (i.e., lozenges formed from two cells of the triangular lattice Λ). Afterward, we use a similar method to get a linear algorithm to tile polygons with m-leaning bars (parallelograms of length m formed from 2m cells of Λ) and equilateral triangles (whose sides have length m) and we produce a quadratic algorithm to tile polygons with m-leaning bars.

The lattice structure of the set of domino tilings of a polygon

Abstract Fix a polygon P with vertical and horizontal sides. We first recall how each tiling of P with dominoes (i.e. rectangles 2×1) can be encoded by a height function. Such an encoding induces a lattice structure on the set TP of the tilings of P. We give some applications of this structure, and we especially describe the order of meet irreducible elements of TP.

Tiling with bars under tomographic constraints

We wish to tile a rectangle or a torus with only vertical and horizontal bars of a given length, such that the number of bars in every column and row equals given numbers. We present results for particular instances and for a more general problem, while leaving open the initial problem.

Characterization of bijective discretized rotations

A discretized rotation is the composition of an Euclidean rotation with the rounding operation. For 0 < α < π/4, we prove that the discretized rotation [ rα] is bijective if and only if there exists a positive integer k such as