The bin packing problem with conflicts consists in packing items in a minimum number of bins of limited capacity while avoiding joint assignments of items that are in conflict. Our study demonstrates the comparatively good performance of a generic Branch-and-Price algorithm for this problem. We made use of our black-box solver BaPCod, relying on its generic branching scheme and primal heuristic...

The packing chromatic number χρ(G) of a graph G is the smallest integer k needed to proper color the vertices of G in such a way that the distance in G between any two vertices having color i be at least i + 1. Goddard et al. [9] found an upper bound for the packing chromatic number of hypercubes Qn. Moreover, they compute χρ(Qn) for n ≤ 5 leaving as an open problem the remaining cases. In this...

The bin-packing problem is a well known NP-Hard optimisation problem, and, over the years, many heuristics have been developed to generate good quality solutions. This paper outlines a genetic programming system which evolves a heuristic that decides whether to put a piece in a bin when presented with the sum of the pieces already in the bin and the size of the piece that is about to be packed....

A major open question in the field of granular materials is the identification of relevant state variables which can predict macroscopic behavior. We experimentally investigate the mixing properties of an idealized granular liquid in the vicinity of its jamming transition, through the generation of ensembles of configurations under various boundary conditions. Our apparatus consists of a two-di...

The circular packing problem (CPP) consists of packing n circles Ci of known radii ri, iAN1⁄4 {1, . . . , n}, into the smallest containing circle C. The objective is to determine the coordinates (xi, yi) of the centre of Ci, iAN, as well as the radius r and centre (x, y) of C. CPP, which is a variant of the two-dimensional open-dimension problem, is NP hard. This paper presents an adaptive algo...

A first-order expansion of the R-vector space structure on R does not define every compact subset of every Rn if and only if topological and Hausdorff dimension coincide on all closed definable sets. Equivalently, if A ⊆ Rk is closed and the Hausdorff dimension of A exceeds the topological dimension of A, then every compact subset of every Rn can be constructed from A using finitely many boolea...

We present a one-parameter family of periodic packings of regular tetrahedra, with the packing fraction 100/117 ≈ 0.8547, that are simple in the sense that they are transitive and their repeating units involve only four tetrahedra. The construction of the packings was inspired from results of a numerical search that yielded a similar packing. We present an analytic construction of the packings ...

Given a bounded sequence of integers {d0,d1,d2, . . .}, 6 ≤ dn ≤M, there is an associated abstract triangulation created by building up layers of vertices so that vertices on the nth layer have degree dn. This triangulation can be realized via a circle packing which fills either the Euclidean or the hyperbolic plane. We give necessary and sufficient conditions to determine the type of the packi...

In this paper, we study online multidimensional bin packing problem when all items are hypercubes. Based on the techniques in one dimensional bin packing algorithm Super Harmonic by Seiden, we give a framework for online hypercube packing problem and obtain new upper bounds of asymptotic competitive ratios. For square packing, we get an upper bound of 2.1439, which is better than 2.24437. For c...

A graph is called a p{polyp if it consists of p simple paths of the same length and one endvertex of all these paths is a common vertex. The Polyp Packing problem is a generalization of the well known Bin Packing problem: How to pack a set of paths with diierent lengths to a set of polyps edge disjointly? It is proved that the Polyp Packing problem is NP-complete and that a modiication of the F...