Bin packing is a well studied problem which has many applications. In this paper we design a robust APTAS for the problem. The robust APTAS receives a single input item to be added to the packing at each step. It maintains an approximate solution throughout this process, by slightly adjusting the solution for each new item. At each step, the total size of items which may migrate between bins mu...

A new short proof is given for Brandt and Harrington's theorem about conformal uniformizations of planar nitely connected domains as domains with boundary components of speciied shapes. This method of proof generalizes to periodic domains. Letting the uniformized domains degenerate in a controlled manner, we deduce the existence of packings of speciied shapes and with speciied combinatorics. Th...

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...

This paper proposes an adaptation, to the two-dimensional irregular bin packing problem of the Djang and Finch heuristic (DJD), originally designed for the one-dimensional bin packing problem. In the two-dimensional case, not only is it the case that the piece’s size is important but its shape also has a significant influence. Therefore, DJD as a selection heuristic has to be paired with a plac...