Orthogonal packing problems are natural multidimensional generalizations of the classical bin packing problem and knapsack problem and occur in many different settings. The input consists of a set I = {r1, . . . , rn} of d-dimensional rectangular items ri = (ai,1, . . . , ai,d) and a space Q. The task is to pack the items in an orthogonal and non-overlapping manner without using rotations into ...

There appear to be two versions of the Dual Bin Packing problem in the literature. In addition, one of the versions has a counterpart in the cutting stock literature, known as the Skiving Stock Problem. This paper outlines branch-andprice algorithms for both. We introduce combinatorial upper bounds and well-performing heuristics from the literature in the branch-and-price framework. Extensive c...

We consider the bin packing problem with d different item sizes and revisit the structure theorem given by Goemans and Rothvoß [GR14] about solutions of the integer cone. We present new techniques on how solutions can be modified and give a new structure theorem that relies on the set of vertices of the underlying integer polytope. As a result of our new structure theorem, we obtain an algorith...

Baskett's "Puzzle" benchmark has been used for almost a decade for the evaluation of hardware architectures, and was included in the "Gabriel" suite of Lisp benchmarks [Gabriel85]. "Puzzle" solves a 3-dimensional packing problem by attempting to pack pieces of 4 different types into a 5x5x5 cube. The class of such packing problems is closely related to the "bin-packing" and "knapsack" problems ...

The Andreev-Koebe-Thurston circle packing theorem is generalized and improved in two ways. First, we get simultaneous circle packings of the map and its dual map so that, in the corresponding straight-line representations of the map and the dual, any two edges dual to each other are perpendicular. Necessary and sufficient condition for a map to have such a primal-dual circle packing representat...

We study the following packing problem: Given a collection of d-dimensional rectangles of specified sizes, pack them into the minimum number of unit cubes. We show that unlike the one-dimensional case, the two-dimensional packing problem cannot have an asymptotic polynomial time approximation scheme (APTAS), unless P = NP . On the positive side, we give an APTAS for the special case of packing ...

We consider a variant of bin packing calledmultiple-choice vector bin packing. In this problem we are given a set of n items, where each item can be selected in one of several D-dimensional incarnations. We are also given T bin types, each with its own cost and D-dimensional size. Our goal is to pack the items in a set of bins of minimum overall cost. The problem is motivated by scheduling in n...

Recently Bansal and Sviridenko [4] proved that for 2-dimensional Orthogonal Rectangle Bin Packing without rotations allowed there is no asymptotic PTAS, unless P = NP. We show that similar approximation hardness results hold for several rectangle packing and covering problems even if rotations by ninety degrees around the axes are allowed. Moreover, for some of these problems we provide explici...

We present a novel heuristic for 2D-packing of rectangles inside a rectangular area where the aesthetics of the resulting packing is amenable to generating large collages of photographs or images. The heuristic works by maintaining a sorted collection of vertical segments covering the area to be packed. The segments define the leftmost boundaries of rectangular and possibly overlapping areas th...

During system synthesis (e.g. task allocation) the transmission of messages between tasks is usually addressed in a simplistic way. If a message is exchanged via an external bus, it is assumed each message is packed in an individual frame. This assumption leads to an over-estimation of bus bandwidth demand and frame response time. For some systems this pessimism is not acceptable (e.g. automoti...