A set of rectangles S is said to be grid packed if there exists a rectangular grid (not necessarily regular) such that every rectangle lies in the grid and there is at most one rectangle of S in each cell. The area of a grid packing is the area of a minimal bounding box that contains all the rectangles in the grid packing. We present an approximation algorithm that given a set S of rectangles a...

We study a wide range of online covering and packing optimization problems. In an online covering problem a linear cost function is known in advance, but the linear constraints that define the feasible solution space are given one by one in an online fashion. In an online packing problem the profit function as well as the exact packing constraints are not fully known in advance. In each round a...

A set of rectangles S is said to be grid packed if there exists a rectangular grid (not necessarily regular) such that every rectangle lies in the grid and there is at most one rectangle of S in each cell. The area of a grid packing is the area of a minimal bounding box that contains all the rectangles in the grid packing. We present an approximation algorithm that given a set S of rectangles a...

Mader’s disjoint S-paths problem is a common generalization of matching and Menger’s disjoint paths problems. Lovász (1980) suggested a polynomial-time algorithm for this problem through a reduction to matroid matching. A more direct reduction to the linear matroid parity problem was given later by Schrijver (2003), which leads to faster algorithms. As a generalization of Mader’s problem, Chudn...

The three dimensional (3D) packing problem is to arrange given rectangular boxes in a rectangular box of the minimum volume without overlapping each other. As an approach, this paper introduces the system of three sequences of the box labels, the sequence-triple, to encode the topology of the 3Dpacking. The topology is the system of relative relations in pairs of boxes such as right-of, above, ...

In this paper we consider the Online Bin Packing Problem in three variants: Circles in Squares, Circles in Isosceles Right Triangles, and Spheres in Cubes. The two first ones receive an online sequence of circles (items) of different radii while the third one receive an online sequence of spheres (items) of different radii, and they want to pack the items into the minimum number of unit squares...

In this paper, we extend the vehicle routing problem with clustered backhauls (VRPCB) to an integrated routing and three-dimensional loading problem, called VRPCB with 3D loading constraints (3L-VRPCB). In the VRPCB each customer is either a linehaul or a backhaul customer and in each route all linehaul customers must be visited before any backhaul customer. In the 3L-VRPCB, each customer deman...