In this article we study a broad class of integer programming problems in variable dimension. We show that these so-termed n-fold integer programming problems are polynomial time solvable. Our proof involves two heavy ingredients discovered recently: the equivalence of linear optimization and so-called directed augmentation, and the stabilization of certain Graver bases. We discuss several appl...

Cutting and packing problems arise in many fields of applications and theory. When dealing with irregular objects, an important subproblem is the identification of the optimal clustering of two objects. Within this paper we consider the case, where two irregular one-connected objects whose frontier formed by circular and/or line segments and which can be free rotated, should be placed such that...

The G12 project is developing a software environment for stating and solving combinatorial problems by mapping a high-level model of the problem to an efficient combination of solving methods. Model annotations are used to control this process. In this paper we explain the mapping to branch-and-price solving. G12 supports the selection of specialised sub-problem solvers, the aggregation of iden...

In this paper, a novel adaptive all-hexahedral (all-hex) mesh generation algorithm is proposed based on a hybrid octree. This hybrid octree consists of polyhedral cells and each grid point is always shared by eight cells by using a cutting procedure. A dual mesh with all-hex elements is then extracted by analyzing each grid point. Next, sphere or ellipsoid bubble packing is applied to improve t...

The Multiple Knapsack problem (MKP) is a hard combinatorial optimization problem with large application, which embraces many practical problems from different domains, like cargo loading, cutting stock, bin-packing, financial and other management, etc. It also arise as a subproblem in several more complex problems like vehicle routing problem and the algorithms to solve these problems will bene...

Abst rac t . The d-dimensional orthogonal knapsack problem (OKP) has a wide range of practical applications, including packing, cutting, and scheduling. We present a new approach to this problem, using a graphtheoretical characterization of feasible packings. This characterization allows us to deal with classes of packings that share a certain combinatorical structure, instead of single ones. C...

In the laser cutting process some well-known parameters, e.g. laser power and cutting speed, play major roles in the performance of the process. Each parameter or a combination of parameters can affect the material removal volume and cutting volume efficiency. The purpose of this research is to study the effect of power and maximum cutting speed on the material removal rate (MRR) and cutting vo...

We consider the one-dimensional bin packing problem with unit-capacity bins and item sizes chosen according to the discrete uniform distribution Ufj; kg, 1 < j k; where each item size in f1=k; 2=k; : : : ; j=kg has probability 1=j of being chosen. Note that for xed j; k as m ! 1 the discrete distributions Ufmj; mkg approach the continuous distribution U(0; j=k], where the item sizes are chosen ...

We consider the 3-stage two-dimensional bin packing problem, which occurs in real-world problems such as glass cutting. For it, we present a new integer linear programming formulation and a branch and price algorithm. Column generation is performed by applying either a greedy heuristic or an Evolutionary Algorithm (EA). Computational experiments show the benefits of the EA-based approach. The h...

Caprara and Fischetti introduced a class of cutting planes, called {0, 1/2}-cuts, which are valid for arbitrary integer linear programs. They also showed that the associated separation problem is strongly NPhard. We show that separation remains strongly NP-hard, even when all integer variables are binary, even when the integer linear program is a set packing problem, and even when the matrix of...