In a previous work, we proposed a new integer programming formulation for the graph coloring problem which, to a certain extent, avoids symmetry. We studied the facet structure of the 0/1-polytope associated with it. Based on these theoretical results, we present now a Branch-and-Cut algorithm for the graph coloring problem. Our computational experiences compare favorably with the well-known exact graph coloring algorithm DSATUR. Keyword: Graph Coloring; Integer Programming; Branch-and-Cut algorithms 1

The one-dimensional cutting stock problem and the two-dimensional two-staged constrained guillotine cutting (knapsack) problem are considered. They can be formulated by column generation. This model has a very tight continuous relaxation which provides a good bound in an LP-based solution approach. We combine a branching scheme and a cutting plane algorithm using Gomory mixed-integer and strengthened Chvatal-Gomory cuts.

Abstract We report on experiments with turning the branch-price-andcut framework SCIP into a generic branch-price-and-cut solver. That is, given a mixed integer program (MIP), our code performs a Dantzig-Wolfe decomposition according to the user’s specification, and solves the resulting re-formulation via branch-and-price. We take care of the column generation subproblems which are solved as MIPs themselves, branch and cut on the original variables (when this is appropriate), aggregate identical subproblems, etc. The charm of building on a well-maintained framework lies in avoiding to re-implement state-of-the-art MIP solving features like pseudo-cost branching, preprocessing, domain propagation, primal heuristics, cutting plane separation etc. 1

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 identical subproblems, automatic disaggregation when required by search, and the use of specialised branching rules. We demonstrate the benefits of the G12 framework on three examples: a trucking problem, cutting stock, and two-dimensional bin packing.

In many mixed integer programs there is some embedded problem structure which can be exploited, often by a decomposition. When the relaxation in each node of a branch-andbound tree is solved by column generation, one speaks of branch-and-price. Optionally, cutting planes can be added in order to strengthen the relaxation, and this is called branchprice-and-cut. We introduce the common concepts of convexification and discretization to arrive at a Dantzig-Wolfe type reformulation of a mixed integer program. The relation between the original and the extended formulations helps us understand how cutting planes should be formulated and how branching decisions can be taken while keeping the column generation subproblems manageable.

The primary objective in cutting and packing problems is trim loss or material input minimization (in stock cutting) or value maximization (when packing into a knapsack). However, in real-life production we usually have many other objectives (costs) and constraints, for example, the number of different patterns. We propose a new simple model for setup minimization (in fact, an extension of the Gilmore-Gomory model for trim loss minimization) and develop a branch-and-price algorithm on its basis. The algorithm is tested on problems with industrially relevant sizes of up to 150 product types. The behavior is investigated on a broad range of problem classes and significant differences between instances of a class are found. Allowing even 0.2% more material input than the minimum significantly improves the results, this tradeoff has not been investigated in the earlier literature. Comparison to a state-of-the art heuristic KOMBI shows mostly better results; to a previous exact approach of Vanderbeck, slightly worse solutions and much worse LP bound, which is a consequence of the simplicity of the model.