François Clautiaux

New reduction procedures and lower bounds for the two-dimensional bin packing problem with fixed orientation

The two-dimensional bin-packing problem (2BP) consists of minimizing the number of identical rectangles used to pack a set of smaller rectangles. In this paper, we propose new lower bounds for 2BP in the discrete case. They are based on the total area of the items after application of dual feasible functions (DFF). We also propose the new concept of data-dependent dual feasible functions (DDFF), which can also be applied to a 2BP instance. We propose two families of Discrete DFF and DDFF and show that they lead to bounds which strictly dominate those obtained previously. We also introduce two new reduction procedures and report computational experiments on our lower bounds. Our bounds improve on the previous best results and close 22 additional instances of a well-known established benchmark derived from literature.

A new constraint programming approach for the orthogonal packing problem

The two-dimensional orthogonal packing problem (2OPP) consists in determining if a set of rectangles can be packed in a larger rectangle of fixed size. We propose an exact method for 2OPP, based on a new constraint-based scheduling model. We provide a generalization of energetic reasoning techniques for the problem under investigation. Feasibility tests requiring the solution of subset-sum problems are described. Computational results confirm the efficiency of our method compared to others in the literature.

A survey of dual-feasible and superadditive functions

Dual-feasible functions are valuable tools that can be used to compute both lower bounds for different combinatorial problems and valid inequalities for integer programs. Several families of functions have been used in the literature. Some of them were defined explicitly, and others not. One of the main objectives of this paper is to survey these functions, and to state results concerning their quality. We clearly identify dominant subsets of functions, i.e. those which may lead to better bounds or stronger cuts. We also describe different frameworks that can be used to create dual-feasible functions. With these frameworks, one can get a dominant function based on other ones. Two new families of dual-feasible functions obtained by applying these methods are proposed in this paper.We also performed a computational comparison on the relative strength of the functions presented in this paper for deriving lower bounds for the bin-packing problem and valid cutting planes for the pattern minimization problem. Extensive experiments on instances generated using methods described in the literature are reported. In many cases, the lower bounds are improved, and the linear relaxations are strengthened.

A new exact method for the two-dimensional orthogonal packing problem

Abstract The two-dimensional orthogonal packing problem (2 OPP ) consists in determining if a set of rectangles (items) can be packed into one rectangle of fixed size (bin). In this paper we propose two exact algorithms for solving this problem. The first algorithm is an improvement on a classical branch&bound method, whereas the second algorithm is based on a new relaxation of the problem. We also describe reduction procedures and lower bounds which can be used within enumerative methods. We report computational experiments for randomly generated benchmarks which demonstrate the efficiency of both methods: the second method is competitive compared to the best previous methods. It can be seen that our new relaxation allows an efficient detection of non-feasible instances.

Heuristic and metaheuristic methods for computing graph treewidth

The notion of treewidth is of considerable interest in relation to NP-hard problems. Indeed, several studies have shown that the tree-decomposition method can be used to solve many basic optimization problems in polynomial time when treewidth is bounded, even if, for arbitrary graphs, computing the treewidth is NP-hard. Several papers present heuristics with computational experiments. For many graphs the discrepancy between the heuristic results and the best lower bounds is still very large. The aim of this paper is to propose two new methods for computing the treewidth of graphs: a heuristic and a metaheuristic. The heuristic returns good results in a short computation time, whereas the metaheuristic (a Tabu search method) returns the best results known to have been obtained so far for all the DIMACS vertex coloring / treewidth benchmarks (a well-known collection of graphs used for both vertex coloring and treewidth problems.) Our results actually improve on the previous best results for treewidth problems in 53% of the cases. Moreover, we identify properties of the triangulation process to optimize the computing time of our method.

New lower and upper bounds for graph treewidth

The notion of tree-decomposition has very strong theoretical interest related to NP-Hard problems. Indeed, several studies show that it can be used to solve many basic optimization problems in polynomial time when the treewidth is bounded. So, given an arbitrary graph, its decomposition and its treewidth have to be determined, but computing the treewidth of a graph is NP-Hard. Hence, several papers present heuristics with computational experiments, but for many instances of graphs, the heuristic results are far from the best lower bounds. The aim of this paper is to propose new lower and upper bounds for the treewidth. We tested them on the well known DIMACS benchmark for graph coloring, so we can compare our results to the best bounds of the literature. We improve the best lower bounds dramatically, and our heuristic method computes good bounds within a very small computing time.

New lower bounds for bin packing problems with conflicts

The bin packing problem with conflicts (BPC) consists of minimizing the number of bins used to pack a set of items, where some items cannot be packed together in the same bin due to compatibility restrictions. The concepts of dual-feasible functions (DFF) and data-dependent dual-feasible functions (DDFF) have been used in the literature to improve the resolution of several cutting and packing problems. In this paper, we propose a general framework for deriving new DDFF as well as a new concept of generalized data-dependent dual-feasible functions (GDDFF), a conflict generalization of DDFF. The GDDFF take into account the structure of the conflict graph using the techniques of graph triangulation and tree-decomposition. Then we show how these techniques can be used in order to improve the existing lower bounds.

A new lower bound for the non-oriented two-dimensional bin-packing problem

We propose a new scheme for computing lower bounds for the non-oriented bin-packing problem when the bin is a square. It leads to bounds that theoretically dominate previous results. Computational experiments show that the bounds are tight. We also discuss the case where the bin is not a square.

A new exact method for the two-dimensional bin-packing problem with fixed orientation

We propose a new exact method for the well-known two-dimensional bin-packing problem. It is based on an iterative decomposition of the set of items into two disjoint subsets. We tested the efficiency of our method against benchmarks of the literature. Computational experiments confirm the efficiency of our method.

Tree-decomposition based heuristics for the two-dimensional bin packing problem with conflicts

This paper deals with the two-dimensional bin packing problem with conflicts (BPC-2D). Given a finite set of rectangular items, an unlimited number of rectangular bins and a conflict graph, the goal is to find a conflict-free packing of the items minimizing the number of bins used. In this paper, we propose a new framework based on a tree-decomposition for solving this problem. It proceeds by decomposing a BPC-2D instance into subproblems to be solved independently. Applying this decomposition method is not straightforward, since merging partial solutions is hard. Several heuristic strategies are proposed to make an effective use of the decomposition. Computational experiments show the practical effectiveness of our approach.