Aziz Moukrim

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. n nThe 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.

A new graph-theoretical model for k-dimensional guillotine-cutting problems

We consider the problem of determining if a given set of rectangular items can be cut in a large rectangle, using guillotine cuts only. We introduce a new class of arc-colored and oriented graphs, named guillotine graphs, which model guillotine patterns. Then we show that an uncolored and non-oriented multigraph is sufficient to obtain any guillotine pattern. We propose linear algorithms for recognizing these graphs, and computing the corresponding patterns. Finally we explain how the model can be used in a constraint programming approach.