Two-dimensional packing with conflicts * Leah Epstein † Asaf Levin ‡ Rob van Stee § Abstract We study the two-dimensional version of the bin packing problem with conflicts. We are given a set of (two-dimensional) squares V = {1, 2,...,n} with sides s 1 ,s 2 ...,s n ∈ [0, 1] and a conflict graph G =(V,E). We seek to find a partition of the items into independent sets of G, where each independent set can be packed into a unit square bin, such that no two squares packed together in one bin overlap. The goal is to minimize the number of independent sets in the partition. This problem generalizes the square packing problem (in which we have E = ∅) and the graph coloring problem (in which s i =0 for all i =1, 2,...,n). It is well known that coloring problems on general graphs are hard to approximate. Following previous work on the one-dimensional problem, we study the problem on specific graph classes, namely, bipartite graphs and perfect graphs. We design a 2+ε-approximation for bipartite graphs, which is almost best possible (unless P = NP ). For perfect graphs, we design a 3.2744-approximation. Topic: Algorithms and data structures 1 Introduction Two-dimensional packing of squares is a well-known problem, with applications in stock cutting and other fields. In the basic problem, the input consists of a set of (two-dimensional) squares of given sides. The goal is to pack the input into bins, which are unit (two-dimensional) squares. A packed item receives a location in the bin so that no pair of squares have an overlap. The goal is to minimize the number of used bins. However, in computer related applications, items often represent processes. These processes may have conflicts due to efficiency, fault tolerance or security reasons. In such cases, the input set of items is accom- panied with a conflict graph where each item corresponds to a vertex. A pair of items that cannot share a bin are represented by an edge in the conflict graph between the two corresponding vertices. Formally, the problem is defined as follows. We are given a set of (two-dimensional) squares V = {1, 2,...,n} whose sides are denoted by s 1 ,s 2 ...,s n and satisfy s i ∈ [0, 1] for all 1 ≤ i ≤ n. We are also given a conflict graph G =(V,E). A valid output is a partition of the items into independent sets of G, together with a packing of the squares of each set into a unit square bin. The packing of a bin is valid if no two squares that are packed together in this bin overlap. The goal is to find such a packing with a minimum number of independent sets. * An extended abstract version of this paper has appeared in Proceedings of the 16th International Symposium on Fundamentals of Computation Theory (FCT 2007), pages 288-299. † Department of Mathematics, University of Haifa, 31905 Haifa, Israel. lea@math.haifa.ac.il. ‡ Department of Statistics, The Hebrew University, Jerusalem 91905, Israel. levinas@mscc.huji.ac.il. § Department of Computer Science, University of Karlsruhe, D-76128 Karlsruhe, Germany. vanstee@ira.uka.de. Re- search supported by the Alexander von Humboldt Foundation. 1