Two-dimensional packing with conflicts

We study the two-dimensional version of the bin packing problem with conflicts. We are given a set of (two-dimensional) squares $V=\{ 1,2, \ldots ,n\}$ with sides $s_1,s_2 \ldots ,s_n \in [0,1]$ 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 $E=\emptyset$) and the graph coloring problem (in which $s_i=0$ for all $i=1,2, \ldots ,n$). 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 $2+\eps$-approximation for bipartite graphs, which is almost best possible (unless ${\mathit P=NP}$). For perfect graphs, we design a 3.2744-approximation