On the complexity of Wafer-to-Wafer Integration

In this paper we consider the Wafer-to-Wafer Integration problem. A wafer is a $$p$$-dimensional binary vector. The input of this problem is described by $$m$$ disjoints sets called \"lots\", where each set contains $$n$$ wafers. The output of the problem is a set of $$n$$ disjoint stacks, where a stack is a set of $$m$$ wafers one wafer from each lot. To each stack we associate a $$p$$-dimensional binary vector corresponding to the bit-wise AND operation of the wafers of the stack. The objective is to maximize the total number of \"1\" in the $$n$$ stacks. We provide $$Om^{1-\\epsilon }$$ and $$Op^{1-\\epsilon }$$ non-approximability results even for $$n= 2$$, as well as a $$\\frac{p}{r}$$-approximation algorithm for any constant $$r$$. Finally, we show that the problem is FPTï¾źwhen parameterized by $$p$$, and we use this FPTï¾źalgorithm to improve the running time of the $$\\frac{p}{r}$$-approximation algorithm.