On the Complexity of Wafer-to-Wafer Integration

Abstract

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 $O(m^{1-\epsilon })$ and $O(p^{1-\epsilon })$ 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 FPT when parameterized by $p$, and we use this FPT algorithm to improve the running time of the $\frac{p}{r}$-approximation algorithm.