On the complexity of Wafer-to-Wafer Integration

In this paper we consider the Wafer-to-Wafer Integration problem. A wafer can be seen as a pp-dimensional binary vector. The input of this problem is described by mm multisets (called “lots”), where each multiset contains nn wafers. The output of the problem is a set of nn disjoint stacks, where a stack is a set of mm wafers (one wafer from each lot). To each stack we associate a pp-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 nn stacks. We provide m1−ϵm1−ϵ and p1−ϵp1−ϵ non-approximability results even for n=2n=2, f(n)f(n) non-approximability for any polynomial-time computable function ff, as well as a View the MathML sourcepr-approximation algorithm for any constant rr. Finally, we show that the problem is View the MathML sourceFPT when parameterized by pp, and we use this View the MathML sourceFPT algorithm to improve the running time of the View the MathML sourcepr-approximation algorithm

Multidimensional Binary Vector Assignment problem: standard, structural and above guarantee parameterizations

In this article we focus on the parameterized complexity of the Multidimensional Binary Vector Assignment problem (called \BVA). An input of this problem is defined by $m$ disjoint sets $V^1, V^2, \dots, V^m$, each composed of $n$ binary vectors of size $p$. An output is a set of $n$ disjoint $m$-tuples of vectors, where each $m$-tuple is obtained by picking one vector from each set $V^i$. To each $m$-tuple we associate a $p$ dimensional vector by applying the bit-wise AND operation on the $m$ vectors of the tuple. The objective is to minimize the total number of zeros in these $n$ vectors. mBVA can be seen as a variant of multidimensional matching where hyperedges are implicitly locally encoded via labels attached to vertices, but was originally introduced in the context of integrated circuit manufacturing. We provide for this problem FPT algorithms and negative results ($ETH$-based results, $W$[2]-hardness and a kernel lower bound) according to several parameters: the standard parameter $k$ i.e. the total number of zeros), as well as two parameters above some guaranteed values.Comment: 16 pages, 6 figure