Multi-dimensional vector assignment problems

Abstract

We consider a special class of axial multi-dimensional assignment problems called multi-dimensional vector assignment (MVA) problems. An instance of the MVA problem is defined by $m$ disjoint sets, each of which contains the same number $n$ of $p$-dimensional vectors with nonnegative integral components, and a cost function defined on vectors. The cost of an $m$-tuple of vectors is defined as the cost of their component-wise maximum. The problem is now to partition the $m$ sets of vectors into $n$   $m$-tuples so that no two vectors from the same set are in the same $m$-tuple and so that the sum of the costs of the $m$-tuples is minimized. The main motivation comes from a yield optimization problem in semi-conductor manufacturing. We consider a particular class of polynomial-time heuristics for MVA, namely the sequential heuristics, and we study their approximation ratio. In particular, we show that when the cost function is monotone and subadditive, sequential heuristics have a finite approximation ratio for every fixed $m$. Moreover, we establish smaller approximation ratios when the cost function is submodular and, for a specific sequential heuristic, when the cost function is additive. We provide examples to illustrate the tightness of our analysis. Furthermore, we show that the MVA problem is APX-hard even for the case $m=3$ and for binary input vectors. Finally, we show that the problem can be solved in polynomial time in the special case of binary vectors with fixed dimension $p$.