Multi-dimensional vector assignment problems

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 mm disjoint sets, each of which contains the same number nn of pp-dimensional vectors with nonnegative integral components, and a cost function defined on vectors. The cost of an mm-tuple of vectors is defined as the cost of their component-wise maximum. The problem is now to partition the mm sets of vectors into nn   mm-tuples so that no two vectors from the same set are in the same mm-tuple and so that the sum of the costs of the mm-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 mm. 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=3m=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 pp.