Multi-dimensional vector assignment problems

Dokka, Trivikram and Crama, Yves and Spieksma, Frits C. R. (2014) Multi-dimensional vector assignment problems. Discrete Optimization, 14. pp. 111-125. ISSN 1572-5286

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Abstract

We consider a special class of axial multi-dimensional assignment problems called multidimensional 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 semiconductor 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.

Item Type:
Journal Article
Journal or Publication Title:
Discrete Optimization
Uncontrolled Keywords:
/dk/atira/pure/subjectarea/asjc/2600/2604
Subjects:
ID Code:
70984
Deposited By:
Deposited On:
25 Sep 2014 15:45
Refereed?:
Yes
Published?:
Published
Last Modified:
24 Nov 2020 12:37