Approximation Algorithms for the Wafer to Wafer Integration Problem

Dokka, Trivikram; Bougeret, Marin; Boudet, Vincent; Giroudeau, Rodolphe; Spieksma, Frits

doi:10.1007/978-3-642-38016-7_23

Cited by 8 publications

(29 citation statements)

References 9 publications

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“…Furthermore, we also noticed in [3] that the natural ILP formulation implied that min 0 and max 1 are polynomial for fixed p. Concerning negative results, the implicit straightforward reduction from k-Dimensional Matching in [3] and made explicit in Appendix A, shows that min 0 is NP-hard, and max 1 is O(…”

Section: Related Workmentioning

confidence: 88%

“…In [3] and [4] we investigated the min 0 problem by providing a 4 3 -approximation algorithm for m = 3 and several f (m)-approximation algorithms for arbitrary m (and for a more general profit function c). Furthermore, we also noticed in [3] that the natural ILP formulation implied that min 0 and max 1 are polynomial for fixed p. Concerning negative results, the implicit straightforward reduction from k-Dimensional Matching in [3] and made explicit in Appendix A, shows that min 0 is NP-hard, and max 1 is O(…”

Section: Related Workmentioning

confidence: 99%

“…The reduction from k-Dimensional Matching (kDM) provided in [3] can be adapted to max 1 instead of min 0 as shown in Appendix A Unlike the case of min 0, the reduction preserves approximability:…”

Section: Hardness Of Max =0 and Max Maxmentioning

confidence: 99%

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On the complexity of Wafer-to-Wafer Integration

Bougeret

Boudet

Dokka

et al. 2016

Discrete Optimization

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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− ) and O(p 1− ) non-approximability results even for n = 2, as well as a 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 p r -approximation algorithm.

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Section: Related Workmentioning

confidence: 88%

Section: Related Workmentioning

confidence: 99%

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On the complexity of Wafer-to-Wafer Integration

Bougeret

Boudet

Dokka

et al. 2016

Discrete Optimization

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“…In [5], authors focus on a generalization of min 0, called Multi Dimensional Vector Assignment, where vectors are not necessary binary vectors. They extend the approximation algorithm of [4] to get a f (m)-approximation algorithm for arbitrary m. They also prove the APXcompleteness of the (min 0) #0≤2 for m = 3. This result was the only known inapproximability result for min 0.…”

Section: Related Workmentioning

confidence: 95%

Approximability and Exact Resolution of the Multidimensional Binary Vector Assignment Problem

Bougeret

Duvillié

Giroudeau

2016

Lecture Notes in Computer Science

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“…The composition of sum and max operators in the cost function (1.1) is superficially reminiscent of max-algebra formulations of assignment problems, such as those discussed in [2] or [4]. To the best of our knowledge, however, the approximability of MVA has only been previously investigated by Dokka et al [6], who mostly focused on the case m = 3 with additive cost functions. The present paper extends to MVA-m, m ≥ 3, and considerably strengthens the results presented in [6].…”

Section: Wafer-to-wafer Integration and Related Workmentioning

confidence: 99%

Multi-dimensional vector assignment problems

Dokka

Crama

Spieksma

2014

Discrete Optimization

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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 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.

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Approximation Algorithms for the Wafer to Wafer Integration Problem

Cited by 8 publications

References 9 publications

On the complexity of Wafer-to-Wafer Integration

On the complexity of Wafer-to-Wafer Integration

Approximability and Exact Resolution of the Multidimensional Binary Vector Assignment Problem

Multi-dimensional vector assignment problems

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