Approximating the multi-level bottleneck assignment problem.

We consider the multi-level bottleneck assignment problem (MBA). This problem is described in the recent book 'Assignment Problems' by Burkard et al. (2009) on pages 188-189. One of the applications described there concerns bus driver scheduling.We view the problem as a special case of a bottleneck m-dimensional multi-index assignment problem. We give approximation algorithms and inapproximability results, depending upon the completeness of the underlying graph. Keywords: bottleneck problem; multidimensional assignment; approximation; computational complexity; efficient algorithm.Bottleneck problem; Multidimensional assignment; Approximation; Computational complexity; Efficient algorithm;

Approximation algorithms for 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 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 total cost of the m-tuples is minimized. The main motivation comes from a yield optimization problem in semi-conductor manufacturing. We consider two classes of polynomial-time heuristics for MVA, namely, hub heuristics and sequential heuristics, and we study their approximation ratio. In particular, we show that when the cost function is monotone and subadditive, hub heuristics, as well as sequential heuristics, have finite approximation ratio for every fixed m. Moreover, we establish better approximation ratios for certain variants of hub heuristics and sequential heuristics when the cost function is monotone and submodular, or when it 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.nrpages: 22status: publishe

On the complexity of separation: the three-index assignment problem

A fundamental step in any cutting plane algorithm is separation: deciding whether a violated inequality exists within a certain class of inequalities. It is customary to express the complexity of a separation algorithm in n, the number of variables. Here, we argue that the input to a separation algorithm can be expressed in jsup(x)j, where sup(x) denotes the vector containing the positive components of x. This input measure allows one to take sparsity into account. We apply this idea to two known classes of valid inequalities for the three-index assignment problem, and we find separation algorithms with a better complexity than the ones known in literature. We also show empirically the performance of our separation algorithms.status: publishe