Average case performance of heuristics for multi-dimensional assignment problems

Beautiful formulas are known for the expected cost of random two-dimensional assignment problems, but in higher dimensions, even the scaling is not known. In 3 dimensions and above, the problem has natural \planar" and \axial" versions, both of which are NP-hard. For 3dimensional Planar random assignment instances of size n, the cost scales as (1 =n), and a main result of the present paper is the rst polynomial-time algorithm that, with high probability, nds a solution of cost O(n 1+" ), for arbitrary positive " (or indeed " going slowly to 0). For 3-dimensional Axial assignment, the lower bound is ( n), and we give a new ecient matchingbased algorithm that returns a solution with expected cost O(n logn).

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