Average-case performance of heuristics for three-dimensional assignment problems

Details Average-case performance of heuristics for three-dimensional assignment problems Average-case performance of heuristics for three-dimensional assignment problems

2010-08-02

arxiv.org/abs/1008.0390v1

Mathematics

Combinatorics

Scientific paper

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 3-dimensional Planar random assignment instances of size $n$, the cost scales as $\Omega(1/n)$, and a main result of the present paper is the first polynomial-time algorithm that, with high probability, finds a solution of cost $O(n^{-1+\e})$, for arbitrary positive $\e$ (or indeed $\e$ going slowly to~0). For 3-dimensional Axial assignment, the lower bound is $\Omega(n)$, and we give a new efficient matching-based algorithm that returns a solution with expected cost $O(n \log n)$.

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