Efficient algorithms for three-dimensional axial and planar random assignment problems

TL;DR

This paper introduces efficient algorithms for three-dimensional axial and planar random assignment problems, providing near-optimal solutions with high probability despite the problems' NP-hardness.

Contribution

It presents the first linear-time algorithm for 3D axial assignment and a new matching-based algorithm for 3D planar assignment, achieving near-optimal costs.

Findings

Axial assignment cost scales as (1/n) with high probability.

Planar assignment cost is (n   n) with high probability.

Algorithms are efficient and work with high probability for large instances.

Abstract

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 three dimensions and above, the problem has natural "Axial" and "Planar" versions, both of which are NP-hard. For 3-dimensional Axial random assignment instances of size $n$, the cost scales as $\Omega(1/n)$, and a main result of the present paper is a linear-time algorithm that, with high probability, finds a solution of cost $O(n^{-1+o(1)})$. For 3-dimensional Planar assignment, the lower bound is $\Omega(n)$, and we give a new efficient matching-based algorithm that with high probability returns a solution with cost $O(n \log n)$.