Georgia Institute of Technology College of Sciences

The ($2$-dimensional) assignment problem is to find, in an edge weighted bipartite graph, an assignment (i.e., a perfect matching) of minimum total weight. Efficient algorithms for this problem have been known since the advent of modern algorithmic analysis. Moreover, if the edge weights are i.i.d. Exp(1) random variables and the host graph is complete bipartite, seminal results of Aldous state that the expected weight of the optimal assignment tends to $\zeta(2)$.

We consider high-dimensional versions of the random assignment problem. Here, we are given a cost array $M$, indexed by $[n]^k$, and with i.i.d. Exp(1) entries. The objective is to find a ${0,1}$-matrix A that minimizes $\sum_{x \in [n]^k} A_xM_x$, subject to the constraint that every axis-parallel line in A contains exactly one 1. This is the planar assignment problem, and when $k=2$ is equivalent to the usual random assignment problem. We prove that the expected cost of an optimal assignment is $\Theta(n^{k-2})$. Moreover, we describe a randomized algorithm that finds such an assignment with high probability. The main tool is iterative absorption, as developed by Glock, Kühn, Lo, and Osthus. The results answer questions of Frieze and Sorkin. The algorithmic result is in contrast to the axial assignment problem (in which A contains exactly one 1 in each axis-parallel co-dimension 1 hyperplane). For the latter, the best known bounds (which are due to Frankston, Kahn, Narayanan, and Park) exploit the connection between ``spread'' distributions and optimal assignments. Due to this reliance, no efficient algorithm is known.

Joint work with Ashwin Sah and Mehtaab Sawhney.