# On random multi-dimensional assignment problems

## TL;DR

This paper investigates the structure and approximation algorithms for random multi-dimensional assignment problems, especially focusing on three dimensions with costs decomposed into sums of independent exponential variables, showing a simple greedy algorithm achieves a near-optimal approximation.

## Contribution

It introduces a new model for 3D assignment problems with costs as sums of independent exponentials and proves a simple greedy algorithm is a (3+o(1))-approximation, improving known bounds.

## Key findings

- Greedy algorithm achieves (3+o(1))-approximation in 3D case.
- Cost structure as sums of independent exponentials impacts approximation quality.
- Contrast with previous results where costs are fully independent exponential variables.

## Abstract

We study random multidimensional assignment problems where the costs decompose into the sum of independent random variables. In particular, in three dimensions, we assume that the costs $W_{i,j,k}$ satisfy $W_{i,j,k}=a_{i,j}+b_{i,k}+c_{j,k}$ where the $a_{i,j},b_{i,k},c_{j,k}$ are independent exponential rate 1 random variables. Our objective is to minimize the total cost and we show that w.h.p. a simple greedy algorithm is a $(3+o(1))$-approximation. This is in contrast to the case where the $W_{i,j,k}$ are independent exponential rate 1 random variables. Here all that is known is an $n^{o(1)}$-approximation, due to Frieze and Sorkin.