The Multi-level Bottleneck Assignment Problem: Complexity and Solution Methods

TL;DR

This paper investigates the complexity of the multi-level bottleneck assignment problem, introduces new solution methods including integer programming and heuristics, and demonstrates improved performance over existing greedy algorithms.

Contribution

It provides a complexity analysis under restrictions, introduces novel solution approaches, and empirically outperforms standard greedy methods.

Findings

New solution methods outperform greedy by around 10% on random instances.

Complexity bounds established via reduction from three-dimensional matching.

Extended greedy, integer programming, and column generation heuristics developed.

Abstract

We study the multi-level bottleneck assignment problem (MBA), which has important applications in scheduling and quantitative finance. Given a weight matrix, the task is to rearrange entries in each column such that the maximum sum of values in each row is as small as possible. We analyze the complexity of this problem in a generalized setting, where there are restrictions in how values in columns can be permuted. We present a lower bound on its approximability by giving a non-trivial gap reduction from three-dimensional matching to MBA. To solve MBA, a greedy method has been used in the literature. We present new solution methods based on an extension of the greedy method, an integer programming formulation, and a column generation heuristic. In computational experiments we show that it is possible to outperform the standard greedy approach by around 10% on random instances.