Close the offline 9/5 vs 3/2 gap for Clos unsplittable flows

Determine whether there exists a polynomial-time algorithm for the offline minimum congestion routing problem for unsplittable flows in Clos networks that achieves worst-case congestion and approximation factor 3/2, thereby closing the gap between the current 9/5-approximation algorithm and the 3/2 lower bound; alternatively, establish stronger lower bounds if a 3/2-approximation is impossible.

Background

The paper presents a polynomial-time algorithm that guarantees congestion at most 9/5 and provides a 9/5-approximation for the minimum congestion routing problem in Clos networks with unsplittable flows. It also proves lower bounds showing that some instances require congestion at least 3/2 and that it is NP-hard to approximate within a factor better than 3/2.

This leaves a quantitative gap between the best-known algorithmic guarantee (9/5) and the hardness/lower bound (3/2). The authors believe the 3/2 lower bound is tight, motivating the problem of either designing a 3/2-approximation algorithm or proving stronger lower bounds if 3/2 is not tight.