Computational Complexity of the Min-Prod Disjoint Paths Problem

Determine the computational complexity of the min-prod disjoint paths problem: given an undirected graph G with source vertex s, terminal vertex t, and constant C ≥ 0, find two internally vertex-disjoint s–t paths P1 and P2 that minimize the objective (|P1| + C)(|P2| + C); in particular, establish whether this problem is NP-hard.

Background

The paper introduces a latency-resilient Layer 3 routing problem and relates it to a novel disjoint-paths objective called the min-prod disjoint paths problem, defined by 2(P1, P2) = (|P1| + C)(|P2| + C) for a constant C ≥ 0. This multiplicative objective couples latency and resiliency and differs from well-studied variants such as min-sum, min-max, and min-min.

The authors note that, unlike standard objectives, the product of path lengths complicates reductions from known NP-hard disjoint-path problems. Although they suspect NP-hardness, they emphasize that the computational complexity of min-prod has not been established in the literature.

The appendix shows that if min-prod is NP-hard, then the latency-resilient Layer 3 routing problem considered in the paper is also NP-hard, underscoring the significance of resolving the complexity of min-prod.