Complexity of the minimum-weight integral cycle basis problem

Determine the computational complexity of the minimum-weight integral cycle basis problem for general graphs, both in the forward and non-forward settings.

Background

The paper studies minimum-weight cycle bases under several structural restrictions, including forward, weakly fundamental, and fundamental bases. It establishes APX-hardness for the minimum-weight forward weakly fundamental cycle basis problem but does not resolve the complexity of the minimum-weight integral cycle basis problem. The authors note that the problem is polynomial-time solvable on planar graphs, while its status for general graphs remains unresolved.

References

While existence and complexity results are well established for many classes of cycle bases , an exception is the computational complexity of the minimum-weight integral cycle basis problem, which remains open.

— Characterizations and Complexity of Minimum Forward and Integer Cycle Bases  (2609.02317 - Riccardi et al., 2 Sep 2026) in Section 1, Introduction; Section 6, Conclusion and future directions