Dependence of 2f-competitive lightness on the fault tolerance

Determine whether 2f-competitive lightness for edge fault-tolerant graph spanners increases or decreases as the fault tolerance parameter f grows, thereby closing the gap between the paper’s upper and lower bounds.

Background

The paper establishes an upper bound proportional to f{1/2} times the corresponding non-fault-tolerant lightness bound and a lower bound with an opposing f-dependence. Consequently, the authors cannot determine whether increasing the number of tolerated edge faults makes the optimal 2f-competitive lightness larger or smaller. Resolving the direction of this dependence would close an important gap in the quantitative theory of light edge fault-tolerant spanners.

References

Note that it remains open whether $2f$-competitive lightness should correlate positively or negatively with $f$. This is an important gap, which we hope can be closed by future work.

— Light Edge Fault Tolerant Graph Spanners  (2502.10890 - Bodwin et al., 15 Feb 2025) in Section 1, subsection “The Bicriteria Price of Fault Tolerance”