Improve the constants in the coarse-equivalence inequalities

Improve the universal constants in the inequalities relating the path-length, path-breadth, strong path-breadth, additive caterpillar distortion, power-AT index, power-cocomparability index, dominating-pair radius, dominating-shortest-path radius, McCarty index, minor fatness index, and the BFS-layering parameters.

Background

The paper establishes that numerous graph parameters are coarsely equivalent to path-length and derives explicit constant-factor inequalities between them. The authors leave unresolved whether these constants can be reduced, which could sharpen approximation guarantees and yield tighter structural characterizations. The concluding question refers collectively to the inequalities proved throughout the paper.