Theoretical explanation for the C-zero-ratio effect

Develop a theoretical explanation for why the proportion of zero entries in the objective coefficient vector, termed the C-zero-ratio, is informative for predicting when Barrier PDLP is likely to outperform PDLP, or derive an analytical result supporting the observed relationship.

Background

The paper reports that the C-zero-ratio—the proportion of zero entries in the objective coefficient vector—appears empirically to distinguish LP instances on which Barrier PDLP (BPDLP) is more likely to outperform the original PDLP method. Numerical performance profiles suggest that BPDLP’s relative advantage generally decreases as the C-zero-ratio increases.

This observation is presented only as an empirical relationship. The paper explicitly states that it lacks both a theoretical explanation for the informativeness of the C-zero-ratio and an analytical result establishing or supporting the observed performance relationship. Consequently, explaining this phenomenon or proving an analytical connection remains unresolved.