Relaxation Gap for Minimum Vertex Cover

Determine the maximum ratio β(G)/β₁(Γ(G)) over all combinatorial graphs G and their corresponding continuous graphs Γ(G), and prove or refute the conjecture that this ratio is at most 2.

Background

The continuous minimum 1-cover problem is polynomially solvable, whereas classical minimum vertex cover is computationally hard. The paper gives complete graphs as examples approaching a ratio of 2 between the classical and continuous parameters and conjectures that 2 is the universal upper bound. The example is intended to show tightness if the conjecture holds.

References

Similarly to the independent set problem above, the following natural open question arises: What is the highest ratio between $\beta(G)$ and $\beta_1(\Gamma(G))$ over all $G$? Following intuition from the matching theory and known results for the minimum vertex cover problem, we conjecture the ratio is at most 2.

— Open Problems in Continuous Graphs  (2501.14554 - Grigoriev et al., 24 Jan 2025) in Section 3, subsection “Vertex cover,” paragraph “Relaxation gap for $r=1$”