Tight Relaxation Gap for Independent Set

Determine whether n/2 is a tight upper bound on the ratio α₁(Γ(G))/α(G) over combinatorial graphs G and their corresponding continuous graphs Γ(G).

Background

The paper compares the continuous 1-independence number α₁(Γ(G)) with the classical independence number α(G). For complete graphs with an even number n of vertices, the authors construct a 1-independent set of size at least n/2 while α(K_n)=1, yielding an unbounded-in-n candidate gap. They do not establish whether this construction is optimal.

References

It is unclear whether or not $n/2$ is a tight upper bound for the ratio $alpha_1(Gamma(G))/alpha(G)$. We leave this as an open question of this section.

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