Exact worst-case ratio of the greedy square-sum algorithm

Determine whether the worst-case approximation ratio of the minimum-degree greedy algorithm for minimizing \(\sum_{v\in V}\cev d(v)^2\) on simple graphs is exactly \(9/7\).

Background

The paper constructs a family of simple graphs consisting of chains of triangles for which the greedy algorithm's approximation ratio tends to $9/7$. Exhaustive experiments on all simple graphs up to 12 vertices found no worse examples within the tested range.

The authors therefore leave unresolved whether this experimentally suggested lower bound is the true worst-case approximation ratio.

References

It remains an open question whether the worst-case approximation ratio of the greedy algorithm for simple graphs is indeed $\frac{9}{7}$.

— Separable convex optimization over indegree polytopes  (2509.06182 - Borsik et al., 7 Sep 2025) in Section 3.2, Greedy approximation algorithm, immediately before Section 4