Constant-factor approximation for greedy square-sum minimization

Determine whether the minimum-degree greedy algorithm for minimizing \(\sum_{v\in V}\cev d(v)^2\) over vertex orderings of simple graphs admits a constant-factor approximation guarantee.

Background

For the special objective $\varphi(z)=z^2$, the paper studies the greedy algorithm that repeatedly removes a minimum-degree vertex and places it last in the ordering. The authors prove approximation guarantees depending on the degeneracy and on the harmonic number.

They exhibit graph families whose ratio approaches $9/7$, but do not establish a constant bound that holds for all simple graphs.