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For the vertex cover problem (Given a graph $G$, find a minimum set $S$ of vertices such that $G - S$ is edgeless), there are two famous greedy algorithms. The edge greedy (finding an edge and taking its both ends) has ratio 2, while the maximum-degree greedy (taking a vertex with the maximum degree) has ratio O(log n). How about the following one:

  • Repeat until the graph is edgeless,
  • take a vertex $v$ with the minimum degree, add $N(v)$ into the solution, and $G \leftarrow G - N[v]$.

I noticed some experiments on this algorithm (e.g., https://doi.org/10.1109/ICPCI.2012.6486444). But my interest is on rigorous analyses. Is there any known ratio for it?

Any suggestion of bad examples for this algorithm is also very appreciated.

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Unfortunately, the algorithm can be arbitrarily bad. In the following example, each vertex $u_i$ has $d$ disjoint neighbors, of which only four are drawn. The optimal solution is $\{u_1, \ldots, u_p, w_1, \ldots, w_d\}$, while the greedy algorithm returns its complement. The ratio is $\frac{d p}{p + d} \approx d$ with $p\to 0$.example

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