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.


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

  • $\begingroup$ I am sorry, why do you say $\frac{dp}{p+d} \approx d$ with $p\rightarrow 0$? In my opinion it is $\frac{dp}{p+d} \approx 1$ with $p\rightarrow 0$ ($p\in \aleph$ and $p\geq 1$) and $d\gg p$. $\endgroup$ – Mario Giambarioli Mar 2 at 13:28

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