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An $(\alpha,\beta)$-ruling set is a set $S$ such that any two nodes in $S$ are at distance at least $\alpha$ from each other, and, for any node $v \notin S$, there exists a node $u \in S$ such that the distance between $u$ and $v$ is at most $\beta$.

Let $G$ be graph which is nearly regular, with all degrees in $[k,2k]$.

Consider the following creation of the set $S$:

each vertex $v$ chooses a random number $r_v$ uniformly in $[0,1]$. If $r_v$ is local minima, namely $r_v < r_u$ for any neighbor $u \in N(v)$, then $v$ joins the set $S$.

I want to show that with high-probability, it has a set $S$ which is a $(2,O(\log n))$ ruling set.

Proving the $\alpha$ part is easy, because from the creation of $S$, only one vertex in $N^{+}(v)$ can be a local minima, and will be in $S$.

Proving the $\beta$ part is more difficult.

I can prove that $Pr[S \cap N^{+}(v) \ne \emptyset]$ for every $v$ with high probability. But not sure how to continue from here.

Help would be appreciated

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I was about to complain that this statement:

Then, I can prove that $Pr[S \cap N^{+}(v) \ne \emptyset]$ for every $v$ with high probability

is probably wrong, because it would imply a stronger result, namely that nearly regular graphs have a $(2, 1)$ ruling set w.h.p.

But then I realised that, not only does this stronger claim actually hold, an even stronger claim holds, and is almost trivial to show:

Theorem: Every graph has a $(2, 1)$ ruling set.

Proof sketch: Choose any vertex $v$, add it to $S$, and remove it and its neighbours (which have distance 1 from $v$). Repeat until no vertices remain. Every pair of vertices in $S$ have distance $\ge 2$ since the only vertices with distance 1 are neighbours, and we removed all of those.

This claim is stronger than the original in three ways: It applies to every graph (not just nearly regular graphs), always (not just w.h.p.), and $1 < \log n$ except on a finite number of graphs.

This makes me wonder if you missed a condition.

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  • $\begingroup$ First of all, thanks for the answer. Moreover, you were right, I missed a condition, the question asks about a specific set $S$ to be a ruling set. $\endgroup$
    – Gabi G
    Commented Jul 13, 2023 at 18:52

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