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Let $G=(V,E)$ be a graph. For a given $\rho \leq |V|$ and $\epsilon$ with $(0<\epsilon<1)$, is there any sublinear query algorithm known/possible to decide if the graph has a clique of size $\rho$, or all the cliques are of size at most $\rho(1-\epsilon)$.

More precisely, for a given probability tolerance $\delta$ and error tolerance $\epsilon$, algorithm queries $f(\frac{1}{\epsilon}, \frac{1}{\delta}).o(n^2)$ many position of the corresponding adjacency matrix of the graph, and accept with probability at least $1-\delta$ if the graph has a clique of size $\rho$, or reject probability at least $1-\delta$ if all the cliques are of size at most $\rho(1-\epsilon).$

Thanks in advance.

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  • $\begingroup$ Did you mean sub exponential? $\endgroup$ – R B Apr 19 '15 at 5:06
  • $\begingroup$ Sorry for the confusion. I meant that algorithm looks only at a small portion of the graph, and could decide with high probability that whether the graph has a clique large size, or all cliques are very small. $\endgroup$ – Ram Apr 19 '15 at 5:16
  • $\begingroup$ This seems highly unlikely. Currently, we have no $n^{o(k)}$ algorithm for deciding $k$-clique, and as this problem $W[1]$-hard w.r.t. $k$, there is no much hope for it either. $\endgroup$ – R B Apr 19 '15 at 9:54
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    $\begingroup$ So I assume you allow exponential time, as long as the query complexity is sublinear, right? $\endgroup$ – Igor Shinkar Apr 20 '15 at 14:31
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    $\begingroup$ @IgorShinkar Yes, I allow exponential running time as long as my query complexity is sublinear. $\endgroup$ – Ram Apr 21 '15 at 3:23
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It's not possible. The reason is that, to distinguish an $n$-vertex graph with no edges (clique size 1) from a graph with a single randomly-chosen edge (clique size 2) requires $\Theta(n^2)$ queries. So it's not possible to get sublinear queries for $\rho=2$ and $\epsilon<\frac12$, but your question asks for a parameterization that remains sublinear for all $\rho$ and all $\epsilon>0$.

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