We consider the problem of sampling  an arbitrary given motif H in a graph G, where access to G is given via queries: degree, neighbor, and pair, as well as uniform edge sample queries. Previous algorithms for the uniform sampling task were based on a decomposition of H into a collection of odd cycles and stars, denoted D^*(H)=\{O_{k_1}, \ldots, O_{k_q}, S_{p_1}, \ldots, S_{p_\ell}\}. These algorithms were shown to be optimal for the case where H is a clique or an odd-length cycle, but no other lower bounds were known. We present a new algorithm for sampling  arbitrary motifs which, up to %DIFDELCMD < \poly%%%  (\log n) factors, for any motif H whose decomposition contains at least two components or at least one star, is always preferable . The main ingredient leading to this improvement is an improved uniform algorithm for sampling stars, which might be of independent interest, as it allows to sample vertices according to the p-th moment of the degree distribution. We further show how to use our sampling algorithm to get an approximate counting algorithm, with essentially the same complexity. Finally, we prove that this algorithm is decomposition-optimal for decompositions that contain at least one odd cycle . That is, we prove that for any decomposition D that contains at least one odd cycle, there exists a motif H_{D . These are the first lower bounds for motifs H with a nontrivial decomposition, i.e., motifs that have more than a single component in their decomposition.
We consider the problem of sampling and approximately counting an arbitrary given motif H in a graph G, where access to G is given via queries: degree, neighbor, and pair, as well as uniform edge sample queries. Previous algorithms for these tasks were based on a decomposition of H into a collection of odd cycles and stars, denoted D^*(H)=\{O_{k_1}, \ldots, O_{k_q}, S_{p_1}, \ldots, S_{p_\ell}\}. These algorithms were shown to be optimal for the case where H is a clique or an odd-length cycle, but no other lower bounds were known. We present a new algorithm for sampling and approximately counting arbitrary motifs which, up to %DIFDELCMD < \poly%%% poly (\log n) factors, is always at least as good as previous results, and for most graphs G is strictly better . The main ingredient leading to this improvement is an improved uniform algorithm for sampling stars, which might be of independent interest, as it allows to sample vertices according to the p-th moment of the degree distribution. Finally, we prove that this algorithm is decomposition-optimal for decompositions that contain at least one odd cycle . These are the first lower bounds for motifs H with a nontrivial decomposition, i.e., motifs that have more than a single component in their decomposition.
A
and approximately counting
R
the uniform sampling task
these tasks
A
and approximately counting
A
poly
R
for any motif H whose decomposition contains at least two components or at least one star, is always preferable
is always at least as good as previous results, and for most graphs G is strictly better
D
We further show how to use our sampling algorithm to get an approximate counting algorithm, with essentially the same complexity.
D
. That is, we prove that for any decomposition D that contains at least one odd cycle, there exists a motif H_{D