Diversity Embeddings and the Hypergraph Sparsest Cut
Adam D. Jozefiak, F. Bruce Shepherd
Good approximations have been attained for the sparsest cut problem by rounding solutions to convex relaxations via low-distortion metric embeddings. Recently, Bryant and Tupper showed that this approach extends to the hypergraph setting by formulating a linear program whose solutions are so-called diversities which are rounded via diversity embeddings into $\ell_1$. Diversities are a generalization of metric spaces in which the nonnegative function is defined on all subsets as opposed to only on pairs of elements. We show that this approach yields a polytime $O(\log{n})$-approximation when either the supply or demands are given by a graph. This result improves upon Plotkin et al.'s $O(\log{(kn)}\log{n})$-approximation, where $k$ is the number of demands, for the setting where the supply is given by a graph and the demands are given by a hypergraph. Additionally, we provide a polytime $O(\min{\{r_G,r_H\}}\log{r_H}\log{n})$-approximation for when the supply and demands are given by hypergraphs whose hyperedges are bounded in cardinality by $r_G$ and $r_H$ respectively. To establish these results we provide an $O(\log{n})$-distortion $\ell_1$ embedding for the class of diversities known as diameter diversities. This improves upon Bryant and Tupper's $O(\log\^2{n})$-distortion embedding. The smallest known distortion with which an arbitrary diversity can be embedded into $\ell_1$ is $O(n)$. We show that for any $ε> 0$ and any $p>0$, there is a family of diversities which cannot be embedded into $\ell_1$ in polynomial time with distortion smaller than $O(n^{1-ε})$ based on querying the diversities on sets of cardinality at most $O(\log^p{n})$, unless $P=NP$. This disproves (an algorithmic refinement of) Bryant and Tupper's conjecture that there exists an $O(\sqrt{n})$-distortion $\ell_1$ embedding based off a diversity's induced metric.
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