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 $\epsilon > 0$ and any $p>0$, there is a family of diversities which cannot be embedded into $\ell_1$ in polynomial time with distortion smaller than $O(n^{1-\epsilon})$ 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.

翻译：稀疏割问题通过低失真度量嵌入对凸松弛解进行取整，已获得良好近似。近期，Bryant和Tupper表明该方法可拓展至超图场景，其关键在于通过线性规划构造所谓的“多样性”解，并借助$\ell_1$多样性嵌入进行取整。多样性是度量空间的推广形式，其非负函数定义于所有子集而非仅元素对。我们证明：当供应或需求由图给出时，该方法可实现多项式时间$O(\log{n})$-近似。相较于Plotkin等人针对供应为图、需求为超图场景提出的$O(\log{(kn)}\log{n})$-近似（其中$k$为需求数量），本结果有所改进。此外，针对供应与需求均由超边基数分别受限于$r_G$与$r_H$的超图给出的情形，我们提出多项式时间$O(\min{\{r_G,r_H\}}\log{r_H}\log{n})$-近似算法。为建立上述结果，我们为被称为“直径多样性”的多样性类构造了$O(\log{n})$-失真的$\ell_1$嵌入，较Bryant和Tupper的$O(\log^2{n})$-失真嵌入更优。任意多样性嵌入$\ell_1$的最小已知失真为$O(n)$。我们证明：对于任意$\epsilon > 0$及$p>0$，存在一族多样性，在仅查询基数不超过$O(\log^p{n})$的子集的情况下，无法在多项式时间内以小于$O(n^{1-\epsilon})$的失真嵌入$\ell_1$（除非$P=NP$）。这否定了Bryant和Tupper关于基于多样性诱导度量可实现$O(\sqrt{n})$-失真$\ell_1$嵌入的猜想（算法精细化版本）。