We consider a generalized poset sorting problem (GPS), in which we are given a query graph $G = (V, E)$ and an unknown poset $\mathcal{P}(V, \prec)$ that is defined on the same vertex set $V$, and the goal is to make as few queries as possible to edges in $G$ in order to fully recover $\mathcal{P}$, where each query $(u, v)$ returns the relation between $u, v$, i.e., $u \prec v$, $v \prec u$ or $u \not \sim v$. This generalizes both the poset sorting problem [Faigle et al., SICOMP 88] and the generalized sorting problem [Huang et al., FOCS 11]. We give algorithms with $\tilde{O}(n\cdot \mathrm{poly}(k))$ query complexity when $G$ is a complete bipartite graph or $G$ is stochastic under the \ER model, where $k$ is the \emph{width} of the poset, and these generalize [Daskalakis et al., SICOMP 11] which only studies complete graph $G$. Both results are based on a unified framework that reduces the poset sorting to partitioning the vertices with respect to a given pivot element, which may be of independent interest. Our study of GPS also leads to a new $\tilde{O}(n^{1 - 1 / (2W)})$ competitive ratio for the so-called weighted generalized sorting problem where $W$ is the number of distinct weights in the query graph. This problem was considered as an open question in [Charikar et al., JCSS 02], and our result makes important progress as it yields the first nontrivial $\tilde{O}(n)$ ratio for general weighted query graphs (and better ratio if $W$ is bounded). We obtain this via an $\tilde{O}(nk + n^{1.5})$ query complexity algorithm for the case where every edge in $G$ is guaranteed to be comparable in the poset, which generalizes the state-of-the-art $\tilde{O}(n^{1.5})$ bound for generalized sorting [Huang et al., FOCS 11].

翻译：我们考虑广义偏序集排序问题（GPS）：给定查询图$G = (V, E)$和定义于同一顶点集$V$上的未知偏序集$\mathcal{P}(V, \prec)$，目标是通过尽可能少的边查询完全恢复$\mathcal{P}$，其中每次查询$(u, v)$返回$u$与$v$的序关系（即$u \prec v$、$v \prec u$或$u \not \sim v$）。该问题同时推广了偏序集排序问题[Faigle等, SICOMP 88]和广义排序问题[Huang等, FOCS 11]。当$G$为完全二部图或$G$在\ER模型下为随机图时，我们提出查询复杂度为$\tilde{O}(n\cdot \mathrm{poly}(k))$的算法，其中$k$为偏序集的宽度，这推广了仅研究完全图$G$的[Daskalakis等, SICOMP 11]。两种算法均基于统一框架，该框架将偏序集排序简化为关于给定枢轴元素的顶点划分，此方法可能具有独立意义。对GPS的研究还引出了所谓的加权广义排序问题的一个新的$\tilde{O}(n^{1 - 1 / (2W)})$竞争比，其中$W$为查询图中不同权重的数量。该问题在[Charikar等, JCSS 02]中被列为开放问题，我们的结果取得了重要进展：首次为一般加权查询图给出非平凡的$\tilde{O}(n)$比率（当$W$有界时比率更优）。这一结果通过$G$中每条边在偏序集中保证可比的$\tilde{O}(nk + n^{1.5})$查询复杂度算法获得，该算法推广了广义排序问题的最新$\tilde{O}(n^{1.5})$界[Huang等, FOCS 11]。