We study the DAG sorting problem: a partial order $\mathcal{P}$ on $n$ keys is to be discovered by querying as few edges of an input graph $G=(V=[n],E)$ as possible. The graph $G$ only contains edges between ordered pairs, and $G$ is promised to contain the transitive reduction of the DAG describing $\mathcal{P}$. We present two technical and one conceptual result. We first show that DAG sorting is closely related to the fundamental problem of sorting with priced information. \emph{Our first technical result} shows the existence of an algorithm with a $\widetilde{O}(n^{3/4})$ competitive ratio for the $\{0,1,n,\infty\}$ cost version. Thus the $\Omega(n)$ lower bound for maximum cannot extend to sorting, reopening the question of the existence of a $o(n)$-competitive algorithm for the general version. \emph{As our main conceptual contribution}, we define a notion of instance-optimality for the specific problem of DAG sorting, and also unify the existing landscape of instance-optimal algorithms for other static problems existing in literature. This includes problems like sorting [Estivill-Castro and Woods, ACM Comput. Surv. 1992], convex hull [Afshani, Barbay and Chan, JACM 2017], and adaptive joins [Demaine, L\'{o}pez-Ortiz and Munro, SODA 2000]. Our unified notion of instance-optimality is also related to FPT algorithms and algorithms with predictions. We consider the special case of DAG sorting where the input graph is bipartite. \emph{As our second technical result}, we show that a recent algorithm for bichromatic sorting [Goswami and Jacob, ITCS 2024] gives an algorithm for bipartite DAG sorting which is instance-optimal to a factor $O(\log^{3}n)$. This generalizes the famous nuts-and-bolts problem to the setting where the number of nuts and bolts are different, and there is no promise of a matching between them, and the resulting order might not be total.

翻译：我们研究有向无环图（DAG）排序问题：通过尽可能少地查询输入图 $G=(V=[n],E)$ 中的边，发现 $n$ 个键上的偏序 $\mathcal{P}$。图 $G$ 仅包含有序对之间的边，且保证包含描述 $\mathcal{P}$ 的 DAG 的传递约简。我们提出两项技术性结果和一项概念性结果。首先，我们证明 DAG 排序与带价格信息排序这一基本问题密切相关。**第一项技术性结果**表明，对于 $\{0,1,n,\infty\}$ 成本版本，存在一个竞争比为 $\widetilde{O}(n^{3/4})$ 的算法。因此，最大值的 $\Omega(n)$ 下界无法推广到排序问题，这重新开启了通用版本是否存在 $o(n)$ 竞争比算法的问题。**作为主要的概念贡献**，我们为 DAG 排序这一特定问题定义了实例最优性概念，并统一了文献中现有其他静态问题的实例最优算法框架。这包括排序[Estivill-Castro 和 Woods, ACM Comput. Surv. 1992]、凸包[Afshani, Barbay 和 Chan, JACM 2017]以及自适应连接[Demaine, López-Ortiz 和 Munro, SODA 2000]等问题。我们统一的实例最优性概念还与 FPT 算法和带预测算法相关。我们考虑输入图为二分图的 DAG 排序特例。**作为第二项技术性结果**，我们证明近期提出的双色排序算法[Goswami 和 Jacob, ITCS 2024]可给出因子为 $O(\log^{3}n)$ 的实例最优二分图 DAG 排序算法。这将会将著名的螺母与螺栓问题推广至螺母与螺栓数量不同、两者之间无匹配保证且所得顺序可能非全序的场景。