We study the problem of sorting under incomplete information, when queries are used to resolve uncertainties. Each of n data items has an unknown value, which is known to lie in a given interval. We can pay a query cost to learn the actual value, and we may allow an error threshold in the sorting. The goal is to find a nearly-sorted permutation by performing a minimum-cost set of queries.

We show that an offline optimum query set can be found in polynomial time, and that both oblivious and adaptive problems have simple query-competitive algorithms. The query-competitiveness for the oblivious problem is n for uniform query costs, and unbounded for arbitrary costs; for the adaptive problem, the ratio is 2.

We then present a unified adaptive strategy for uniform query costs that yields the following improved results: (i) a 3/2-query-competitive randomized algorithm; (ii) a 5/3-query-competitive deterministic algorithm if the dependency graph has no 2-components after some preprocessing, which has query-competitive ratio $3/2+\mathrm{O}(1/k)$ if the components obtained have size at least k; and (iii) an exact algorithm if the intervals constitute a laminar family. The first two results have matching lower bounds, and we have a lower bound of 7/5 for large components.

We also give a randomized adaptive algorithm with query-competitive factor $1+\frac{4}{3\sqrt{3}}\approx 1.7698$ for arbitrary query costs, and we show that the 2-query competitive deterministic adaptive algorithm can be generalized for queries returning intervals and for a more general graph problem (which is also a generalization of the vertex cover problem), by using the local ratio technique. Furthermore, we prove that the advice complexity of the adaptive problem is $\lfloor n/2\rfloor$ if no error threshold is allowed, and $\lceil n/3\cdot \lg 3\rceil$ for the general case.

Finally, we present some graph-theoretical results regarding co-threshold tolerance graphs, and we discuss uncertainty variants of some classical interval problems.

当查询用于解决不确定性时，我们研究了在不完整信息下进行排序的问题。n个数据项中的每个数据项都有一个未知值，已知该值位于给定间隔中。我们可以支付查询费用来学习实际值，并且可以在排序中允许错误阈值。目标是通过执行一组成本最低的查询来找到几乎分类的排列。

我们表明，可以在多项式时间内找到离线最佳查询集，并且遗忘和自适应问题都具有简单的查询竞争算法。遗忘问题的查询竞争力为n表示统一查询成本，无限制为任意成本；对于自适应问题，比率为2。

然后，我们针对统一查询成本提出了一种统一的自适应策略，该策略可产生以下改进结果：（i）3/2查询竞争竞争性随机算法；（ii）如果依赖图经过一些预处理后没有2分量，则该5/3查询竞争性确定性算法具有查询竞争率$3/\mathrm{2个}+\mathrm{Ø}（\mathrm{1个}/ķ）$如果获得的分量的大小至少为k ; （iii）如果间隔构成一个层流族，则采用精确算法。前两个结果具有匹配的下界，对于大型组件，我们的下界为7/5。

我们还给出了具有查询竞争因子的随机自适应算法 $\mathrm{1个}+\frac{4}{3\sqrt{3}}\approx 1.7698$对于任意查询成本，我们证明了使用局部比率技术，可以将2-查询竞争确定性自适应算法推广到查询返回间隔和更一般的图问题（这也是顶点覆盖问题的推广）。 。此外，我们证明了自适应问题的建议复杂度为$\lfloor ñ/\mathrm{2个}\rfloor$ 如果不允许错误阈值，并且 $\lceil ñ/3\cdot \lg 3\rceil$ 对于一般情况。

最后，我们提出了一些关于共阈值公差图的图论结果，并讨论了一些经典区间问题的不确定性变体。