We study an interval ordering problem introduced by Dürr et al. [Discrete Appl. Math. 2012] which is motivated by applications in bioinformatics. The task is to order a given set of n intervals with the goal of minimizing a certain objective which is defined via a given cost function $f$ which assigns a cost to the exposed part of each interval (that is, the pieces not covered by previous intervals). We develop a dynamic programming approach which solves the problem with $O(2^n\text{poly}(n))$ oracle calls to $f$ and arithmetic operations. Moreover, our approach yields polynomial-time algorithms for all cost functions $f$ such that $f-f(0)$ is subadditive or superadditive. This answers an open question for the function $f(x)=2^x$. We contrast these results by proving a running time lower bound of $2^{n-1}$ for any algorithm that solves the problem for every function $f$ (with oracle access only) and further proving NP-hardness for some classes of simple functions. Thus, we significantly narrow the gap regarding the computational complexity of the problem.

翻译：我们研究了由Dürr等人[Discrete Appl. Math. 2012]提出的一个区间排序问题，该问题源于生物信息学中的实际应用。任务是对给定的n个区间进行排序，以最小化由给定代价函数$f$定义的特定目标值，该函数为每个区间的暴露部分（即未被先前区间覆盖的片段）分配代价。我们提出了一种动态规划方法，通过$O(2^n\text{poly}(n))$次对$f$的预言机调用和算术运算来求解该问题。此外，我们的方法对所有满足$f-f(0)$为次可加或超可加的代价函数$f$均能给出多项式时间算法。这回答了针对函数$f(x)=2^x$的一个开放性问题。我们通过证明对于任意函数$f$（仅通过预言机访问）求解该问题的任何算法都需要$2^{n-1}$的运行时间下界，并进一步证明某些简单函数类别的NP难度，从而与这些结果形成了对比。因此，我们显著缩小了该问题计算复杂度的认知差距。