Computational Complexity of the Interval Ordering Problem
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arXiv
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| Natura: | Preprint |
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2026
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| _version_ | 1866917466211352576 |
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| author | Pawlowski, Simeon Froese, Vincent |
| author_facet | Pawlowski, Simeon Froese, Vincent |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_24237 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Computational Complexity of the Interval Ordering Problem Pawlowski, Simeon Froese, Vincent Data Structures and Algorithms 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. |
| title | Computational Complexity of the Interval Ordering Problem |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2604.24237 |