Computational Complexity of the Interval Ordering Problem

TL;DR

This paper investigates the computational complexity of an interval ordering problem motivated by bioinformatics, providing algorithms for certain cost functions and establishing hardness results for others.

Contribution

It introduces a dynamic programming approach for the problem, solves it efficiently for specific cost functions, and proves NP-hardness and lower bounds for general cases.

Findings

Polynomial-time algorithms for subadditive or superadditive cost functions.

Solution for the specific exponential cost function $f(x)=2^x$.

NP-hardness established for some simple cost functions.

Abstract

We study an interval ordering problem introduced by D\"urr 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.