An instance of the NP-hard Quadratic Shortest Path Problem (QSPP) is called linearizable iff it is equivalent to an instance of the classic Shortest Path Problem (SPP) on the same input digraph. The linearization problem for the QSPP (LinQSPP) decides whether a given QSPP instance is linearizable and determines the corresponding SPP instance in the positive case. We provide a novel linear time algorithm for the LinQSPP on acyclic digraphs which runs considerably faster than the previously best algorithm. The algorithm is based on a new insight revealing that the linearizability of the QSPP for acyclic digraphs can be seen as a local property. Our approach extends to the more general higher-order shortest path problem.

翻译：NP难的二次最短路径问题（QSPP）的一个实例被称为可线性化的当且仅当它在相同的输入有向图上等价于经典最短路径问题（SPP）的一个实例。QSPP的线性化问题（LinQSPP）判定给定的QSPP实例是否可线性化，并在肯定情况下确定对应的SPP实例。我们针对无环有向图提出了一种新颖的线性时间算法，其运行速度显著快于先前的最优算法。该算法基于一个新的发现：无环有向图中QSPP的可线性化性可被看作一种局部性质。我们的方法可推广至更一般的高阶最短路径问题。