A linear time algorithm for linearizing quadratic and higher-order shortest path problems

TL;DR

This paper introduces a new linear time algorithm for determining linearizability of quadratic shortest path problems on acyclic graphs, significantly improving efficiency over previous methods and extending to higher-order problems.

Contribution

It presents a novel linear time algorithm for the linearization problem of quadratic shortest path problems on acyclic digraphs, based on a local property insight, and extends to higher-order problems.

Findings

The algorithm runs in linear time on acyclic digraphs.

Linearizability can be characterized as a local property.

Extension of the approach to higher-order shortest path problems.

Abstract

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.