Special cases of the quadratic shortest path problem

TL;DR

This paper investigates special cases of the quadratic shortest path problem, providing complexity results, polynomial-time solutions for certain matrix types, and algorithms for linearizability on specific graph classes.

Contribution

It offers new proofs, characterizations, and algorithms for solving or simplifying the quadratic shortest path problem in special cases.

Findings

Adjacent QSPP on cyclic digraphs is hard to approximate.

QSPP with symmetric weak sum matrices reduces to shortest path.

QSPP with symmetric product matrices is solvable in polynomial time.

Abstract

The quadratic shortest path problem (QSPP) is \textcolor{black}{the problem of finding a path with prespecified start vertex $s$ and end vertex $t$ in a digraph} such that the sum of weights of arcs and the sum of interaction costs over all pairs of arcs on the path is minimized. We first consider a variant of the QSPP known as the adjacent QSPP. It was recently proven that the adjacent QSPP on cyclic digraphs cannot be approximated unless P=NP. Here, we give a simple proof for the same result. We also show that if the quadratic cost matrix is a symmetric weak sum matrix \textcolor{black}{ and all $s$-$t$ paths have the same length,} then an optimal solution for the QSPP can be obtained by solving the corresponding instance of the shortest path problem. Similarly, it is shown that the QSPP with a symmetric product cost matrix is solvable in polynomial time. Further, we provide sufficient and necessary conditions for a QSPP instance on a complete symmetric digraph with four vertices to be linearizable. We also characterize linearizable QSPP instances on complete symmetric digraphs with more than four vertices. Finally, we derive an algorithm that examines whether a QSPP instance on the directed grid graph $G_{pq}$ ($p,q\geq 2$) is linearizable. The complexity of this algorithm is ${\mathcal{O}(p^{3}q^{2}+p^{2}q^{3})}$.