Efficient labeling algorithms for adjacent quadratic shortest paths

TL;DR

This paper introduces efficient algorithms, aqD and aqA*, for solving the Adjacent Quadratic Shortest Path Problem, demonstrating significant speed improvements over existing methods through extensive numerical experiments on large-scale graphs.

Contribution

It extends Dijkstra's algorithm to polynomial-time solutions for AQSPP and develops an adjacent quadratic A* algorithm with backward search for faster performance.

Findings

aqA* outperforms all other algorithms, being about 75 times faster than aqD.

The algorithms maintain efficiency despite variations in quadratic costs.

Both algorithms are faster than benchmark methods on random graph instances.

Abstract

In this article, we study the Adjacent Quadratic Shortest Path Problem (AQSPP), which consists in finding the shortest path on a directed graph when its total weight component also includes the impact of consecutive arcs. We provide a formal description of the AQSPP and propose an extension of Dijkstra's algorithm (that we denote aqD) for solving AQSPPs in polynomial-time and provide a proof for its correctness under some mild assumptions. Furthermore, we introduce an adjacent quadratic A* algorithm (that we denote aqA*) with a backward search for cost-to-go estimation to speed up the search. We assess the performance of both algorithms by comparing their relative performance with benchmark algorithms from the scientific literature and carry out a thorough collection of sensitivity analysis of the methods on a set of problem characteristics using randomly generated graphs. Numerical results suggest that: (i) aqA* outperforms all other algorithms, with a performance being about 75 times faster than aqD and the fastest alternative; (ii) the proposed solution procedures do not lose efficiency when the magnitude of quadratic costs vary; (iii) aqA* and aqD are fastest on random graph instances, compared with benchmark algorithms from scientific literature. We conclude the numerical experiments by presenting a stress test of the AQSPP in the context of real grid graph instances, with sizes up to $16 \times 10^6$ nodes, $64 \times 10^6$ arcs, and $10^9$ quadratic arcs.