Multi Purpose Routing: New Perspectives and Approximation Algorithms

TL;DR

This paper introduces the $L_p$ TSP, a generalized vehicle routing problem focusing on minimizing delay using Minkowski norms, and provides new approximation algorithms with theoretical guarantees.

Contribution

It develops efficient combinatorial and LP algorithms for $L_p$ TSP, including approximation ratios for single and multi-vehicle cases, and addresses scenarios with unknown cost functions.

Findings

4.27 and 10.92-approximation algorithms for $L_2$ TSP

An 8-approximation and 1.78 inapproximability for All-Norm TSP

Algorithms applicable to general metric spaces

Abstract

The cost due to delay in services may be intrinsically different for various applications of vehicle routing such as medical emergencies, logistical operations, and ride-sharing. We study a fundamental generalization of the Traveling Salesman Problem, namely $L_p$ TSP, where the objective is to minimize an aggregated measure of the delay in services, quantified by the Minkowski $p$-norm of the delay vector. We present efficient combinatorial and Linear Programming algorithms for approximating $L_p$ TSP on general metrics. We provide several approximation algorithms for the $L_p$ TSP problem, including $4.27$ & $10.92$-approximation algorithms for single & multi vehicle $L_2$ TSP, called the Traveling Firefighter Problem. Among other contributions, we provide an $8$-approximation and a $1.78$ inapproximability for All-Norm TSP problem, addressing scenarios where one does not know the ideal cost function, or is seeking simultaneous approximation with respect to any cost function.