Additional Results and Extensions for the paper "Probabilistic bounds on the $k-$Traveling Salesman Problem and the Traveling Repairman Problem''

TL;DR

This paper extends probabilistic bounds for the $k$-TSP and TRP to more general distributions and introduces a fairness-based utility measure with corresponding bounds, enhancing the theoretical understanding of these problems.

Contribution

It provides new probabilistic bounds for the $k$-TSP with general density distributions and introduces a fairness-based utility for the TRP with derived bounds.

Findings

Extended bounds to general density distributions for $k$-TSP.

Derived constant-factor bounds for a fairness-based utility in TRP.

Enhanced theoretical framework for probabilistic analysis of TSP variants.

Abstract

This technical report provides additional results for the main paper ``Probabilistic bounds on the $k-$Traveling Salesman Problem ($k-$TSP) and the Traveling Repairman Problem (TRP)''. For the $k-$TSP, we extend the probabilistic bounds derived in the main paper to the case of distributions with general densities. For the TRP, we propose a utility-based notion of fairness and derive constant-factor probabilistic bounds for this objective, thus extending the TRP bounds from the main paper to non-linear utilities.