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

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.