# Wasserstein Distributionally Robust Shortest Path Problem

## TL;DR

This paper introduces a novel distributionally robust shortest path model that minimizes worst-case travel time using Wasserstein balls, offering improved out-of-sample performance and computational efficiency.

## Contribution

It develops a tractable reformulation of the DRSP model as a mixed 0-1 convex problem and extends it to bi-criteria shortest path and minimum cost flow problems.

## Key findings

- Model achieves better out-of-sample performance.
- Reformulation as a mixed 0-1 convex problem.
- Extensions to bi-criteria and flow problems.

## Abstract

This paper proposes a data-driven distributionally robust shortest path (DRSP) model where the distribution of the travel time in the transportation network can only be partially observed through a finite number of samples. Specifically, we aim to find an optimal path to minimize the worst-case $\alpha$-reliable mean-excess travel time (METT) over a Wasserstein ball, which is centered at the empirical distribution of the sample dataset and the ball radius quantifies the level of its confidence. In sharp contrast to the existing DRSP models, our model is equivalently reformulated as a tractable mixed 0-1 convex problem, e.g., 0-1 linear program or 0-1 second-order cone program. Moreover, we also explicitly derive the distribution achieving the worst-case METT by simply perturbing each sample. Experiments demonstrate the advantages of our DRSP model in terms of the out-of-sample performance and computational complexity. Finally, our DRSP model is easily extended to solve the DR bi-criteria shortest path problem and the minimum cost flow problem.