Approximation Algorithms for P2P Orienteering and Stochastic Vehicle Routing Problem

We consider the P2P orienteering problem on general metrics and present a (2+{\epsilon}) approximation algorithm. In the stochastic P2P orienteering problem we are given a metric and each node has a fixed reward and random size. The goal is to devise a strategy for visiting the nodes so as to maximize the expected value of the reward without violating the budget constraints. We present an approximation algorithm for the non-adaptive variant of the P2P Stochastic orienteering. As an implication of the approximation to the stochastic P2P orienteering problem, we define a stochastic vehicle routing problem with time-windows and present a constant factor approximation solution.

💡 Research Summary

The paper tackles two closely related combinatorial optimization problems—point‑to‑point (P2P) orienteering and a stochastic vehicle routing problem (VRP) with time windows—by developing constant‑factor approximation algorithms that work on general metric spaces.

1. P2P Orienteering on General Metrics
The classic P2P orienteering problem asks for a path from a source s to a destination t whose total length does not exceed a budget B, while maximizing the sum of rewards collected at visited vertices. Existing work achieved a (2 + ε) approximation but relied on special metric properties. The authors present a clean reduction that works for any metric: they split the s‑t path into two directed sub‑problems (forward and backward), apply a Lagrangian‑relaxation based orienteering algorithm to each, and then combine the two solutions. Careful analysis of the overlap shows that the combined route respects the budget and guarantees a (2 + ε) approximation to the optimal reward.

2. Stochastic P2P Orienteering (Non‑Adaptive Variant)
In the stochastic version each vertex i carries a fixed reward r_i and a random size (or service time) s_i drawn from a known distribution. The total consumed budget must not be exceeded in any realization. The paper focuses on non‑adaptive policies—i.e., a fixed ordering of vertices decided before any randomness is revealed. The key technical contribution is a “expected‑size metric transformation”: each vertex is assigned its expected size μ_i = E