Distributed and Adaptive Algorithms for Vehicle Routing in a Stochastic and Dynamic Environment

In this paper we present distributed and adaptive algorithms for motion coordination of a group of m autonomous vehicles. The vehicles operate in a convex environment with bounded velocity and must service demands whose time of arrival, location and on-site service are stochastic; the objective is to minimize the expected system time (wait plus service) of the demands. The general problem is known as the m-vehicle Dynamic Traveling Repairman Problem (m-DTRP). The best previously known control algorithms rely on centralized a-priori task assignment and are not robust against changes in the environment, e.g. changes in load conditions; therefore, they are of limited applicability in scenarios involving ad-hoc networks of autonomous vehicles operating in a time-varying environment. First, we present a new class of policies for the 1-DTRP problem that: (i) are provably optimal both in light- and heavy-load condition, and (ii) are adaptive, in particular, they are robust against changes in load conditions. Second, we show that partitioning policies, whereby the environment is partitioned among the vehicles and each vehicle follows a certain set of rules in its own region, are optimal in heavy-load conditions. Finally, by combining the new class of algorithms for the 1-DTRP with suitable partitioning policies, we design distributed algorithms for the m-DTRP problem that (i) are spatially distributed, scalable to large networks, and adaptive to network changes, (ii) are within a constant-factor of optimal in heavy-load conditions and stabilize the system in any load condition. Simulation results are presented and discussed.

💡 Research Summary

The paper addresses the m‑Vehicle Dynamic Traveling Repairman Problem (m‑DTRP), a stochastic‑dynamic vehicle routing model where m autonomous agents must continuously service randomly arriving demands in a bounded convex planar region. Demands arrive according to a homogeneous spatio‑temporal Poisson process with intensity λ and spatial density f(x); each demand requires an on‑site service time drawn i.i.d. from a distribution with mean (\bar{s}). The load factor (\rho = \lambda \bar{s} / m) must be less than one for any stabilizing policy to exist.

The authors first focus on the single‑vehicle case (1‑DTRP). They introduce two novel policies:

1. Divide & Conquer (DC) – The environment is partitioned into a grid of r cells. Within each cell the vehicle computes an optimal (or near‑optimal) Traveling Salesman Problem (TSP) tour over the pending requests and services them before moving to the next cell. The design parameter r controls the granularity. As r → ∞ the policy becomes asymptotically optimal in both light‑load (ρ → 0) and heavy‑load (ρ → 1) regimes; for r = 1 the policy is provably optimal in light‑load and within a factor of two of optimal in heavy‑load, while requiring no prior knowledge of λ or f.

2. Receding Horizon (RH) – At each decision epoch the vehicle solves a TSP over all currently waiting demands but only executes a short horizon τ of that tour before replanning. RH is shown to be optimal in light‑load, stabilizing for any ρ < 1, and empirically near‑optimal in heavy‑load.

Both policies rely on well‑known asymptotic properties of Euclidean TSP (β_TSP,2 ≈ 0.712) and on Jensen’s inequality for performance bounds.

The second major contribution extends these ideas to the multi‑vehicle setting. The authors construct an equitable partition of the workspace into m subregions that are simultaneously equitable with respect to the spatial demand density f and the service‑time distribution. Each vehicle is assigned one subregion and runs independently either DC or RH within it. This partitioning yields a partitioning policy that is provably optimal in the heavy‑load limit: the expected system time scales as (\frac{C}{1-\rho}) with a constant C that matches the lower bound derived from the continuous multi‑median problem and TSP asymptotics. Moreover, the policy is distributed (vehicles need only local information and the partition map), scalable (performance improves linearly with the number of vehicles), and adaptive (the partition can be recomputed on‑the‑fly when vehicles join/leave, when the environment changes, or when the demand statistics vary).

Theoretical analysis is supported by extensive simulations. In a square domain, the authors vary λ from low to near‑saturation values and compare DC (r = ∞), DC (r = 1), RH, and a benchmark centralized assignment policy. Results show that the proposed distributed policies achieve the same or better average system time in light‑load, and stay within a factor of two of the optimal lower bound in heavy‑load, while requiring far less communication and computation. Increasing the number of vehicles reduces the average waiting time roughly proportionally to 1/m, confirming the scalability claim.

In summary, the paper delivers a comprehensive framework for stochastic dynamic vehicle routing that simultaneously satisfies three critical practical requirements: (i) distribution – no central controller, only local decisions; (ii) scalability – performance degrades gracefully with load and improves with more agents; (iii) adaptivity – robustness to changes in demand rate, service‑time distribution, vehicle count, and workspace geometry. The Divide & Conquer and Receding Horizon policies for the single‑vehicle case are notable for their simplicity and for requiring no a‑priori knowledge of demand statistics, making them immediately applicable to real‑world autonomous fleet operations such as UAV surveillance, on‑demand delivery robots, and smart‑city service platforms. The rigorous proofs of optimality in both light‑ and heavy‑load regimes, together with the constructive partitioning scheme for multi‑vehicle systems, represent a significant advance over prior centralized, static‑assignment approaches.