Distributed Multi-Depot Routing without Communications

We consider and formulate a class of distributed multi-depot routing problems, where servers are to visit a set of requests, with the aim of minimizing the total distance travelled by all servers. These problems fall into two categories: distributed offline routing problems where all the requests that need to be visited are known from the start; distributed online routing problems where the requests come to be known incrementally. A critical and novel feature of our formulations is that communications are not allowed among the servers, hence posing an interesting and challenging question: what performance can be achieved in comparison to the best possible solution obtained from an omniscience planner with perfect communication capabilities? The worst-case (over all possible request-set instances) performance metrics are given by the approximation ratio (offline case) and the competitive ratio (online case). Our first result indicates that the online and offline problems are effectively equivalent: for the same request-set instance, the approximation ratio and the competitive ratio differ by at most an additive factor of 2, irrespective of the release dates in the online case. Therefore, we can restrict our attention to the offline problem. For the offline problem, we show that the approximation ratio given by the Voronoi partition is m (the number of servers). For two classes of depot configurations, when the depots form a line and when the ratios between the distances of pairs of depots are upper bounded by a sublinear function f(m) (i.e., f(m) = o(m)), we give partition schemes with sublinear approximation ratios O(log m) and {\Theta}(f(m)) respectively. We also discuss several interesting open problems in our formulations: in particular, how our initial results (on the two deliberately chosen classes of depots) shape our conjecture on the open problems.

💡 Research Summary

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The paper introduces a novel class of distributed multi‑depot vehicle routing problems in which no communication is allowed among the servers. Each server starts and ends at a distinct depot, and a static partition of the ambient metric space—determined solely from the depot locations—assigns every request to exactly one server. The objective is to minimize the total distance traveled by all servers. Two variants are considered: an offline version, where the entire request set is known in advance, and an online version, where requests appear incrementally.

The first major result (Theorem 2.1) shows that for any given request set the approximation ratio of the offline problem and the competitive ratio of the online problem differ by at most an additive constant of 2. Consequently, the online case can be reduced to the offline case without loss of generality, and the rest of the analysis focuses on static partitions for the offline problem.

The paper then evaluates several partition schemes. The most natural one, the Voronoi partition (assign each request to the nearest depot), yields an approximation ratio of m, where m is the number of servers. This bound is tight: in the worst case the total travel distance can be m times the optimal offline solution, because the partition does not adapt to the distribution of requests.

To improve on this baseline, the authors study two restricted depot configurations.

1. Depots on a line (Section 3.2). By constructing a hierarchical interval decomposition that respects the linear order of the depots, they obtain a partition with an O(log m) approximation ratio. The construction essentially groups nearby depots into logarithmically sized clusters, ensuring that any request is assigned to a depot that is not too far from the optimal one.

2. Bounded distance ratios (Section 3.3). Suppose that for every pair of depots the ratio of their distances is bounded by a sub‑linear function f(m) (i.e., f(m)=o(m)). Under this assumption the authors design a partition whose approximation ratio is Θ(f(m)). When f(m)=√m, for example, the total travel distance is at most a constant factor times √m times the optimal distance, a substantial improvement over the trivial m‑approximation.

All partition schemes are static, computable in polynomial time, and require no runtime communication among servers. The paper also discusses the relevance of these results to practical settings such as robot fleets, autonomous drones, or any distributed system where communication is costly or unreliable.

Finally, the authors present a suite of open problems. The most prominent question is whether a sub‑linear approximation ratio can be achieved for arbitrary depot placements without any geometric restriction. Other directions include: (i) exploiting probabilistic information about request locations to design better partitions, (ii) allowing limited, possibly delayed, communication to improve performance, (iii) extending the analysis to capacitated vehicles with replenishment at depots, and (iv) considering alternative objectives such as minimizing the makespan (maximum individual tour length) rather than the sum of distances.

In summary, the paper establishes a foundational framework for routing without communication, proves that online and offline formulations are essentially equivalent, and demonstrates that, under modest geometric assumptions on depot locations, static partitions can achieve sub‑linear approximation guarantees. These insights open a new line of research on communication‑free distributed routing, with potential impact on a wide range of autonomous multi‑agent systems.