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

Distributed Multi-Depot Routing without Communications