Multi-depot traveling salesmen location problems on networks with special structure

Suppose that on any day, calls for service are generated by nodes of a transportation network independently with known probabilities. The calls are allocated to p service units that visit the allocated customers on shortest closed tours. That is, each service unit performs an optimal traveling salesman tour on its set of allocated calls (customers). We need to find optimal locations for the service units to minimize the expected travel distance. For this problem on a path, we present an O ( n 4 ) algorithm, where n is the number of nodes of the network. We also present an O ( n 5 ) algorithm for the problem on a tree with p = 2 .