Local search algorithms for multiple-depot vehicle routing and for multiple traveling salesman problems with proved performance guarantees

We consider two related problems: the multiple-depot vehicle routing problem (MDVRP) and the Multiple traveling salesman problem (mTSP). In both of them, given is the complete graph on n vertices (Formula presented.) with nonnegative edge lengths that form a metric on V. Also given is a positive integer k. In typical applications, V represents locations of customers and k represents the number of available vehicles. In MDVPR, we are also given a set of k depots (Formula presented.), and the goal is to find a minimum-length cycle cover of G of size k, that is, a collection of k (possibly empty) cycles such that each (Formula presented.) is in exactly one cycle, and each cycle in the cover contains exactly one depot. In mTSP, no depots are given, so the goal is to find (any) minimum-length cycle cover of G of size k. We present local search algorithms for both problems, and we prove that their approximation ratio is 2.