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Let G = (V, E) be a graph and w : E {yields} R{sup +} be a length function. Given S {improper_subset} V, a Steiner tour is a cycle passing through each vertex of S. In this paper, we investigate naturally submodular graphs: graphs for which the length function of the Steiner tours is submodular. Graphs of this nature have applications in determining near optimal schedules for the one warehouse multiretailer distribution problem. We provide two characterizations of naturally submodular graphs, as well as an O(n) time algorithm for identifying such graphs. In addition, we present an O(n) time algorithm for solving the Steiner traveling salesman problem in naturally submodular graphs based on one of our characterizations together with Cornuejols, fonlupt, and Naddef`s [CFN85] algorithm for finding a shortest Steiner tour (i.e. solving the Steiner traveling salesman problem) in series-parallel graphs.

Characterizations of naturally submodular graphs: a polynomially solvable class of the TSP