Characterizations of Natural Submodular Graphs: a Polynomially Solvable Class of the Tsp

Let G = (V, E) be a graph and w: E -» R+ be a length function. Given S C V, a Steiner tour is a cycle passing at least once 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. We provide two characterizations of naturally submodular graphs, an 0(n) time algorithm for identifying such graphs, and an O(n) time algorithm for solving the Steiner traveling salesman problem on such graphs. We study the relationship between submodular functions and the length of traveling salesman tours on a graph. Our study was motivated by results in the production/distribution literature. See for example [FQZ92], [Q85], [HR90], [R86], [HP93]. A set function G(S) that maps subsets of V to $K is said to be submodular if G(S U L u M) G(S U L) < G(S U M) G(S) for all disjoint S, L, M c V. We use a slightly liberalized definition of a traveling salesman tour called the Steiner traveling salesman tour [CFN85]. For an excellent review of the classical traveling salesman problem see Lawler et al. [LLRS85]. In the Steiner traveling salesman tour on a subset of vertices S, the tour must visit the vertices in S at least once, but it may also visit some vertices not in S one or more times. Our Steiner traveling salesman tour differs from the one in [CFN85] since we require it to visit a special vertex called the central warehouse. If for a particular graph the lengths of the Steiner traveling salesman tours are submodular for all nonnegative weight functions and all choices of the central warehouse, then the graph is termed naturally submodular. Note that there are graphs which are not naturally submodular but that the tour lengths are submodular for certain weight functions. We need to include the central warehouse on the Steiner traveling salesman tours because otherwise (assuming the length of the tour through the empty set is defined to be zero) only trivial graphs are naturally submodular. Received by the editors May 25, 1993; presented during the 15th International Symposium on Mathematical Programming, Ann Arbor, MI, August 15-19, 1994. 1991 Mathematics Subject Classification. Primary 05C38, 05C45, 05C75, 05C85.