# The traveling salesman problem on a graph and some related integer polyhedra

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## Abstract

Given a graph*G = (N, E)* and a length function*l: E* → ℝ, the Graphical Traveling Salesman Problem is that of finding a minimum length cycle going*at least* once through each node of*G.* This formulation has advantages over the traditional formulation where each node must be visited exactly once. We give some facet inducing inequalities of the convex hull of the solutions to that problem. In particular, the so-called comb inequalities of Grötschel and Padberg are generalized. Some related integer polyhedra are also investigated. Finally, an efficient algorithm is given when*G* is a series-parallel graph.

## Key words

Traveling Salesman Problem Integer Polyhedron Facet Graph Series-Parallel Graph Steiner Tree Polynomial Algorithm## Preview

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## Copyright information

© The Mathematical Programming Society, Inc. 1985