On the general routing polytope

Authors:
Gerhard Reinelt;Dirk Oliver Theis
Affiliations:
Institute of Computer Science, University of Heidelberg, Im Neuenheimer Feld, Heidelberg, Germany;Faculty of Mathematics, University of Heidelberg, Im Neuenheimer Feld, Heidelberg, Germany
Venue:
Discrete Applied Mathematics
Year:
2008

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Abstract

The General Routing Problem (GRP) consists of finding a minimum length closed walk in an edge-weighted undirected graph, subject to containing certain sets of required nodes and edges. It is related to the Rural Postman Problem and the Graphical Traveling Salesman Problem. We examine the 0/1-polytope associated with the GRP introduced by Ghiani and Laporte [A branch-and-cut algorithm for the Undirected Rural Postman Problem, Math. Program. Ser. A 87 (3) (2000) 467-481]. We show that whenever it is not full-dimensional, the set of equations and facets can be characterized, and the polytope is isomorphic to the full-dimensional polytope associated with another GRP instance which can be obtained in polynomial time. We also offer a node-lifting method. Both results are applied to prove the facet-defining property of some classes of valid inequalities. As a tool, we study more general polyhedra associated to the GRP.