Abstract
Given a directed graph G = (V, E), with distinguished nodes s and t, a k-walk from s to t is a walk with exactly k arcs. In this paper we consider polyhedral aspects of the problem of finding a minimum-weight k-walk from s to t. We describe an extended linear programming formulation, in which the number of inequalities and variables is polynomial in the size of G(for fixed k),and all the coefficients have values 0,−1, +1. We apply the projection method on the extended formulation and give two natural linear programming formulations, each having one variable for each arc of G. These correspond to two natural polyhedra: the convex hull of the (s,t)-k-walk incidence vectors, and the dominant of that poly tope. We focus on the dominant. We give three classes of facet constraints, showing that the dominant has an exponential number of facet constraints and that some facet constraints have coefficients as large as the size of V.
Supported by National Science Foundation Grant No. DDM-91-96083.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
E. Balas and W.R. Pulleyblank [ 1983 ], The perfect matchable subgraph poly tope of a bipartite graph, Networks, Vol. 13 (1983) 495–516.
E. Balas and W.R. Pulleyblank [ 1987 ], The perfect matchable subgraph polytope of an arbitrary graph, Research Report CORR87-33(l987), Department of Combinatorics and Optimization, University of Waterloo, to appear in Combinatorica.
M. O. Ball, W-G. Liu and W.R. Pullyblank [1987], Two terminal Steiner tree polyhedra, Research Report CORR87-33(1987),Department of Combinatorics and Optimization, University of Waterloo. Appeared In Proceedings of CORE 20th Anniversary Conference.
A. Schrijver [ 1986 ], Theory of Linear and Integer Programming, John Wiley and Sons Ltd., Chichester, 1986.
F. Barahona and A.R Mahjoub [ 1987 ], Compositions of graphs and polyhedra II: stablesets, Research Report CORR 87-47 1987, Department of Combinatorics and Optimization, University of Waterloo.
R.K. Martin, R.L. Rardin and B.A. Campbell [ 1987 ], Polyhedral characterization of discrete dynamic programming, Research Report CC-87-24, School of Industrial Engineering, Purdue University.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1994 Kluwer Academic Publishers
About this chapter
Cite this chapter
Coullard, C.R., Gamble, A.B., Liu, J. (1994). The K-Walk Polyhedron. In: Du, DZ., Sun, J. (eds) Advances in Optimization and Approximation. Nonconvex Optimization and Its Applications, vol 1. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-3629-7_2
Download citation
DOI: https://doi.org/10.1007/978-1-4613-3629-7_2
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4613-3631-0
Online ISBN: 978-1-4613-3629-7
eBook Packages: Springer Book Archive