Abstract
Many applications of the traveling salesman problem require the introduction of additional constraints. One of the most frequently occurring classes of such constraints are those requiring that certain cities be visited before others (precedence constraints). In this paper we study the Precedence-Constrained Asymmetric Traveling Salesman (PCATS) polytope, i.e. the convex hull of incidence vectors of tours in a precedence-constrained directed graph. We derive several families of valid inequalities, and give polynomial time separation algorithms for important subfamilies. We then establish the dimension of the PCATS polytope and show that, under reasonable assumptions, the two main classes of inequalities derived are facet inducing.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
N. Ascheuer, L.F. Escudero, M. Grötschel and M. Stoer, “On identifying in polynomial time violated subtour elimination and precedence forcing constraints for the sequential ordering problem,” in: R. Kannan and W.R. Pulleyblank, eds.,Integer Programming and Combinatioral Optimization (University of Waterloo Press, 1990) 19–28.
E. Balas and M. Fischetti, “A lifting procedure for the asymmetric traveling salesman polytope and a large new class of facets,”Mathematical Programming 58 (1993) 325–352.
L. Escudero, M. Guignard and K. Malik, “On identifying and lifting valid cuts for the sequential ordering problem with precedence relationships and deadlines,” University of Madrid and IBM Spain, Madrid, Spain (1991).
R. Gomory and T.C. Hu, “Multiterminal network flows,”SIAM Journal on Applied Mathematics 9 (1961) 551–556.
M.W. Padberg and M. Grötschel, “Polyhedral computations,” in: E.L. Lawler, J.K. Lenstra, A. Rinnooy Kan and D.B. Shmoys, eds.,The Traveling Salesman Problem (Wiley, 1985) 307–360.
M.W. Padberg and G. Rinaldi, “Facet identification for the symmetric traveling salesman polytope,”Mathematical Programming 47 (1990) 219–257.
H.N. Psaraftis, “A dynamic programming solution to the single-vehicle many-to-many immediate request dial-a-ride problem,”Transportation Science 14 (1980) 130–154.
M.T. Fiala Timlin, “Precedence constrained routing and helicopter scheduling,” M.Sc. Thesis, Department of Combinatorics and Optimization, University of Waterloo (1989).
M.T. Fiala Timlin and W.R. Pulleyblank, “Precedence constrained routing and helicopter scheduling: heuristic design,”Interfaces 22 (1992) 100–111.
Author information
Authors and Affiliations
Additional information
Corresponding author.
The work of this author was supported by MURST, Italy.
Rights and permissions
About this article
Cite this article
Balas, E., Fischetti, M. & Pulleyblank, W.R. The precedence-constrained asymmetric traveling salesman polytope. Mathematical Programming 68, 241–265 (1995). https://doi.org/10.1007/BF01585767
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01585767