DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

https://doi.org/10.7151/dmgt

Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory 20(2) (2000) 231-242
DOI: 10.7151/dmgt.1122

PERSISTENCY IN THE TRAVELING SALESMAN PROBLEM ON HALIN GRAPHS

Vladimír Lacko

Department of Geometry and Algebra
P.J. Safárik University
Jesenná 5, 041 54 Košice, Slovakia
e-mail: lackov@Košice.upjs.sk

Abstract

For the Traveling Salesman Problem (TSP) on Halin graphs with three types of cost functions: sum, bottleneck and balanced and with arbitrary real edge costs we compute in polynomial time the persistency partition EAll,ESome,ENone of the edge set E, where:

EAll = {e ∈ E,e belongs to all optimum solutions},
ENone = {e ∈ E,e does not belong to any optimum solution} and
ESome = {e ∈ E,e belongs to some but not to all optimum solutions}.

Keywords: persistency, traveling salesman problem, Halin graph, polynomial algorithm.

2000 Mathematics Subject Classification: 05C45, 68Q25.

References

[1] K. Cechlárová, Persistency in the assignment and transportation problems, Math. Methods of Operations Research 47 (1998) 234-254.
[2] K. Cechlárová and V. Lacko, Persistency in some combinatorial optimization problems, in: Proc. Mathematical Methods in Economy 99 (Jindrichúv Hradec, 1999) 53-60.
[3] K. Cechlárová and V. Lacko, Persistency in combinatorial optimization problems on matroids, to appear in Discrete Applied Math.
[4] G. Cornuéjols, D. Naddef and W.R. Pulleyblank, Halin graphs and the Traveling salesman problem, Mathematical Programming 26 (1983) 287-294, doi: 10.1007/BF02591867.
[5] M.C. Costa, Persistency in maximum cardinality bipartite matchings, Operations Research Letters 15 (1994) 143-149, doi: 10.1016/0167-6377(94)90049-3.
[6] V. Lacko, Persistency in optimization problems on graphs and matroids, Master Thesis, UPJS Košice, 1998.
[7] V. Lacko, Persistency in the matroid product problem, in: Proc. CEEPUS Modern Applied Math. Workshop (AGH Kraków, 1999), 47-51.

Received 11 January 2000
Revised 29 March 2000