Variantes del problema del cartero mixto que se pueden resolver usando programación lineal

None; None

Resúmenes

Given a connected mixed graph with costs on its edges and arcs, the mixed postman problem consists of finding a minimum cost closed tour of the mixed graph traversing all of its edges and arcs. It is well-known that this problem is NPhard. However, under certain conditions, the problem can be solved in polynomial time using linear programming, in other words, the corresponding polyhedra are integral. Some of these conditions are: the mixed graph is series-parallel or the mixed graph is even. Also, we show that if we add the constraint that each edge is traversed exactly once then the problem can be solved in polynomial time if the set of arcs forms a forest.

Gráfica mixta; problema de cartero; programación lineal

Dada una gráfica mixta y conexa con costos en sus aristas y arcos, el problema del cartero mixto consiste en encontrar un circuito cerrado de la gráfica mixta que recorra sus aristas y arcos a costo mínimo. Se sabe que este problema es NP-duro. Sin embargo, bajo ciertas condiciones adicionales, el problema se puede resolver en tiempo polinomial usando programación lineal, en otras palabras, los poliedros correspondientes son enteros. Algunas de estas condiciones son: la gráfica mixta es serie paralelo o la gráfica mixta tiene grado total par en todos sus vértices. Además, mostramos que si agregamos la restricción adicional de que cada arista se recorra exactamente una vez entonces el problema se puede resolver en tiempo polinomial si el conjunto de arcos forma un bosque.

Gráfica mixta; problema de cartero; programación lineal

Variants of the mixed postman problem solvable using linear programming

Variantes del problema del cartero mixto que se pueden resolver usando programación lineal

Francisco Javier Zaragoza Martínez^*+
Rafael López Bracho^†*

*^{Dirección para correspondencia}
Abstract

Given a connected mixed graph with costs on its edges and arcs, the mixed postman problem consists of finding a minimum cost closed tour of the mixed graph traversing all of its edges and arcs. It is well-known that this problem is NPhard. However, under certain conditions, the problem can be solved in polynomial time using linear programming, in other words, the corresponding polyhedra are integral. Some of these conditions are: the mixed graph is series-parallel or the mixed graph is even. Also, we show that if we add the constraint that each edge is traversed exactly once then the problem can be solved in polynomial time if the set of arcs forms a forest.

Keywords: Mixed graph, postman problem, linear programming.

Resumen

Dada una gráfica mixta y conexa con costos en sus aristas y arcos, el problema del cartero mixto consiste en encontrar un circuito cerrado de la gráfica mixta que recorra sus aristas y arcos a costo mínimo. Se sabe que este problema es NP-duro. Sin embargo, bajo ciertas condiciones adicionales, el problema se puede resolver en tiempo polinomial usando programación lineal, en otras palabras, los poliedros correspondientes son enteros. Algunas de estas condiciones son: la gráfica mixta es serie paralelo o la gráfica mixta tiene grado total par en todos sus vértices. Además, mostramos que si agregamos la restricción adicional de que cada arista se recorra exactamente una vez entonces el problema se puede resolver en tiempo polinomial si el conjunto de arcos forma un bosque.

Palabras clave: Gráfica mixta, problema de cartero, programación lineal.

Mathematics Subject Classification: 05C45, 90C35.

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References

Referencias bibliográficas

[1] Guan, M.G. (1960) “Graphic programming using odd or even points”, Chinese Math. 1: 273–277.
[2] Edmonds, J.; Johnson, E.L. (1973) “Matching, Euler tours and the Chinese postman”, Math. Programming 5: 88–124.
[3] Papadimitriou, C.H. (1976) “On the complexity of edge traversing”, J. ACM 23: 544–554.
[4] Duffin, R.J. (1965) “Topology of series-parallel networks”, Journal of Mathematical Analysis and Applications 10: 303–318.
[5] Kappauf, C.H.; Koehler, G.J. (1979) “The mixed postman problem”, Discrete Appl. Math. 1: 89–103.
[6] Christofides, N.; Benavent, E.; Campos, V.; Corberán, Á. ; Mota, E. (1984) “An optimal method for the mixed postman problem”, in: P. Thoft-Christensen (Ed.) Lecture Notes in Control and Inform. Sci. 59 Springer, Berlin: 641–649.
[7] Grötschel, M.; Win, Z. (1992) “A cutting plane algorithm for the windy postman problem”, Math. Programming 55: 339–358.
[8] Ralphs, T.K. (1993) “On the mixed Chinese postman problem”, Oper. Res. Lett. 14: 123–127.
[9] Veblen, O. (1912/1913) “An application of modular equations in analysis situs”, Ann. of Math. 2: 86–94.
[10] Ford, L.R., Jr.; Fulkerson, D.R. (1962) Flows in Networks. Princeton University Press, Princeton, N.J.
[11] Zaragoza Martínez, F.J. (2005) “Linear programming relaxations of the mixed postman problem”, Morfismos9: 21–34.
[12] Win, Z. (1989) “On the windy postman problem on Eulerian graphs”, Math. Programming 44: 97–112.
[13] Grötschel, M.; Lovász, L.; Schrijver, A. (1981) “The ellipsoid method and its consequences in combinatorial optimization”, Combinatorica 1: 169–197.
[14] Zaragoza Martínez, F. J. (2008) “Series-parallel graphs are windy postman perfect”, Discrete Mathematics 308: 1366–1374.
[15] Fernandes, C.G.; Lee, O.; Wakabayashi, Y. (2009) “Minimum cycle cover and chinese postman problems on mixed graphs with bounded tree-width”, Discrete Applied Mathematics 157: 272 – 279.
[16] Tohyama, H.; Adachi, A. (1996) “Complexity of a restricted Chinese postman problem”, Trans. Inform. Process. Soc. Japan 37: 1886– 1896.
[17] Zaragoza Martínez, F.J. (2003) Postman Problems on Mixed Graphs. Ph.D. Thesis, University of Waterloo, Waterloo, On.
[18] Khachiyan, L.G. (1979) “A polynomial algorithm in linear programming”, Dokl. Akad. Nauk SSSR 244: 1093–1096.

^*Correspondencia a:

Francisco Javier Zaragoza Martínez.

Universidad Autónoma Metropolitana Unidad Azcapotzalco, Departamento de Sistemas. Av. San Pablo 180, México, D.F., México 02200. E-mail franz@correo.azc.uam.mx

Rafael López Bracho.

Universidad Autónoma Metropolitana Unidad Azcapotzalco, Departamento de Sistemas. Av. San Pablo 180, México, D.F., México 02200. E-mail rlb@correo.azc.uam.mx

^*Universidad Autónoma Metropolitana Unidad Azcapotzalco, Departamento de Sistemas. Av. San Pablo 180, México, D.F., México 02200. E-mail franz@correo.azc.uam.mx

^†Universidad Autónoma Metropolitana Unidad Azcapotzalco, Departamento de Sistemas. Av. San Pablo 180, México, D.F., México 02200. E-mail rlb@correo.azc.uam.mx

Received: 27 Nov 2009; Revised: 27 Aug 2011; Accepted: 10 May 2012

Fechas de Publicación

Publicación en esta colección
03 Dic 2012
Fecha del número
Jul 2012

Histórico

Recibido
27 Nov 2009
Revisado
27 Ago 2011
Acepto
10 Mayo 2011

[1] [1] Guan, M.G. (1960) “Graphic programming using odd or even points”, Chinese Math. 1: 273–277.

[2] [2] Edmonds, J.; Johnson, E.L. (1973) “Matching, Euler tours and the Chinese postman”, Math. Programming 5: 88–124.

[3] [3] Papadimitriou, C.H. (1976) “On the complexity of edge traversing”, J. ACM 23: 544–554.

[4] [4] Duffin, R.J. (1965) “Topology of series-parallel networks”, Journal of Mathematical Analysis and Applications 10: 303–318.

[5] [5] Kappauf, C.H.; Koehler, G.J. (1979) “The mixed postman problem”, Discrete Appl. Math. 1: 89–103.

[6] [6] Christofides, N.; Benavent, E.; Campos, V.; Corberán, Á. ; Mota, E. (1984) “An optimal method for the mixed postman problem”, in: P. Thoft-Christensen (Ed.) Lecture Notes in Control and Inform. Sci. 59 Springer, Berlin: 641–649.

[7] [7] Grötschel, M.; Win, Z. (1992) “A cutting plane algorithm for the windy postman problem”, Math. Programming 55: 339–358.

[8] [8] Ralphs, T.K. (1993) “On the mixed Chinese postman problem”, Oper. Res. Lett. 14: 123–127.

[9] [9] Veblen, O. (1912/1913) “An application of modular equations in analysis situs”, Ann. of Math. 2: 86–94.

[10] [10] Ford, L.R., Jr.; Fulkerson, D.R. (1962) Flows in Networks. Princeton University Press, Princeton, N.J.

[11] [11] Zaragoza Martínez, F.J. (2005) “Linear programming relaxations of the mixed postman problem”, Morfismos9: 21–34.

[12] [12] Win, Z. (1989) “On the windy postman problem on Eulerian graphs”, Math. Programming 44: 97–112.

[13] [13] Grötschel, M.; Lovász, L.; Schrijver, A. (1981) “The ellipsoid method and its consequences in combinatorial optimization”, Combinatorica 1: 169–197.

[14] [14] Zaragoza Martínez, F. J. (2008) “Series-parallel graphs are windy postman perfect”, Discrete Mathematics 308: 1366–1374.

[15] [15] Fernandes, C.G.; Lee, O.; Wakabayashi, Y. (2009) “Minimum cycle cover and chinese postman problems on mixed graphs with bounded tree-width”, Discrete Applied Mathematics 157: 272 – 279.

[16] [16] Tohyama, H.; Adachi, A. (1996) “Complexity of a restricted Chinese postman problem”, Trans. Inform. Process. Soc. Japan 37: 1886– 1896.

[17] [17] Zaragoza Martínez, F.J. (2003) Postman Problems on Mixed Graphs. Ph.D. Thesis, University of Waterloo, Waterloo, On.

[18] [18] Khachiyan, L.G. (1979) “A polynomial algorithm in linear programming”, Dokl. Akad. Nauk SSSR 244: 1093–1096.