An updated survey on the linear ordering problem for weighted or unweighted tournaments

Annals of Operations Research

Volumn 175, Issue 1, 2010, Pages 107-158

  Charon, Irène ^a   Hudry, Olivier ^a  

^a TELECOM PARISTECH   (France)

Author keywords

Acyclic subgraph; Aggregation of preferences; Combinatorial optimization; Combinatorics; Complexity; Feedback arc set; Graph theory; Kemeny's problem; Linear ordering problem; Median order; Optimal triangulation; Reversing set; Slater's problem; Social choice; Tournament solutions; Voting theory

Indexed keywords

EID: 76849096444     PISSN: 02545330     EISSN: 15729338     Source Type: Journal    
DOI: 10.1007/s10479-009-0648-7     Document Type: Article

References (294)

1. Adám, A. (1964). Problem. In Theory of graphs and its applications. Proc. Coll. Smolenice, Czech. Acad. Sci. Publ.
2. Adler, I., Alon, N., & Ross, S. M. (2001). On the number of Hamiltonian paths in tournaments. Random Structures and Algorithms, 18, 291-296.
3. Ailon, N., & Alon, N. (2007). Hardness of fully dense problems. Information and Computation, 205, 117-1129.
4. Ailon, N., Charikar, M., & Newman, A. (2005). Aggregating inconsistent information: ranking and clustering. In Proceedings of the 37th annual ACM symposium on theory of computing (STOC) (pp. 684-693).
5. Ali, I., Cook, W. D., & Kress, M. (1986). On the minimum violations ranking of a tournament. Management Science, 32, 660-674.
6. Alon, N. (1990). The maximum number of Hamiltonian paths in tournaments. Combinatorica, 10, 319-324.
7. Alon, N. (2006). Ranking tournaments. SIAM Journal on Discrete Mathematics, 20(1), 137-142.
8. Alon, N., & Spencer, J. (2000). The probabilistic method (2nd edn.). New York: Wiley.
9. Alspach, B. (1967). Cycles of each length in regular tournaments. Canadian Mathematical Bulletin, 10, 283-286.
10. Alspach, B. (1968). A combinatorial proof of a conjecture of Goldberg and Moon. Canadian Mathematical Bulletin, 11, 655-661.
11. Arditti, D. (1984). Un nouvel algorithme de recherche d'un ordre induit par des comparaisons par paires. In E. Diday et al. (Eds.), Data analysis and informatics III (pp. 323-343). Amsterdam: North-Holland.
12. Arora, S., Frieze, A., & Kaplan, H. (1996). A new rounding procedure for the assignment problem with applications to dense graph arrangement problems. In Proceedings of the 37th annual IEEE symposium on foundations of computer science (FOCS) (pp. 24-33).
13. Ausiello, G., Crescenzi, P., Gambosi, G., Kann, V., Marchetti-Spaccamela, A., & Protasi, M. (2003). Complexity and approximation (2nd edn.). Berlin: Springer.
14. Baker, F. B., & Hubert, L. J. (1977). Applications of combinatorial programming to data analysis: seriation using asymmetric proximity measures. British Journal of Mathematical & Statistical Psychology, 30, 154-164.
15. Bang-Jensen, J., & Gutin, G. (2001). Digraphs: theory, algorithms, and applications. London, Berlin, Heidelberg: Springer.
16. Banks, J. (1985). Sophisticated voting outcomes and agenda control. Social Choice and Welfare, 2, 295-306.
17. Banks, J., Bordes, G., & Le Breton, M. (1991). Covering relations, closest orderings and Hamiltonian bypaths in tournaments. Social Choice and Welfare, 8, 355-363.
18. Bar-Noy, A., & Naor, J. (1990). Sorting, minimal feedback sets, and Hamilton paths in tournaments. SIAM Journal on Discrete Mathematics, 3(1), 7-20.
19. Barahona, F., Fonlupt, J., & Mahjoub, A. R. (1994). Compositions of graphs and polyhedra IV: acyclic spanning subgraph. SIDMA, 7(3), 390-402.
20. Barbut, M. (1966). Note sur les ordres totaux à distance minimum d'une relation binaire donnée. Mathématiques et Sciences humaines, 17, 47-48.
21. Barthélemy, J.-P., & Monjardet, B. (1981). The median procedure in cluster analysis and social choice theory. Mathematical Social Sciences, 1, 235-267.
22. Barthélemy, J.-P., & Monjardet, B. (1988). The median procedure in data analysis: new results and open problems. In H. H. Bock (Ed.), Classification and related methods of data analysis. Amsterdam: North-Holland.
23. Barthélemy, J.-P., Guénoche, A., & Hudry, O. (1989). Median linear orders: heuristics and a branch and bound algorithm. European Journal of Operational Research, 41, 313-325.
24. Barthélemy, J.-P., Hudry, O., Isaak, G., Roberts, F. S., & Tesman, B. (1995). The reversing number of a digraph. Discrete Applied Mathematics, 60, 39-76.
25. Barthélemy, J.-P., Cohen, G., & Lobstein, A. (1996). Algorithmic complexity and communication problems. London: UCL Press.
26. Bartholdi, J. J. III, Tovey, C. A., & Trick, M. A. (1989). Voting schemes for which it can be difficult to tell who won the election. Social Choice and Welfare, 6, 157-165.
27. Becker, O. (1967). Das Helmstädtersche Reihenfolgeproblem-die Effizienz verschiedener Näherungsverfahren. In: Computers uses in the social science. Vienna.
28. Belloni, A., & Lucena, A. (2004). Lagrangian heuristics for the linear ordering problem. In M. G. C. Resende & J. P. de Sousa (Eds.), Metaheuristics: computer decision making (pp. 37-63). Dordrecht: Kluwer Academic.
29. Berge, C. (1985). Graphs. Amsterdam: North-Holland.
30. Berger, B., & Shor, P. W. (1997). Tight bounds for the maximum acyclic subgraph problem. Journal of Algorithms, 25, 1-18.
31. Berger, B., & Shor, P. W. (1990). Approximation algorithms for the maximum acyclic subgraph problem. In Proceedings of the first annual ACM-SIAM symposium on Discrete algorithms (SODA) (pp. 236-243).
32. Bermond, J.-C. (1972). Ordres à distance minimum d'un tournoi et graphes partiels sans circuits maximaux. Mathématiques et Sciences humaines, 37, 5-25.
33. Bermond, J.-C. (1975). The circuit-hypergraph of a tournament. In Infinite and finite sets, proceedings of the colloquia mathematica societatis János Bolyai (Vol. 10, pp. 165-180). Amsterdam: North-Holland,.
34. Bermond, J.-C., & Kodratoff, Y. (1976). Une heuristique pour le calcul de l'indice de transitivité d'un tournoi. RAIRO, 10(3), 83-92.
35. Bertacco, L., Brunetta, L., & Fischetti, M. (2008). The Linear Ordering Problem with Cumulative Costs. European Journal of Operational Research, 189, 1345-1357.
36. Black, D. (1958). The theory of committees and elections. Cambridge: Cambridge University Press.
37. Blin, J. M., & Whinston, A. B. (1974). A note on majority rule under transitivity constraints. Management Science, 20, 1439-1440.
38. Blin, J. M., & Whinston, A. B. (1975). Discriminant functions and majority voting. Management Science, 21, 1029-1041.
39. Boenchendorf, K. (1982). Mathematical systems in economics: Vol. 74. Reihenfolgenprobleme/mean-flow-time sequencing. Königstein: Verlagsgruppe Athanäum/Hain/Scriptor/Hanstein.
40. Bolotashvili, G., Kovalev, M., & Girlich, E. (1999). New facets of the linear ordering polytope. SIAM Journal on Discrete Mathematics, 12(3), 326-336.
41. Borda, J.-C. (chevalier de) (1784). Mémoire sur les élections au scrutin. Histoire de l'Académie Royale des Sciences pour 1781, Paris.
42. Bowman, V. J., & Colantoni, C. S. (1973). Majority rule under transitivity constraints. Management Science, 19, 1029-1041.
43. Brualdi, R. A., & Qiao, L. (1983). Upsets in round robin tournaments. Journal of Combinatorial Theory Series B, 35, 62-77.
44. Brualdi, R. A., & Qiao, L. (1984). The interchange graph of tournaments with the same score vector. In Progress in graph theory (pp. 129-151). San Diego: Academic Press.
45. Buchheim, C., Wiegele, A., & Zheng, L. (2009). Exact algorithms for the quadratic linear ordering problem. Informs Journal on Computing (to appear).
46. Burkard, R. E., Çela, E., Pardalos, P. M., & Pitsoulis, L. S. (1998). The quadratic assignment problem. In P. M. Pardalos & D.-Z. Du (Eds.), Handbook of combinatorial optimization (pp. 241-338). Dordrecht: Kluwer Academic.
47. Burkov, V. N., & Groppen, V. O. (1972). Branch cuts in strongly connected graphs and permutation potentials. Automation and Remote Control, 6, 111-119.
48. Busch, A. H. (2006). A note on the number of Hamiltonian paths in strong tournaments. Electronic Journal of Combinatorics, 13, #N3.
49. Campos, V., Laguna, M., & Martí, R. (1999). Scatter search for the linear ordering problem. In D. Corne, M. Dorigo, & F. Glover (Eds.), New ideas in optimization (pp. 331-339). New York: McGraw-Hill.
50. Campos, V., Glover, F., Laguna, M., & Martí, R. (2001). An experimental evaluation of a scatter search for the linear ordering problem. Journal of Global Optimization, 21(4), 397-414.
51. Campos, V., Laguna, M., & Martí, R. (2005). Context-independent scatter and tabu search for permutation problems. INFORMS Journal on Computing, 17(1), 331-339.
52. Caspard, N., Monjardet, B., & Leclerc, B. (2007). Ensembles ordonnés finis: concepts, résultats et usages. Berlin: Springer.
53. Chanas, S., & Kobylanski, P. (1996). A new heuristic algorithm solving the linear ordering problem. Computational Optimization and Applications, 6, 191-205.
54. Chaovalitwongse, W., Pardalos, P. M., Resende, M. G. C., & Grundel, D. A. (2009). Revised GRASP with path-relinking for the linear ordering problem. Submitted for publication to the Journal of Combinatorial Optimization, and AT& T Labs Research technical report.
55. Charbit, P., Thomasse, S., & Yeo, A. (2007). The minimum feedback arc set problem is NP-hard for tournaments. Combinatorics, Probability and Computing, 16(1), 1-4.
56. Charon, I., & Hudry, O. (1998). Lamarckian genetic algorithms applied to the aggregation of preferences. Annals of Operations Research, 80, 281-297.
57. Charon, I., & Hudry, O. (2000). Slater orders and Hamiltonian paths of tournaments. Electronic Notes in Discrete Mathematics, 5, 60-63.
58. Charon, I., & Hudry, O. (2001a). The noising methods: a generalization of some metaheuristics. European Journal of Operational Research, 135(1), 86-101.
59. Charon, I., & Hudry, O. (2001b). Metod vetvei i granits dlia recheniia zadatchi o lineinom poriadke na vzvechennikh tournirakh. Discretii Analiz i Issledovanie Operatsii, Seriia 2, 8(2), 73-91 (in Russian).
60. Charon, I., & Hudry, O. (2002). The noising methods: a survey. In P. Hansen & C. C. Ribeiro (Eds.), Essays and surveys in metaheuristics (pp. 245-261). Dordrecht: Kluwer Academic.
61. Charon, I., & Hudry, O. (2003). Links between the Slater index and the Ryser index of tournaments. Graphs and Combinatorics, 19(3), 309-322.
62. Charon, I., & Hudry, O. (2006). A branch and bound algorithm to solve the linear ordering problem for weighted tournaments. Discrete Applied Mathematics, 154, 2097-2116.
63. Charon, I., & Hudry, O. (2007). A survey on the linear ordering problem for weighted or unweighted tournaments, 4OR, 5(1), 5-60.
64. Charon, I., & Hudry, O. (2009a). Self-tuning of the noising methods. Optimization, 58(7), 1-21.
65. Charon, I., & Hudry, O. (2009b). Branch and bound. In V. Paschos (Ed.) Handbook of combinatorial optimization. Wiley, New York (to appear).
66. Charon, I., & Hudry, O. (2009c). Maximum distance between Slater orders and Copeland orders of tournaments (submitted for publication).
67. Charon, I., & Hudry, O. (2009d). How different can Slater's solutions and Kemeny's solutions of tournaments be? (submitted for publication).
68. Charon, I., Germa, A., & Hudry, O. (1992a). Encadrement de l'indice de Slater d'un tournoi à l'aide de ses scores. Mathématiques, Informatique et Sciences humaines, 118, 53-68.
69. Charon, I., Germa, A., & Hudry, O. (1992b). Utilisation des scores dans des méthodes exactes déterminant les ordres médians de tournois. Mathématiques, Informatique et Sciences humaines, 119, 53-74.
70. Charon, I., Germa, A., & Hudry, O. (1996a). Random generation of tournaments and asymmetric digraphs with given out-degrees. European Journal of Operational Research, 95, 411-419.
71. Charon, I., Guénoche, A., Hudry, O., & Woirgard, F. (1996b). A bonsaï branch and bound method applied to voting theory. In Ordinal and symbolic data analysis (pp. 309-318). Berlin: Springer.
72. Charon, I., Hudry, O., & Woirgard, F. (1996c). Ordres médians et ordres de Slater des tournois. Mathématiques, Informatique et Sciences humaines, 133, 23-56.
73. Charon, I., Guénoche, A., Hudry, O., & Woirgard, F. (1997a). New results on the computation of median orders. Discrete Mathematics, 165-166, 139-154.
74. Charon, I., Hudry, O., & Woirgard, F. (1997b). A 16-vertex tournament for which Banks set and Slater set are disjoint. Discrete Applied Mathematics, 80(2-3), 211-215.
75. Chartrand, G., Geller, D., & Hedetniemi, S. (1971). Graphs with forbidden subgraphs. Journal of Combinatorial Theory, Series B, 10(1), 12-41.
76. Chenery, H. B., & Watanabe, T. (1958). International comparisons of the structure of production. Econometrica, 26(4), 487-521.
77. Chiarini, B., Chaovalitwongse, W., & Pardalos, P. M. (2004). A new algorithm for the triangulation of input-output tables. In P. M. Pardalos, A. Migdalas, & G. Baourakis (Eds.), Supply chain and finance (pp. 254-273). Singapore: World Scientific.
78. Christof, T., & Reinelt, G. (1996). Combinatorial optimization and small polytopes. Top, 4(1), 1-64.
79. Christof, T., & Reinelt, G. (1997a). Low-dimensional linear ordering polytopes. Working paper, University of Heidelberg.
80. Christof, T., & Reinelt, G. (1997b). Small instances relaxations for solving linear ordering problems by branch-and-cut (Technical report). University of Heidelberg.
81. Christof, T., & Reinelt, G. (2001). Algorithmic aspects of using small instance relaxations in parallel branch-and-cut. Algorithmica, 30(4), 507-629.
82. Christophe, J., Doignon, J.-P., & Fiorini, S. (2004). The biorder. Order, 21/1, 61-82.
83. Cohen, W., Schapire, R., & Singer, Y. (1999). Learning to order things. Journal of Artificial Intelligence Research, 10, 213-270.
84. Condorcet, M. J. A. N. Caritat (marquis de) (1785). Essai sur l'application de l'analyse à la probabilité des décisions rendues à la pluralité des voix, Paris.
85. Congram, R. K. (2000). Polynomially searchable exponential neighbourhoods for sequencing problems in combinatorial optimisation. Ph. D. thesis, University of Southampton.
86. Conitzer, V. (2006). Computing Slater rankings using similarities among candidates. In Proceedings of the 21st national conference on artificial intelligence (AAAI-06), Boston, MA, USA (pp. 613-619). Early version: IBM research report RC23748, 2005.
87. Cook, W. D., & Saipe, A. L. (1976). Committee approach to priority planning: the median ranking method. Cahiers du Centre d'Études et de Recherche Opérationnelle18(3), 337-352.
88. Cook, W. D., Golan, I., & Kress, M. (1988). Heuristics for ranking players in a round robin tournament. Computers and Operations Research, 15(2), 135-144.
89. Cook, W. D., Doyle, J., Green, R., & Kress, M. (1996). Ranking players in multiple tournaments. Computers and Operations Research, 23(9), 869-880.
90. Copeland, A. H. (1951). A "reasonable" social welfare function. Seminar on applications of mathematics to social sciences, University of Michigan.
91. Coppersmith, D., & Winograd, S. (1990). Matrix multiplication via arithmetic progressions. Journal of Symbolic Computation, 9, 251-280.
92. Coppersmith, D., Fleischer, L., & Rudra, A. (2006). Ordering by weighted number of wins gives a good ranking for weighted tournaments. In Proceedings of the seventeenth annual ACM-SIAM symposium on discrete algorithm (SODA'06) (pp. 776-782).
93. Czygrinow, A., Poljak, S., & Rödl, V. (1999). Constructive quasi-Ramsey numbers and tournament ranking. SIAM Journal on Discrete Mathematics, 12(1), 48-63.
94. Davenport, A., & Kalagnanam, J. (2004). A computational study of the Kemeny rule for preference aggregation. In Proceedings of the national conference on artificial intelligence (AAAI) (pp. 697-702).
95. de Cani, J. S. (1969). Maximum likelihood paired comparison ranking by linear programming. Biometrika, 3, 537-545.
96. de Cani, J. S. (1972). A branch and bound algorithm for maximum likelihood paired comparison ranking. Biometrika, 59, 131-135.
97. de la Vega, W. F. (1983). On the maximal cardinality of a consistent set of arcs in a random tournament. Journal of Combinatorial Theory Series B, 35, 328-332.
98. Debord, B. (1987a). Caractérisation des matrices de préférences nettes et méthodes d'agrégation associées. Mathématiques et Sciences humaines, 97, 5-17.
99. Debord, B. (1987b). Axiomatisation de procédures d'agrégation de préférences. Ph. D. thesis, Université scientifique technologique et médicale de Grenoble.
100. Dixon, J. D. (1967). The maximum order of the group of a tournament. Canadian Mathematical Bulletin, 10, 503-505.
101. Dodgson, C. L. (1876). A method of taking votes on more than two issues. Oxford: Clarendon.
102. Doignon, J.-P., Fiorini, S., & Joret, G. (2006). Facets of the linear ordering polytope: a unification for the fence family through weighted graphs. Journal of Mathematical Psychology, 50/3, 251-262.
103. Dom, M., Guo, J., Hüffner, F., Niedermeier, R., & Truß, A. (2006). Lecture notes in computer science: Vol. 3998. Fixed-parameter tractability results for feedback set problems in tournaments (pp. 320-331). Heidelberg: Springer.
104. Downey, R. G., & Fellows, M. R. (1999). Parameterized complexity. Berlin: Springer.
105. Dréo, J., Petrowski, A., Taillard, E., & Siarry, P. (2006). Metaheuristics for hard optimization. Methods and case studies. Berlin: Springer.
106. Duarte, A., Laguna, M., & Marti, R. (2009). Tabu search for the linear ordering problem with cumulative costs. Computational Optmization and Applications (to appear).
107. Dugat, V. (1990). Décomposition de tournois réguliers: théorie et application aux algorithmes de tests d'isomorphisme. Ph. D. thesis, Université Paul Sabatier, Toulouse.
108. Dutta, B. (1988). Covering sets and a new Condorcet choice correspondence. Journal of Economic Theory, 44, 63-80.
109. Dwork, C., Kumar, R., Naor, M., & Sivakumar, D. (2001). Rank aggregation methods for the Web. In Proceedings of the 10th international conference on World Wide Web (WWW10) (pp. 613-622).
110. Eades, P., & Lin, X. (1995). A new heuristic for the feedback arc set problem. Australian Journal of Combinatorics, 12, 15-26.
111. Eades, P., Lin, X., & Smyth, W. F. (1993). A fast and effective heuristic for the feedback arc set problem. Information Processing Letters, 47(6), 319-323.
112. Erdös, P. & Moon, J. W. (1965). On sets of consistent arcs in a tournament. Canadian Mathematical Bulletin, 8, 269-271.
113. Erdös, P., & Moser, L. (1964). On the representation of directed graphs as unions of orderings. Magyar Tudományos Akadémia Matematikai Kutató Intézetének Közleményei. 9, 125-132.
114. Even, G., Naor, J. S., Sudan, M., & Schieber, B. (1998). Approximating minimum feedback sets and multicuts in directed graphs. Algorithmica, 20(2), 151-174.
115. Even, G., Naor, J. S., Rao, S., & Schieber, B. (2000). Divide-and-conquer approximation algorithms via spreading metrics. Journal of the ACM, 47, 585-616.
116. Fagin, R., Kumar, R., Mahdian, M., Sivakumar, D., & Vee, E. (2005). Rank aggregation: an algorithmic perspective. Unpublished manuscript.
117. Festa, P., Pardalos, P., & Resende, M. (1999). Feedback set problems. In D. Z. Du & P. M. Pardalos (Eds.), Handbook of combinatorial optimization (Vol. 4, pp. 209-258). Dordrecht: Kluwer Academic.
118. Fiorini, S. (2001a). Polyhedral combinatorics of order polytopes. Ph. D. thesis, Université libre de Bruxelles.
119. Fiorini, S. (2001b). Determining the automorphism group of the linear ordering polytope. Discrete Applied Mathematics, 112/1-3, 121-128.
120. Fiorini, S. (2006a). How to recycle your facets. Discrete Optimization, 3/2, 136-153.
121. Fiorini, S. (2006b). {0, 1/2}-cuts and the linear ordering problem: surfaces that define facets. SIAM Journal on Discrete Mathematics, 20/4, 893-912.
122. Fiorini, S., & Fishburn, P. (2003). Facets of linear signed order polytopes. Discrete Applied Mathematics, 131/3, 597-610.
123. Fishburn, P. (1977). Condorcet social choice functions. SIAM Journal of Applied Mathematics, 33, 469-489.
124. Fishburn, P. (1992). Induced binary probabilities and the linear ordering polytope: a status report. Mathematical Social Sciences, 23, 67-80.
125. Flood, M. M. (1990). Exact and heuristic algorithms for the weighted feedback arc set problem: a special case of the skew-symmetric quadratic assignment problem. Networks, 20, 1-23.
126. Flueck, J. A., & Korsh, J. F. (1974). A branch search algorithm for maximum likelihood paired comparison ranking. Biometrika, 61(3), 621-626.
127. Fulkerson, D. R. (1965). Upsets in round robin tournaments. Canadian Journal of Mathematics, 17, 957-969.
128. Gamboa, D., Rego, C., & Glover, F. (2006). A relax-and-cut RAMP approach for the linear ordering problem. In the abstracts booklet of ECCO XIX/CO2006 Joint Meeting, Porto, Portugal, p. 40.
129. Garcia, C. G., Perez-Brito, D., Campos, V., & Marti, R. (2006). Variable neighborhood search for the linear ordering problem. Computers and Operations Research, 33(12), 3549-3565.
130. Garey, M. R., & Johnson, D. S. (1979). Computers and intractability, a guide to the theory of NP-completeness. New York: Freeman.
131. Girlich, E., Kovalev, M., & Nalivaiko, V. (1998). A note on the extension of facet-defining digraphs (Technical report 23/98). Faculty of Mathematics, University of Magdeburg.
132. Glover, F., & Kochenberger, G. A. (2003). Handbook of metaheuristics. Dordrecht: Kluwer Academic.
133. Glover, F., Klastorin, T., & Klingman, D. (1974). Optimal weighted ancestry relationships. Management Science, 20, 1190-1193.
134. Goddard, S. T. (1983). Tournament rankings. Management Science, 29(12), 1385-1392.
135. Goemans, M. X., & Hall, L. A. (1996). The strongest facets of the acyclic subgraph polytope are unknown. In W. H. Cunningham, S. T. McCormick, & M. Queyranne (Eds.), Lecture notes in computer science: Vol. 1084. Integer programming and optimization (pp. 415-429). Berlin: Springer.
136. Goldberg, M. (1966). Results on the automorphism group of a graph. M. Sc. Thesis, university of Alberta.
137. Goldberg, D. E. (1989). Genetic algorithms in search, optimization and machine learning. Reading: Addison-Wesley.
138. González, C. G., & Pérez-Brito, D. (2001). A variable neighborhood search for solving the linear ordering problem. In Proceedings of MIC'2001-4th metaheuristics international conference (pp. 181-185).
139. Greistorfer, P. (2004). Experimental pool design: input, output and combination strategies for scatter search. In M. G. C. Resende & J. P. de Sousa (Eds.), Metaheuristics: computer decision making (pp. 279-300). Dordrecht: Kluwer Academic.
140. Grindberg, E., & Dambit, Y. (1965). Some properties of graphs containing circuits. Latvijskij Matematičeskij Ežegodnik (pp. 65-70) (in Russian).
141. Grötschel, M., Jünger, M., & Reinelt, G. (1984a). A cutting plane algorithm for the linear ordering problem. Operations Research, 32, 1195-1220.
142. Grötschel, M., Jünger, M., & Reinelt, G. (1984b). Optimal triangulation of large real-world input-output-matrices. Statistische Hefte, 25, 261-295.
143. Grötschel, M., Jünger, M., & Reinelt, G. (1985a). On the acyclic subgraph polytope. Mathematical Programming, 33, 1-27.
144. Grötschel, M., Jünger, M., & Reinelt, G. (1985b). Facets of the linear ordering polytope. Mathematical Programming, 33, 43-60.
145. Grötschel, M., Jünger, M., & Reinelt, G. (1985c). Acyclic subdigraphs and linear orderings: polytopes, facets, and a cutting plane algorithm. In I. Rival (Ed.), Graphs and orders (pp. 217-264). Dordrecht: Reidel.
146. Guénoche, A. (1977). Un algorithme pour pallier l'effet Condorcet. RAIRO, 11(1), 77-83.
147. Guénoche, A. (1986). A. B. C. D.: logiciel d'analyses booléennes et combinatoires des données, notice d'utilisation. G. R. T. C., Marseille.
148. Guénoche, A. (1988). Order at minimum distance of a valued tournament. Communication to Modélisation, analyse et agrégation des préférences et des choix (TRAP 3), Marseille-Luminy.
149. Guénoche, A. (1996). Vainqueurs de Kemeny et tournois difficiles. Mathématiques, Informatique et Sciences humaines, 133, 57-66.
150. Guénoche, A., Vandeputte-Riboud, B., & Denis, J.-B. (1994). Selecting varieties using a series of trials and a combinatorial ordering method. Agronomie, 14, 363-375.
151. Guénoche, A. (1995). How to choose according to partial evaluations. In B. Bouchon-Meunier et al. (Ed.), Lecture notes in computer sciences: Vol. 945. Advances in intelligent computing, IPMU'94 (pp. 611-618). Berlin-Heidelberg: Springer.
152. Guilbaud, G. T. (1952). Les théories de l'intérêt général et le problème logique de l'agrégation. Économie Appliquée 5(4). Reprint in Éléments de la théorie des jeux, Dunod, Paris, 1968.
153. Guy, R. K. (1967). A coarseness conjecture of Erdös. Journal of Combinatorial Theory, 3, 38-42.
154. Hansen, M. (1989). Approximation algorithms for geometric embeddings in the plane with applications to parallel processing problems. In Proceedings of the 30th annual symposium on foundations of computer science (pp. 604-609). Los Alamitos: IEEE Computer Society.
155. Hardouin Duparc, J. (1975). Quelques résultats sur l' « indice de transitivité » de certains tournois. Mathématiques et Sciences humaines, 51, 35-41.
156. Hassin, R., & Rubinstein, S. (1994). Approximations for the maximum acyclic subgraph problem. Information Processing Letters, 51(3), 133-140.
157. Hemaspaandra, L. (2000). Complexity classes. In K. H. Rosen (Ed.), Handbook of discrete and combinatorial mathematics (pp. 1085-1090). Boca Raton: CRC Press.
158. Hemaspaandra, E., Spakowski, H., & Vogel, J. (2005). The complexity of Kemeny elections. Theoretical Computer Science, 349, 382-391.
159. Huang, G., & Lim, A. (2003). Designing a hybrid genetic algorithm for the linear ordering problem. In LNCS: Vol. 2723. Proceedings of "genetic and evolutionary computation conference (GECCO) 2003", part I (pp. 1053-1064). Berlin: Springer.
160. Huber, L. (1976). Seriation using asymmetric proximity measures. British Journal of Mathematical & Statistical Psychology, 29, 32-52.
161. Hubert, L., & Schulz, J. (1975). Maximum likehood paired-comparison ranking and quadratic assignment. Biometrika, 62(3), 655-659.
162. Hudry, O. (1989). Recherche d'ordres médians: complexité, algorithmique et problèmes combinatoires. Ph. D. thesis, ENST, Paris.
163. Hudry, O. (1997a). Algorithms for the aggregation of ordinal preferences: a review. In Proceedings of the first conference on operations and quantitative management (ICOQM), Jaipur, India (pp. 169-176).
164. Hudry, O. (1997b). Nombre maximum d'ordres de Slater des tournois T vérifiant σ(T)=1. Mathématiques, Informatique et Sciences humaines, 140, 51-58.
165. Hudry, O. (1998). Tournois et optimisation combinatoire. Habilitation à diriger des recherches, université Paris 6, Paris, 1998.
166. Hudry, O. (2004). A note on "Banks winners in tournaments are difficult to recognize" by G. J. Woeginger. Social Choice and Welfare, 23, 113-114.
167. Hudry, O. (2008). NP-hardness results on the aggregation of linear orders into median orders. Annals of Operations Research, 163(1), 63-88.
168. Hudry, O. (2009a). A survey on the complexity of tournament solutions. Mathematical Social Sciences, 57, 292-303.
169. Hudry, O. (2009b). Complexity of voting procedures. In R. Meyers (Ed.), Encyclopedia of complexity and systems science. New York: Springer.
170. Hudry, O. (2009c). Complexity of Kemeny's problems (submitted for publication).
171. Hudry, O. (2010). On the complexity of Slater's problems. European Journal of Operational Research203, 216-221.
172. Hudry, O., Leclerc, B., Monjardet, B., & Barthélemy, J.-P. (2009). Metric and latticial medians. In D. Dubois, M. Pirlot, D. Bouyssou and H. Prade (Eds.), Concepts and methods for decision aid (pp. 763-803). Hermès (to appear).
173. Isaak, G. (1995). Tournaments as feedback arc sets. The Electronic Journal of Combinatorics, 2, R20.
174. Isaak, G., & Tesman, B. (1991). The weighted reversing number of a digraph. Congressus Numerantium, 83, 115-124.
175. Jacquet-Lagrèze É. (1969). L'agrégation des opinions individuelles. Informatique et Sciences humaines, 4, 1-21.
176. Jung, H. A. (1970). On subgraphs without cycles in a tournament. In P. Erdös, A. Renyi, & V. T. Sös (Eds.), Combinatorial theory and its applications II (pp. 675-677). Amsterdam: North-Holland.
177. Jünger, M. (1985). Polyhedral combinatorics and the acyclic subdigraph problem. Berlin: Heldermann.
178. Kaas, R. (1981). A branch and bound algorithm for the acyclic subgraph problem. European Journal of Operational Research, 8, 355-362.
179. Kadane, J. B. (1966). Some equivalence classes in paired comparisons. Annals of Mathematical Statistics, 37, 488-494.
180. Kano, M., & Sakamoto, A. (1985). Ranking the vertices of a paired comparison digraph. SIAM Journal on Algebraic and Discrete Methods, 6(1), 79-92.
181. Karp, R. (1972). Reducibility among combinatorial problems. In R. E. Miller & J. W. Tatcher (Eds.), Complexity of computer computations (pp. 85-103). New York: Plenum.
182. Kaykobad, M., Ahmed, Q. N. U., Shafiqul Khalid, A. T. M., & Bakhtiar, R.-A. (1995). A new algorithm for ranking players of a round-robin tournament. Computers and Operations Research, 22(2), 221-226.
183. Kemeny, J. G. (1959). Mathematics without numbers. Daedalus, 88, 577-591.
184. Kendall, M. G. (1938). Rank correlation methods. New York: Hafner.
185. Kendall, M. G., & Babington Smith, B. (1940). On the method of paired comparisons. Biometrika, 33, 239-251.
186. Klamler, C. (2003). Kemeny's rule and Slater's rule: a binary comparison. Economics Bulletin, 4(35), 1-7.
187. Klamler, C. (2004). The Dodgson ranking and its relation to Kemeny's method and Slater's rule. Social Choice and Welfare, 23, 91-102.
188. Koppen, M. (1995). Random utility representation of binary choice probabilities: critical graphs yielding critical necessary conditions. Journal of Mathematical Psychology, 19, 21-39.
189. Korte, B. (1979). Approximation algorithms for discrete optimization problems. Annals of Discrete Mathematics, 4, 85-120.
190. Korte, B., & Oberhofer, W. (1968). Zwei Algorithmen zur Lösung eines komplexen Reihenfolgeproblems. Unternehmensforschung, 12, 217-231.
191. Korte, B., & Oberhofer, W. (1969). Zur Triangulation von Input-Output-Matrizen. Jahrbucher für Nationaloekonomie und Statistik, 182, 398-433.
192. Kotzig, A. (1975). On the maximal order of cyclicity of antisymmetric directed graphs. Discrete Mathematics, 12, 17-25.
193. Lad, A., Ghani, R., Yang, Y., & Kisiel, B. (2009). Towards optimal ordering of prediction tasks. SIAM international conference on data mining (SDM09) (pp. 883-893).
194. Laffond, G., & Laslier, J.-F. (1991). Slater's winners of a tournament may not be in the Banks set. Social Choice and Welfare, 8, 355-363.
195. Laffond, G., Laslier, J.-F., & Le Breton, M. (1993). The Bipartisan set of a tournament game. Games and Economic Behavior, 5, 182-201.
196. Laffond, G., Laslier, J.-F., & Le Breton, M. (1991a). Choosing from a tournament: a progress report and some new results (Technical report). CNAM, Paris.
197. Laffond, G., Laslier, J.-F., & Le Breton, M. (1991b). A game-theoretical method for ranking the participants in a tournament (Technical report). CNAM, Paris.
198. Laguna, M., Marti, R., & Campos, V. (1999). Intensification and diversification with elite tabu search solutions for the linear ordering problem. Computers and Operations Research, 26(12), 1217-1230.
199. Landau, H. G. (1953). On dominance relations and the structure of animal societies III. The condition for a score structure. Bulletin of Mathematical Biophysics, 13, 1-19.
200. Laslier, J.-F. (1997). Tournament solutions and majority voting. Berlin, Heidelberg, New York: Springer.
201. Lawler, E. L. (1964). A comment on minimum feedback arc sets. IEEE Transactions on Circuit Theory, 11, 296-297.
202. Leighton, T., & Rao, S. (1988). An approximation max-flow min-cut theorem for uniform multicommodity flow problems with applications to approximation algorithms. In Proceedings of the 29th annual symposium on foundations of computer science (pp. 422-431).
203. Lemaréchal, C. (2003). The omnipresence of Lagrange. 4OR, 1(1), 7-25.
204. Lempel, A., & Cederbaum, I. (1966). Minimum feedback arc and vertex sets of a directed graph. IEEE Transactions on Circuit Theory, 13, 399-403.
205. Lenstra, J. K. (1977). Mathematical centre tracts: Vol. 69. Sequencing by enumerative methods. Amsterdam: Mathematisch Centrum.
206. Lenstra, H. W. Jr. (1973). The acyclic subgraph problem (Technical report BW26). Mathematisch Centrum, Amsterdam.
207. Leung, J., & Lee, J. (1994). More facets from fences for linear ordering and acyclic subgraph polytopes. Discrete Applied Mathematics, 50, 185-200.
208. Loiseau, I., Mendez-Dias, I., & Nasini, G. (1993). Determinacion del rango disyuntivo de facetas del problema de ordinacion lineal. Anales XXII JAIIO (pp. 124-130).
209. Marcotorchino, J.-F., & Michaud, P. (1979). Optimisation en analyse ordinale de données. Paris: Masson.
210. Matoušek, J., & Nešetřil, J. (1998). Invitation to discrete mathematics. London: Clarendon Press, Oxford University Press.
211. McGarvey, D. (1953). A theorem on the construction of voting paradoxes. Econometrica, 21, 608-610.
212. McKey, B. (2006). http://cs. anu. edu. au/~bdm/data/digraphs. html.
213. McLean, I. (1995). The first golden age of social choice, 1784-1803. In W. A. Barnett, H. Moulin, M. Salles, & N. J. Schofield (Eds.), Social choice, welfare, and ethics: proceedings of the eighth international symposium in economic theory, econometrics (pp. 13-33). Cambridge: Cambridge University Press.
214. McLean, I., & Hewitt, F. (1994). Condorcet: foundations of social choice and political theory. Cheltenham Glos: Edward Elgar.
215. McLean, I., & Urken, A. (1995). Classics of social choice. Ann Arbor: University of Michigan Press.
216. McLean, I., & Urken, A. (1997). La réception des œuvres de Condorcet sur le choix social (1794-1803): Lhuilier, Morales et Daunou. In A.-M. Chouillet, P. Crépel (Eds.), Condorcet, homme des lumières et de la révolution (pp. 147-160). ENS éditions, Fontenay-aux-roses.
217. McLean, I., Lorrey, H., & Colomer, J. M. (2008). Social choice in medieval Europe. Electronic Journal for History of Probability and Statistics, 4(1).
218. Méndez-Díaz, I., Nasini, G., & Zabala, P. (2009). The disjunctive rank and cutting plane algorithms using of facets of the linear ordering polytope (submitted for publication).
219. Mendonça, D., & Raghavachari, M. (2000). Comparing the efficacy of ranking methods for multiple round-robin tournaments. European Journal of Operational Research, 123, 593-605.
220. Merchant, D. K., & Rao, M. R. (1976). Majority decisions and transitivity: some special cases. Management Science, 23(2), 125-130.
221. Miller, N. (1980). A new solution set for tournaments and majority voting: Further graph-theoretical approaches to the theory of voting. American Journal of Political Science, 24(1), 68-96.
222. Mitchell, J. E. (2000). Computational experience with an interior point cutting plane algorithm. SIAM Journal on Optimization, 10(4), 1212-1227.
223. Mitchell, J. E. (2007). Generating linear ordering problems. http://www. rpi. edu/~mitchj/generators/linord/.
224. Mitchell, J. E., & Borchers, B. (1996). Solving real world linear ordering problems using a primal-dual interior point cutting plane method. Annals of Operations Research, 62, 253-276.
225. Mitchell, J. E., & Borchers, B. (2000). Solving linear ordering problems with a combined interior point/simplex cutting plane algorithm. In H. L. Frenk, K. Roos, T. Terlaky, & S. Zhang (Eds.), High performance optimization (pp. 349-366). Dordrecht: Kluwer Academic.
226. Monjardet, B. (1973). Tournois et ordres médians pour une opinion. Mathématiques et Sciences humaines, 43, 55-73.
227. Monjardet, B. (1979). Relations à éloignement minimum de relations binaires, note bibliographique. Mathématiques et Sciences humaines, 67, 115-122.
228. Monjardet, B. (1990). Sur diverses formes de la "règle de Condorcet" d'agrégation des préférences. Mathématiques, Informatique et Sciences humaines, 111, 61-71.
229. Monsuur, H., & Storcken, T. (1997). Measuring intransitivity. Mathematical Social Sciences, 34, 125-152.
230. Moon, J. W. (1968). Topics on tournaments. New York: Holt, Rinehart and Winston.
231. Nalivaiko, V. (1997). The linear ordering polytope. Preprint 39/97, Faculty of Mathematics of the Otto-von-Guericke-University of Magdeburg.
232. Nalivaiko, V. (1999). Das lineare Ordnungsproblem. Ph. D. Thesis, Falkultät für Mathematik, Universität Magdeburg, Magdeburg, Germany, 1999.
233. Newman, A. (2000). Approximating the maximum acyclic subgraph. Master's thesis, MIT, USA.
234. Newman, A. (2004). Cuts and orderings: on semidefinite relaxations for the linear ordering problem. In K. Jansen, S. Khanna, J. D. P. Rolim, & D. Ron (Eds.), Lectures notes in computer science (Vol. 3122, pp. 195-206). Berlin: Springer.
235. Newman, A., & Vempala, S. (2001). Fences are futile: on relaxations for the linear ordering problem. In Proceedings of the 8th conference on integer programming and combinatorial optimization (IPCO) (pp. 333-347).
236. Nishihara, O., Kumamoto, H., & Inoue, K. (1989). The new formulations of minimum feedback arc set problem. In C. Brexinski (Ed.), Numerical and applied mathematics. J. C. Baltzer AG, Scientific Publishing Co.
237. Nutov, Z., & Penn, M. (1995). On the integral dicycle packings and covers and the linear ordering polytope. Discrete Applied Mathematics, 60, 293-309.
238. Nutov, Z., & Penn, M. (1996). On non (0, 1/2, 1) extreme points of the generalized transitive tournament polytope. Linear Algebra and Its Applications, 233, 149-159.
239. Orlin, J. B. (1981). Unpublished manuscript.
240. Östergård, P. R. J., & Vaskelainen, V. P. (2009). A tournament of order 14 with disjoint Banks and Slater sets. Discrete Applied Mathematics (to appear).
241. Papadimitriou, C., & Yannakakis, M. (1991). Optimization, approximation, and complexity classes. Journal of Computer System Science, 43, 425-440.
242. Phillips, J. P. N. (1967). A procedure for determining Slater's i and all nearest adjoining orders. British Journal of Mathematical and Statistical Psychology, 20, 217-225.
243. Phillips, J. P. N. (1969). A further procedure for determining Slater's i and all nearest adjoining orders. British Journal of Mathematical and Statistical Psychology, 22, 97-101.
244. Phillips, J. P. N. (1976). On an algorithm of Smith and Payne for determining Slater's i and all nearest adjoining orders. British Journal of Mathematical and Statistical Psychology, 29, 126-127.
245. Poljak, S., & Turzík, D. (1986). A polynomial time heuristic for certain subgraph optimization problems with guaranteed lower bound. Discrete Mathematics, 58, 99-104.
246. Poljak, S., Rödl, V., & Spencer, J. (1988). Tournament ranking with expected profit in polynomial time. SIAM Journal Discrete Mathematics, 1(3), 372-376.
247. Raman, V., & Saurabh, S. (2006). Parameterized algorithms for feedback set problems and their duals in tournaments. Theoretical Computer Science, 351, 446-458.
248. Rao, S., & Richa, A. W. (2004). New approximation techniques for some linear ordering problems. SIAM Journal on Computing, 34(2), 388-404.
249. Ratliff, T. C. (2001). A comparison of the Dodgson method and Kemeny's rule. Social Choice and Welfare, 18, 79-90.
250. Rédei, L. (1934). Ein kombinatorischer Satz. Acta. Litt. Szeged, 7, 39-43.
251. Reid, K. B. (1969). On set of arcs containing no cycles in tournaments. Canadian Mathematical Bulletin, 12, 261-264.
252. Reid, K. B. (1983). Monochromatic reachability, complementary cycles and single arc reversals in tournaments. In Lecture notes in mathematics: Vol. 1073. Graph theory, proceedings of the first Southeast Asian graph theory colloquium held in Singapore (May 1983) (pp. 11-21). Berlin: Springer.
253. Reid, K. B. (2004). Tournaments. In J. L. Gross & J. Yellen (Eds.), Handbook of graph theory (pp. 156-184). Boca Raton: CRC Press.
254. Reid, K. B., & Beineke, L. W. (1978). Tournaments. In L. W. Beineke & R. J. Wilson (Eds.), Selected topics in graph theory (pp. 169-204). San Diego: Academic Press.
255. Reinelt, G. (1985). Research and exposition in mathematics: Vol. 8. The linear ordering problem: algorithms and applications. Berlin: Heldermann.
256. Reinelt, G. (1993). A note on small linear-ordering polytopes. Discrete & Computational Geometry, 10, 67-78.
257. Reinelt, G. (2002). Linear ordering library (LOLIB). http://www. iwr. uni-heidelberg. de/iwr/comopt/soft/LOLIB/LOLIB. html.
258. Remage, R., & Thompson, W. A. (1964). Rankings from paired comparison. Annals of Mathematical Statistics, 35, 739-747.
259. Remage, R., & Thompson, W. A. (1966). Maximum likelihood paired comparison rankings. Biometrika, 53, 143-149.
260. Righini G. (2008). A branch-and-bound algorithm for the linear ordering problem with cumulative costs. European Journal of Operational Research, 186(3), 965-971.
261. Rubinstein, A. (1980). Ranking the participants in a tournament. SIAM Journal of Applied Mathematics, 98, 108-111.
262. Ryser, H. J. (1964). Matrices of zeros and ones in combinatorial mathematics. In Recent advances in matrix theory (pp. 103-124). Madison: University of Wisconsin Press.
263. Sakuraba, C. S., & Yagiura, M. (2009). Efficient local search algorithms for the linear ordering problem (submitted for publication).
264. Schiavinotto, T., & Stützle, T. (2003). Search space analysis of the linear ordering problem. In G. R. Raidl et al. (Eds.), Lecture notes in computer science: Vol. 2611. Applications of evolutionary computing (pp. 322-333). Berlin: Springer.
265. Schiavinotto, T., & Stützle, T. (2004). The linear ordering problem: instances, search space analysis and algorithms. Journal of Mathematical Modelling and Algorithms, 3(4), 367-402.
266. Schrijver, A. (2003). Combinatorial optimization. Polyhedra and efficiency. Berlin: Springer.
267. Schwartz, T. (1990). Cyclic tournaments and cooperative majority voting: a solution. Social Choice and Welfare, 7, 19-29.
268. Seymour, P. (1995). Packing directed circuits fractionally. Combinatorica, 15(2), 281-288.
269. Slater, P. (1961). Inconsistencies in a schedule of paired comparisons. Biometrika, 48, 303-312.
270. Smith, W. D. (2007). http://rangevoting. org/PuzzDG. html.
271. Smith, A. F. M., & Payne, C. D. (1974). An algorithm for determining Slater's i and all nearest adjoining orders. British Journal of Mathematical and Statistical Psychology, 27, 49-52.
272. Spencer, J. (1971). Optimal ranking of tournaments. Networks, 1, 135-138.
273. Spencer, J. (1978). Nonconstructive methods in discrete mathematics. In G. C. Rota (Ed.), Studies in combinatorics (pp. 142-178). Washington: Mathematical Association of America.
274. Spencer, J. (1987). CBMS-NSF regional conference series in applied mathematics: Vol. 52. Ten lectures on the probabilistic method. Philadelphia: SIAM.
275. Stearns, R. (1959). The voting problem. American Mathematical Monthly, 66, 761-763.
276. Szele, T. (1943). Kombinatorikai vizsgálatok az irányított teljes gráffal kapesolatban. Matematikai és Fizikai Lapok, 50, 223-256. German translation: Untersuchungen über gerichtete vollständige Graphen. Publicationes Mathematical Debrecen, 13, 145-168 (1966).
277. Thomassen, C. (1975). Transversals of circuits in the lexicographic product of directed graphs. Mathématiques et Sciences humaines, 51, 43-45.
278. Thomassen, C. (1987). Counterexamples to Adám's conjecture on arc reversals in directed graphs. Journal of Combinatorial Theory Series B, 42, 128-130.
279. Tucker, A. W. (1960). On directed graphs and integer programs. In 1960 Symposium on combinatorial problems. Princeton: Princeton University.
280. Tüshaus, U. (1983). Mathematical systems in economics: Vol. 82. Aggregation binärer Relationen in der qualitativen Datenanalyse. Königstein: Verlagsgruppe Athenäum/Hain/Hanstein.
281. van Zuylen, A. (2005). Deterministic approximation algorithms for ranking and clusterings (Cornell ORIE Tech. Report No. 1431).
282. van Zuylen, A., & Williamson, D. P. (2008). Deterministic algorithms for rank aggregation and other ranking and clustering problems. In C. Kaklamanis & M. Skutella (Eds.), LNCS: Vol. 4927. Approximation and online algorithms (pp. 260-273). Berlin: Springer.
283. Vazirani, V. V. (2003). Approximation algorithms. Berlin: Springer.
284. Wakabayashi, Y. (1986). Aggregation of binary relations: algorithmic and polyhedral investigations. Ph. D. thesis, Augsburg university.
285. Wakabayashi, Y. (1998). The complexity of computing medians of relations. Resenhas, 3(3), 323-349.
286. Wei, T. (1952). The algebraic foundations of ranking theory. Ph. D. thesis, Cambridge University.
287. Wessels, H. (1981). Triangulation und Blocktriangulation von Input-Output Tabellen. Deutsches Institut für Wirtschaftsforschung: Beiträge zur Strukturforshung, Heft 63, Berlin.
288. Woeginger, G. J. (2003). Banks winners in tournaments are difficult to recognize. Social Choice and Welfare, 20(3), 523-528.
289. Woirgard, F. (1997). Recherche et dénombrement des ordres médians des tournois. Ph. D. thesis, ENST, Paris.
290. Wolsey, L. (1998). Integer programming. New York: Wiley.
291. Young, H. P. (1978). On permutations and permutation polytopes. Mathematical Programming Study, 8, 128-140.
292. Young, H. P. (1988). Condorcet theory of voting. American Political Science Review, 82, 1231-1244.
293. Younger, D. H. (1963). Minimum feedback arc sets for a directed graph. IEEE Transactions on Circuit Theory, 10(2), 238-245.
294. Zermelo, E. (1929). Die Berechnung der Turnier-Ergebnisse als ein maximal Problem der Warscheinlichkeistsrechnung. Mathematische Zeitung, 29, 436-460.