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

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Abstract

In this paper, we survey some results, conjectures and open problems dealing with the combinatorial and algorithmic aspects of the linear ordering problem. This problem consists in finding a linear order which is at minimum distance from a (weighted or not) tournament. We show how it can be used to model an aggregation problem consisting of going from individual preferences defined on a set of candidates to a collective ranking of these candidates.

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References

  • Adám, A. (1964). Problem. In Theory of graphs and its applications. Proc. Coll. Smolenice, Czech. Acad. Sci. Publ.

  • Adler, I., Alon, N., & Ross, S. M. (2001). On the number of Hamiltonian paths in tournaments. Random Structures and Algorithms, 18, 291–296.

    Google Scholar 

  • Ailon, N., & Alon, N. (2007). Hardness of fully dense problems. Information and Computation, 205, 117–1129.

    Google Scholar 

  • 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).

  • Ali, I., Cook, W. D., & Kress, M. (1986). On the minimum violations ranking of a tournament. Management Science, 32, 660–674.

    Google Scholar 

  • Alon, N. (1990). The maximum number of Hamiltonian paths in tournaments. Combinatorica, 10, 319–324.

    Google Scholar 

  • Alon, N. (2006). Ranking tournaments. SIAM Journal on Discrete Mathematics, 20(1), 137–142.

    Google Scholar 

  • Alon, N., & Spencer, J. (2000). The probabilistic method (2nd edn.). New York: Wiley.

    Google Scholar 

  • Alspach, B. (1967). Cycles of each length in regular tournaments. Canadian Mathematical Bulletin, 10, 283–286.

    Google Scholar 

  • Alspach, B. (1968). A combinatorial proof of a conjecture of Goldberg and Moon. Canadian Mathematical Bulletin, 11, 655–661.

    Google Scholar 

  • 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.

    Google Scholar 

  • 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).

  • Ausiello, G., Crescenzi, P., Gambosi, G., Kann, V., Marchetti-Spaccamela, A., & Protasi, M. (2003). Complexity and approximation (2nd edn.). Berlin: Springer.

    Google Scholar 

  • 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.

    Google Scholar 

  • Bang-Jensen, J., & Gutin, G. (2001). Digraphs: theory, algorithms, and applications. London, Berlin, Heidelberg: Springer.

    Google Scholar 

  • Banks, J. (1985). Sophisticated voting outcomes and agenda control. Social Choice and Welfare, 2, 295–306.

    Google Scholar 

  • Banks, J., Bordes, G., & Le Breton, M. (1991). Covering relations, closest orderings and Hamiltonian bypaths in tournaments. Social Choice and Welfare, 8, 355–363.

    Google Scholar 

  • Bar-Noy, A., & Naor, J. (1990). Sorting, minimal feedback sets, and Hamilton paths in tournaments. SIAM Journal on Discrete Mathematics, 3(1), 7–20.

    Google Scholar 

  • Barahona, F., Fonlupt, J., & Mahjoub, A. R. (1994). Compositions of graphs and polyhedra IV: acyclic spanning subgraph. SIDMA, 7(3), 390–402.

    Google Scholar 

  • Barbut, M. (1966). Note sur les ordres totaux à distance minimum d’une relation binaire donnée. Mathématiques et Sciences humaines, 17, 47–48.

    Google Scholar 

  • Barthélemy, J.-P., & Monjardet, B. (1981). The median procedure in cluster analysis and social choice theory. Mathematical Social Sciences, 1, 235–267.

    Google Scholar 

  • 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.

    Google Scholar 

  • 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.

    Google Scholar 

  • 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.

    Google Scholar 

  • Barthélemy, J.-P., Cohen, G., & Lobstein, A. (1996). Algorithmic complexity and communication problems. London: UCL Press.

    Google Scholar 

  • 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.

    Google Scholar 

  • Becker, O. (1967). Das Helmstädtersche Reihenfolgeproblem—die Effizienz verschiedener Näherungsverfahren. In: Computers uses in the social science. Vienna.

  • 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.

    Google Scholar 

  • Berge, C. (1985). Graphs. Amsterdam: North-Holland.

    Google Scholar 

  • Berger, B., & Shor, P. W. (1997). Tight bounds for the maximum acyclic subgraph problem. Journal of Algorithms, 25, 1–18.

    Google Scholar 

  • 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).

  • Bermond, J.-C. (1972). Ordres à distance minimum d’un tournoi et graphes partiels sans circuits maximaux. Mathématiques et Sciences humaines, 37, 5–25.

    Google Scholar 

  • 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,

    Google Scholar 

  • Bermond, J.-C., & Kodratoff, Y. (1976). Une heuristique pour le calcul de l’indice de transitivité d’un tournoi. RAIRO, 10(3), 83–92.

    Google Scholar 

  • Bertacco, L., Brunetta, L., & Fischetti, M. (2008). The Linear Ordering Problem with Cumulative Costs. European Journal of Operational Research, 189, 1345–1357.

    Google Scholar 

  • Black, D. (1958). The theory of committees and elections. Cambridge: Cambridge University Press.

    Google Scholar 

  • Blin, J. M., & Whinston, A. B. (1974). A note on majority rule under transitivity constraints. Management Science, 20, 1439–1440.

    Google Scholar 

  • Blin, J. M., & Whinston, A. B. (1975). Discriminant functions and majority voting. Management Science, 21, 1029–1041.

    Google Scholar 

  • Boenchendorf, K. (1982). Mathematical systems in economics : Vol. 74. Reihenfolgenprobleme/mean-flow-time sequencing. Königstein: Verlagsgruppe Athanäum/Hain/Scriptor/Hanstein.

    Google Scholar 

  • Bolotashvili, G., Kovalev, M., & Girlich, E. (1999). New facets of the linear ordering polytope. SIAM Journal on Discrete Mathematics, 12(3), 326–336.

    Google Scholar 

  • Borda, J.-C. (chevalier de) (1784). Mémoire sur les élections au scrutin. Histoire de l’Académie Royale des Sciences pour 1781, Paris.

  • Bowman, V. J., & Colantoni, C. S. (1973). Majority rule under transitivity constraints. Management Science, 19, 1029–1041.

    Google Scholar 

  • Brualdi, R. A., & Qiao, L. (1983). Upsets in round robin tournaments. Journal of Combinatorial Theory Series B, 35, 62–77.

    Google Scholar 

  • 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.

    Google Scholar 

  • Buchheim, C., Wiegele, A., & Zheng, L. (2009). Exact algorithms for the quadratic linear ordering problem. Informs Journal on Computing (to appear).

  • 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.

    Google Scholar 

  • Burkov, V. N., & Groppen, V. O. (1972). Branch cuts in strongly connected graphs and permutation potentials. Automation and Remote Control, 6, 111–119.

    Google Scholar 

  • Busch, A. H. (2006). A note on the number of Hamiltonian paths in strong tournaments. Electronic Journal of Combinatorics, 13, #N3.

    Google Scholar 

  • 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.

    Google Scholar 

  • 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.

    Google Scholar 

  • Campos, V., Laguna, M., & Martí, R. (2005). Context-independent scatter and tabu search for permutation problems. INFORMS Journal on Computing, 17(1), 331–339.

    Google Scholar 

  • Caspard, N., Monjardet, B., & Leclerc, B. (2007). Ensembles ordonnés finis : concepts, résultats et usages. Berlin: Springer.

    Google Scholar 

  • Chanas, S., & Kobylanski, P. (1996). A new heuristic algorithm solving the linear ordering problem. Computational Optimization and Applications, 6, 191–205.

    Google Scholar 

  • 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.

  • 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.

    Google Scholar 

  • Charon, I., & Hudry, O. (1998). Lamarckian genetic algorithms applied to the aggregation of preferences. Annals of Operations Research, 80, 281–297.

    Google Scholar 

  • Charon, I., & Hudry, O. (2000). Slater orders and Hamiltonian paths of tournaments. Electronic Notes in Discrete Mathematics, 5, 60–63.

    Google Scholar 

  • Charon, I., & Hudry, O. (2001a). The noising methods: a generalization of some metaheuristics. European Journal of Operational Research, 135(1), 86–101.

    Google Scholar 

  • 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).

    Google Scholar 

  • 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.

    Google Scholar 

  • Charon, I., & Hudry, O. (2003). Links between the Slater index and the Ryser index of tournaments. Graphs and Combinatorics, 19(3), 309–322.

    Google Scholar 

  • 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.

    Google Scholar 

  • Charon, I., & Hudry, O. (2007). A survey on the linear ordering problem for weighted or unweighted tournaments, 4OR, 5(1), 5–60.

    Google Scholar 

  • Charon, I., & Hudry, O. (2009a). Self-tuning of the noising methods. Optimization, 58(7), 1–21.

    Google Scholar 

  • Charon, I., & Hudry, O. (2009b). Branch and bound. In V. Paschos (Ed.) Handbook of combinatorial optimization. Wiley, New York (to appear)

  • Charon, I., & Hudry, O. (2009c). Maximum distance between Slater orders and Copeland orders of tournaments (submitted for publication).

  • Charon, I., & Hudry, O. (2009d). How different can Slater’s solutions and Kemeny’s solutions of tournaments be? (submitted for publication).

  • 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.

    Google Scholar 

  • 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.

    Google Scholar 

  • 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.

    Google Scholar 

  • 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

    Google Scholar 

  • Charon, I., Hudry, O., & Woirgard, F. (1996c). Ordres médians et ordres de Slater des tournois. Mathématiques, Informatique et Sciences humaines, 133, 23–56.

    Google Scholar 

  • Charon, I., Guénoche, A., Hudry, O., & Woirgard, F. (1997a). New results on the computation of median orders. Discrete Mathematics, 165–166, 139–154.

    Google Scholar 

  • 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.

    Google Scholar 

  • Chartrand, G., Geller, D., & Hedetniemi, S. (1971). Graphs with forbidden subgraphs. Journal of Combinatorial Theory, Series B, 10(1), 12–41.

    Google Scholar 

  • Chenery, H. B., & Watanabe, T. (1958). International comparisons of the structure of production. Econometrica, 26(4), 487–521.

    Google Scholar 

  • 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.

    Google Scholar 

  • Christof, T., & Reinelt, G. (1996). Combinatorial optimization and small polytopes. Top, 4(1), 1–64.

    Google Scholar 

  • Christof, T., & Reinelt, G. (1997a). Low-dimensional linear ordering polytopes. Working paper, University of Heidelberg.

  • Christof, T., & Reinelt, G. (1997b). Small instances relaxations for solving linear ordering problems by branch-and-cut (Technical report). University of Heidelberg.

  • Christof, T., & Reinelt, G. (2001). Algorithmic aspects of using small instance relaxations in parallel branch-and-cut. Algorithmica, 30(4), 507–629.

    Google Scholar 

  • Christophe, J., Doignon, J.-P., & Fiorini, S. (2004). The biorder. Order, 21/1, 61–82.

    Google Scholar 

  • Cohen, W., Schapire, R., & Singer, Y. (1999). Learning to order things. Journal of Artificial Intelligence Research, 10, 213–270.

    Google Scholar 

  • 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.

  • Congram, R. K. (2000). Polynomially searchable exponential neighbourhoods for sequencing problems in combinatorial optimisation. Ph.D. thesis, University of Southampton.

  • 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.

  • Cook, W. D., & Saipe, A. L. (1976). Committee approach to priority planning: the median ranking method. Cahiers du Centre d’Études et de Recherche Opérationnelle 18(3), 337–352.

    Google Scholar 

  • 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.

    Google Scholar 

  • Cook, W. D., Doyle, J., Green, R., & Kress, M. (1996). Ranking players in multiple tournaments. Computers and Operations Research, 23(9), 869–880.

    Google Scholar 

  • Copeland, A. H. (1951). A “reasonable” social welfare function. Seminar on applications of mathematics to social sciences, University of Michigan.

  • Coppersmith, D., & Winograd, S. (1990). Matrix multiplication via arithmetic progressions. Journal of Symbolic Computation, 9, 251–280.

    Google Scholar 

  • 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).

  • Czygrinow, A., Poljak, S., & Rödl, V. (1999). Constructive quasi-Ramsey numbers and tournament ranking. SIAM Journal on Discrete Mathematics, 12(1), 48–63.

    Google Scholar 

  • 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).

  • de Cani, J. S. (1969). Maximum likelihood paired comparison ranking by linear programming. Biometrika, 3, 537–545.

    Google Scholar 

  • de Cani, J. S. (1972). A branch and bound algorithm for maximum likelihood paired comparison ranking. Biometrika, 59, 131–135.

    Google Scholar 

  • 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.

    Google Scholar 

  • Debord, B. (1987a). Caractérisation des matrices de préférences nettes et méthodes d’agrégation associées. Mathématiques et Sciences humaines, 97, 5–17.

    Google Scholar 

  • Debord, B. (1987b). Axiomatisation de procédures d’agrégation de préférences. Ph.D. thesis, Université scientifique technologique et médicale de Grenoble.

  • Dixon, J. D. (1967). The maximum order of the group of a tournament. Canadian Mathematical Bulletin, 10, 503–505.

    Google Scholar 

  • Dodgson, C. L. (1876). A method of taking votes on more than two issues. Oxford: Clarendon.

    Google Scholar 

  • 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.

    Google Scholar 

  • 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.

    Google Scholar 

  • Downey, R. G., & Fellows, M. R. (1999). Parameterized complexity. Berlin: Springer.

    Google Scholar 

  • Dréo, J., Petrowski, A., Taillard, E., & Siarry, P. (2006). Metaheuristics for hard optimization. Methods and case studies. Berlin: Springer.

    Google Scholar 

  • Duarte, A., Laguna, M., & Marti, R. (2009). Tabu search for the linear ordering problem with cumulative costs. Computational Optmization and Applications (to appear).

  • Dugat, V. (1990). Décomposition de tournois réguliers: théorie et application aux algorithmes de tests d’isomorphisme. Ph.D. thesis, Université Paul Sabatier, Toulouse.

  • Dutta, B. (1988). Covering sets and a new Condorcet choice correspondence. Journal of Economic Theory, 44, 63–80.

    Google Scholar 

  • 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).

  • Eades, P., & Lin, X. (1995). A new heuristic for the feedback arc set problem. Australian Journal of Combinatorics, 12, 15–26.

    Google Scholar 

  • 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.

    Google Scholar 

  • Erdös, P. & Moon, J. W. (1965). On sets of consistent arcs in a tournament. Canadian Mathematical Bulletin, 8, 269–271.

    Google Scholar 

  • 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.

    Google Scholar 

  • Even, G., Naor, J. S., Sudan, M., & Schieber, B. (1998). Approximating minimum feedback sets and multicuts in directed graphs. Algorithmica, 20(2), 151–174.

    Google Scholar 

  • 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.

    Google Scholar 

  • Fagin, R., Kumar, R., Mahdian, M., Sivakumar, D., & Vee, E. (2005). Rank aggregation: an algorithmic perspective. Unpublished manuscript.

  • 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.

    Google Scholar 

  • Fiorini, S. (2001a). Polyhedral combinatorics of order polytopes. Ph.D. thesis, Université libre de Bruxelles.

  • Fiorini, S. (2001b). Determining the automorphism group of the linear ordering polytope. Discrete Applied Mathematics, 112/1–3, 121–128.

    Google Scholar 

  • Fiorini, S. (2006a). How to recycle your facets. Discrete Optimization, 3/2, 136–153.

    Google Scholar 

  • 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.

    Google Scholar 

  • Fiorini, S., & Fishburn, P. (2003). Facets of linear signed order polytopes. Discrete Applied Mathematics, 131/3, 597–610.

    Google Scholar 

  • Fishburn, P. (1977). Condorcet social choice functions. SIAM Journal of Applied Mathematics, 33, 469–489.

    Google Scholar 

  • Fishburn, P. (1992). Induced binary probabilities and the linear ordering polytope: a status report. Mathematical Social Sciences, 23, 67–80.

    Google Scholar 

  • 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.

    Google Scholar 

  • Flueck, J. A., & Korsh, J. F. (1974). A branch search algorithm for maximum likelihood paired comparison ranking. Biometrika, 61(3), 621–626.

    Google Scholar 

  • Fulkerson, D. R. (1965). Upsets in round robin tournaments. Canadian Journal of Mathematics, 17, 957–969.

    Google Scholar 

  • 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.

  • 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.

    Google Scholar 

  • Garey, M. R., & Johnson, D. S. (1979). Computers and intractability, a guide to the theory of NP-completeness. New York: Freeman.

    Google Scholar 

  • 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.

  • Glover, F., & Kochenberger, G. A. (2003). Handbook of metaheuristics. Dordrecht: Kluwer Academic.

    Google Scholar 

  • Glover, F., Klastorin, T., & Klingman, D. (1974). Optimal weighted ancestry relationships. Management Science, 20, 1190–1193.

    Google Scholar 

  • Goddard, S. T. (1983). Tournament rankings. Management Science, 29(12), 1385–1392.

    Google Scholar 

  • 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.

    Google Scholar 

  • Goldberg, M. (1966). Results on the automorphism group of a graph. M.Sc. Thesis, university of Alberta.

  • Goldberg, D. E. (1989). Genetic algorithms in search, optimization and machine learning. Reading: Addison-Wesley.

    Google Scholar 

  • 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).

  • 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.

    Google Scholar 

  • Grindberg, E., & Dambit, Y. (1965). Some properties of graphs containing circuits. Latvijskij Matematičeskij Ežegodnik (pp. 65–70) (in Russian).

  • Grötschel, M., Jünger, M., & Reinelt, G. (1984a). A cutting plane algorithm for the linear ordering problem. Operations Research, 32, 1195–1220.

    Google Scholar 

  • Grötschel, M., Jünger, M., & Reinelt, G. (1984b). Optimal triangulation of large real-world input-output-matrices. Statistische Hefte, 25, 261–295.

    Google Scholar 

  • Grötschel, M., Jünger, M., & Reinelt, G. (1985a). On the acyclic subgraph polytope. Mathematical Programming, 33, 1–27.

    Google Scholar 

  • Grötschel, M., Jünger, M., & Reinelt, G. (1985b). Facets of the linear ordering polytope. Mathematical Programming, 33, 43–60.

    Google Scholar 

  • 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.

    Google Scholar 

  • Guénoche, A. (1977). Un algorithme pour pallier l’effet Condorcet. RAIRO, 11(1), 77–83.

    Google Scholar 

  • Guénoche, A. (1986). A. B. C. D. : logiciel d’analyses booléennes et combinatoires des données, notice d’utilisation. G. R. T. C., Marseille.

  • 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.

  • Guénoche, A. (1996). Vainqueurs de Kemeny et tournois difficiles. Mathématiques, Informatique et Sciences humaines, 133, 57–66.

    Google Scholar 

  • 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.

    Google Scholar 

  • 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.

    Google Scholar 

  • 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.

  • Guy, R. K. (1967). A coarseness conjecture of Erdös. Journal of Combinatorial Theory, 3, 38–42.

    Google Scholar 

  • 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.

    Google Scholar 

  • Hardouin Duparc, J. (1975). Quelques résultats sur l’ «  indice de transitivité » de certains tournois. Mathématiques et Sciences humaines, 51, 35–41.

    Google Scholar 

  • Hassin, R., & Rubinstein, S. (1994). Approximations for the maximum acyclic subgraph problem. Information Processing Letters, 51(3), 133–140.

    Google Scholar 

  • Hemaspaandra, L. (2000). Complexity classes. In K. H. Rosen (Ed.), Handbook of discrete and combinatorial mathematics (pp. 1085–1090). Boca Raton: CRC Press.

    Google Scholar 

  • Hemaspaandra, E., Spakowski, H., & Vogel, J. (2005). The complexity of Kemeny elections. Theoretical Computer Science, 349, 382–391.

    Google Scholar 

  • 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.

    Google Scholar 

  • Huber, L. (1976). Seriation using asymmetric proximity measures. British Journal of Mathematical & Statistical Psychology, 29, 32–52.

    Google Scholar 

  • Hubert, L., & Schulz, J. (1975). Maximum likehood paired-comparison ranking and quadratic assignment. Biometrika, 62(3), 655–659.

    Google Scholar 

  • Hudry, O. (1989). Recherche d’ordres médians : complexité, algorithmique et problèmes combinatoires. Ph.D. thesis, ENST, Paris.

  • 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).

  • Hudry, O. (1997b). Nombre maximum d’ordres de Slater des tournois T vérifiant σ(T)=1. Mathématiques, Informatique et Sciences humaines, 140, 51–58.

    Google Scholar 

  • Hudry, O. (1998). Tournois et optimisation combinatoire. Habilitation à diriger des recherches, université Paris 6, Paris, 1998.

  • 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.

    Google Scholar 

  • Hudry, O. (2008). NP-hardness results on the aggregation of linear orders into median orders. Annals of Operations Research, 163(1), 63–88.

    Google Scholar 

  • Hudry, O. (2009a). A survey on the complexity of tournament solutions. Mathematical Social Sciences, 57, 292–303.

    Google Scholar 

  • Hudry, O. (2009b). Complexity of voting procedures. In R. Meyers (Ed.), Encyclopedia of complexity and systems science. New York: Springer.

    Google Scholar 

  • Hudry, O. (2009c). Complexity of Kemeny’s problems (submitted for publication).

  • Hudry, O. (2010). On the complexity of Slater’s problems. European Journal of Operational Research 203, 216–221.

    Google Scholar 

  • 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).

  • Isaak, G. (1995). Tournaments as feedback arc sets. The Electronic Journal of Combinatorics, 2, R20.

    Google Scholar 

  • Isaak, G., & Tesman, B. (1991). The weighted reversing number of a digraph. Congressus Numerantium, 83, 115–124.

    Google Scholar 

  • Jacquet-Lagrèze É. (1969). L’agrégation des opinions individuelles. Informatique et Sciences humaines, 4, 1–21.

    Google Scholar 

  • 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.

    Google Scholar 

  • Jünger, M. (1985). Polyhedral combinatorics and the acyclic subdigraph problem. Berlin: Heldermann.

    Google Scholar 

  • Kaas, R. (1981). A branch and bound algorithm for the acyclic subgraph problem. European Journal of Operational Research, 8, 355–362.

    Google Scholar 

  • Kadane, J. B. (1966). Some equivalence classes in paired comparisons. Annals of Mathematical Statistics, 37, 488–494.

    Google Scholar 

  • Kano, M., & Sakamoto, A. (1985). Ranking the vertices of a paired comparison digraph. SIAM Journal on Algebraic and Discrete Methods, 6(1), 79–92.

    Google Scholar 

  • 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.

    Google Scholar 

  • 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.

    Google Scholar 

  • Kemeny, J. G. (1959). Mathematics without numbers. Daedalus, 88, 577–591.

    Google Scholar 

  • Kendall, M. G. (1938). Rank correlation methods. New York: Hafner.

    Google Scholar 

  • Kendall, M. G., & Babington Smith, B. (1940). On the method of paired comparisons. Biometrika, 33, 239–251.

    Google Scholar 

  • Klamler, C. (2003). Kemeny’s rule and Slater’s rule: a binary comparison. Economics Bulletin, 4(35), 1–7.

    Google Scholar 

  • Klamler, C. (2004). The Dodgson ranking and its relation to Kemeny’s method and Slater’s rule. Social Choice and Welfare, 23, 91–102.

    Google Scholar 

  • Koppen, M. (1995). Random utility representation of binary choice probabilities: critical graphs yielding critical necessary conditions. Journal of Mathematical Psychology, 19, 21–39.

    Google Scholar 

  • Korte, B. (1979). Approximation algorithms for discrete optimization problems. Annals of Discrete Mathematics, 4, 85–120.

    Google Scholar 

  • Korte, B., & Oberhofer, W. (1968). Zwei Algorithmen zur Lösung eines komplexen Reihenfolgeproblems. Unternehmensforschung, 12, 217–231.

    Google Scholar 

  • Korte, B., & Oberhofer, W. (1969). Zur Triangulation von Input-Output-Matrizen. Jahrbucher für Nationaloekonomie und Statistik, 182, 398–433.

    Google Scholar 

  • Kotzig, A. (1975). On the maximal order of cyclicity of antisymmetric directed graphs. Discrete Mathematics, 12, 17–25.

    Google Scholar 

  • 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).

  • 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.

    Google Scholar 

  • Laffond, G., Laslier, J.-F., & Le Breton, M. (1993). The Bipartisan set of a tournament game. Games and Economic Behavior, 5, 182–201.

    Google Scholar 

  • Laffond, G., Laslier, J.-F., & Le Breton, M. (1991a). Choosing from a tournament: a progress report and some new results (Technical report). CNAM, Paris.

  • Laffond, G., Laslier, J.-F., & Le Breton, M. (1991b). A game-theoretical method for ranking the participants in a tournament (Technical report). CNAM, Paris.

  • 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.

    Google Scholar 

  • 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.

    Google Scholar 

  • Laslier, J.-F. (1997). Tournament solutions and majority voting. Berlin, Heidelberg, New York: Springer.

    Google Scholar 

  • Lawler, E. L. (1964). A comment on minimum feedback arc sets. IEEE Transactions on Circuit Theory, 11, 296–297.

    Google Scholar 

  • 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).

  • Lemaréchal, C. (2003). The omnipresence of Lagrange. 4OR, 1(1), 7–25.

    Google Scholar 

  • Lempel, A., & Cederbaum, I. (1966). Minimum feedback arc and vertex sets of a directed graph. IEEE Transactions on Circuit Theory, 13, 399–403.

    Google Scholar 

  • Lenstra, J. K. (1977). Mathematical centre tracts : Vol. 69. Sequencing by enumerative methods. Amsterdam: Mathematisch Centrum.

    Google Scholar 

  • Lenstra, H. W. Jr. (1973). The acyclic subgraph problem (Technical report BW26). Mathematisch Centrum, Amsterdam.

  • Leung, J., & Lee, J. (1994). More facets from fences for linear ordering and acyclic subgraph polytopes. Discrete Applied Mathematics, 50, 185–200.

    Google Scholar 

  • 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).

  • Marcotorchino, J.-F., & Michaud, P. (1979). Optimisation en analyse ordinale de données. Paris: Masson.

    Google Scholar 

  • Matoušek, J., & Nešetřil, J. (1998). Invitation to discrete mathematics. London: Clarendon Press, Oxford University Press.

    Google Scholar 

  • McGarvey, D. (1953). A theorem on the construction of voting paradoxes. Econometrica, 21, 608–610.

    Google Scholar 

  • McKey, B. (2006). http://cs.anu.edu.au/~bdm/data/digraphs.html.

  • 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.

    Google Scholar 

  • McLean, I., & Hewitt, F. (1994). Condorcet: foundations of social choice and political theory. Cheltenham Glos: Edward Elgar.

    Google Scholar 

  • McLean, I., & Urken, A. (1995). Classics of social choice. Ann Arbor: University of Michigan Press.

    Google Scholar 

  • 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.

  • McLean, I., Lorrey, H., & Colomer, J. M. (2008). Social choice in medieval Europe. Electronic Journal for History of Probability and Statistics, 4(1).

  • 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).

  • Mendonça, D., & Raghavachari, M. (2000). Comparing the efficacy of ranking methods for multiple round-robin tournaments. European Journal of Operational Research, 123, 593–605.

    Google Scholar 

  • Merchant, D. K., & Rao, M. R. (1976). Majority decisions and transitivity: some special cases. Management Science, 23(2), 125–130.

    Google Scholar 

  • 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.

    Google Scholar 

  • Mitchell, J. E. (2000). Computational experience with an interior point cutting plane algorithm. SIAM Journal on Optimization, 10(4), 1212–1227.

    Google Scholar 

  • Mitchell, J. E. (2007). Generating linear ordering problems. http://www.rpi.edu/~mitchj/generators/linord/.

  • 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.

    Google Scholar 

  • 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.

    Google Scholar 

  • Monjardet, B. (1973). Tournois et ordres médians pour une opinion. Mathématiques et Sciences humaines, 43, 55–73.

    Google Scholar 

  • Monjardet, B. (1979). Relations à éloignement minimum de relations binaires, note bibliographique. Mathématiques et Sciences humaines, 67, 115–122.

    Google Scholar 

  • Monjardet, B. (1990). Sur diverses formes de la “règle de Condorcet” d’agrégation des préférences. Mathématiques, Informatique et Sciences humaines, 111, 61–71.

    Google Scholar 

  • Monsuur, H., & Storcken, T. (1997). Measuring intransitivity. Mathematical Social Sciences, 34, 125–152.

    Google Scholar 

  • Moon, J. W. (1968). Topics on tournaments. New York: Holt, Rinehart and Winston.

    Google Scholar 

  • Nalivaiko, V. (1997). The linear ordering polytope. Preprint 39/97, Faculty of Mathematics of the Otto-von-Guericke-University of Magdeburg.

  • Nalivaiko, V. (1999). Das lineare Ordnungsproblem. Ph.D. Thesis, Falkultät für Mathematik, Universität Magdeburg, Magdeburg, Germany, 1999.

  • Newman, A. (2000). Approximating the maximum acyclic subgraph. Master’s thesis, MIT, USA.

  • 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.

    Google Scholar 

  • 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).

  • 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.

  • Nutov, Z., & Penn, M. (1995). On the integral dicycle packings and covers and the linear ordering polytope. Discrete Applied Mathematics, 60, 293–309.

    Google Scholar 

  • 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.

    Google Scholar 

  • Orlin, J. B. (1981). Unpublished manuscript.

  • Östergård, P. R. J., & Vaskelainen, V. P. (2009). A tournament of order 14 with disjoint Banks and Slater sets. Discrete Applied Mathematics (to appear).

  • Papadimitriou, C., & Yannakakis, M. (1991). Optimization, approximation, and complexity classes. Journal of Computer System Science, 43, 425–440.

    Google Scholar 

  • 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.

    Google Scholar 

  • 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.

    Google Scholar 

  • 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.

    Google Scholar 

  • Poljak, S., & Turzík, D. (1986). A polynomial time heuristic for certain subgraph optimization problems with guaranteed lower bound. Discrete Mathematics, 58, 99–104.

    Google Scholar 

  • Poljak, S., Rödl, V., & Spencer, J. (1988). Tournament ranking with expected profit in polynomial time. SIAM Journal Discrete Mathematics, 1(3), 372–376.

    Google Scholar 

  • Raman, V., & Saurabh, S. (2006). Parameterized algorithms for feedback set problems and their duals in tournaments. Theoretical Computer Science, 351, 446–458.

    Google Scholar 

  • Rao, S., & Richa, A. W. (2004). New approximation techniques for some linear ordering problems. SIAM Journal on Computing, 34(2), 388–404.

    Google Scholar 

  • Ratliff, T. C. (2001). A comparison of the Dodgson method and Kemeny’s rule. Social Choice and Welfare, 18, 79–90.

    Google Scholar 

  • Rédei, L. (1934). Ein kombinatorischer Satz. Acta. Litt. Szeged, 7, 39–43.

    Google Scholar 

  • Reid, K. B. (1969). On set of arcs containing no cycles in tournaments. Canadian Mathematical Bulletin, 12, 261–264.

    Google Scholar 

  • 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.

    Google Scholar 

  • Reid, K. B. (2004). Tournaments. In J. L. Gross & J. Yellen (Eds.), Handbook of graph theory (pp. 156–184). Boca Raton: CRC Press.

    Google Scholar 

  • 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.

    Google Scholar 

  • Reinelt, G. (1985). Research and exposition in mathematics : Vol. 8. The linear ordering problem: algorithms and applications. Berlin: Heldermann.

    Google Scholar 

  • Reinelt, G. (1993). A note on small linear-ordering polytopes. Discrete & Computational Geometry, 10, 67–78.

    Google Scholar 

  • Reinelt, G. (2002). Linear ordering library (LOLIB). http://www.iwr.uni-heidelberg.de/iwr/comopt/soft/LOLIB/LOLIB.html.

  • Remage, R., & Thompson, W. A. (1964). Rankings from paired comparison. Annals of Mathematical Statistics, 35, 739–747.

    Google Scholar 

  • Remage, R., & Thompson, W. A. (1966). Maximum likelihood paired comparison rankings. Biometrika, 53, 143–149.

    Google Scholar 

  • 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.

    Google Scholar 

  • Rubinstein, A. (1980). Ranking the participants in a tournament. SIAM Journal of Applied Mathematics, 98, 108–111.

    Google Scholar 

  • 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.

    Google Scholar 

  • Sakuraba, C. S., & Yagiura, M. (2009). Efficient local search algorithms for the linear ordering problem (submitted for publication).

  • 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.

    Google Scholar 

  • 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.

    Google Scholar 

  • Schrijver, A. (2003). Combinatorial optimization. Polyhedra and efficiency. Berlin: Springer.

    Google Scholar 

  • Schwartz, T. (1990). Cyclic tournaments and cooperative majority voting: a solution. Social Choice and Welfare, 7, 19–29.

    Google Scholar 

  • Seymour, P. (1995). Packing directed circuits fractionally. Combinatorica, 15(2), 281–288.

    Google Scholar 

  • Slater, P. (1961). Inconsistencies in a schedule of paired comparisons. Biometrika, 48, 303–312.

    Google Scholar 

  • Smith, W. D. (2007). http://rangevoting.org/PuzzDG.html.

  • 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.

    Google Scholar 

  • Spencer, J. (1971). Optimal ranking of tournaments. Networks, 1, 135–138.

    Google Scholar 

  • Spencer, J. (1978). Nonconstructive methods in discrete mathematics. In G. C. Rota (Ed.), Studies in combinatorics (pp. 142–178). Washington: Mathematical Association of America.

    Google Scholar 

  • Spencer, J. (1987). CBMS-NSF regional conference series in applied mathematics : Vol. 52. Ten lectures on the probabilistic method. Philadelphia: SIAM.

    Google Scholar 

  • Stearns, R. (1959). The voting problem. American Mathematical Monthly, 66, 761–763.

    Google Scholar 

  • 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).

    Google Scholar 

  • Thomassen, C. (1975). Transversals of circuits in the lexicographic product of directed graphs. Mathématiques et Sciences humaines, 51, 43–45.

    Google Scholar 

  • Thomassen, C. (1987). Counterexamples to Adám’s conjecture on arc reversals in directed graphs. Journal of Combinatorial Theory Series B, 42, 128–130.

    Google Scholar 

  • Tucker, A. W. (1960). On directed graphs and integer programs. In 1960 Symposium on combinatorial problems. Princeton: Princeton University.

    Google Scholar 

  • Tüshaus, U. (1983). Mathematical systems in economics : Vol. 82. Aggregation binärer Relationen in der qualitativen Datenanalyse. Königstein: Verlagsgruppe Athenäum/Hain/Hanstein.

    Google Scholar 

  • van Zuylen, A. (2005). Deterministic approximation algorithms for ranking and clusterings (Cornell ORIE Tech. Report No. 1431).

  • 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.

    Google Scholar 

  • Vazirani, V. V. (2003). Approximation algorithms. Berlin: Springer.

    Google Scholar 

  • Wakabayashi, Y. (1986). Aggregation of binary relations: algorithmic and polyhedral investigations. Ph.D. thesis, Augsburg university.

  • Wakabayashi, Y. (1998). The complexity of computing medians of relations. Resenhas, 3(3), 323–349.

    Google Scholar 

  • Wei, T. (1952). The algebraic foundations of ranking theory. Ph.D. thesis, Cambridge University.

  • Wessels, H. (1981). Triangulation und Blocktriangulation von Input-Output Tabellen. Deutsches Institut für Wirtschaftsforschung: Beiträge zur Strukturforshung, Heft 63, Berlin.

  • Woeginger, G. J. (2003). Banks winners in tournaments are difficult to recognize. Social Choice and Welfare, 20(3), 523–528.

    Google Scholar 

  • Woirgard, F. (1997). Recherche et dénombrement des ordres médians des tournois. Ph.D. thesis, ENST, Paris.

  • Wolsey, L. (1998). Integer programming. New York: Wiley.

    Google Scholar 

  • Young, H. P. (1978). On permutations and permutation polytopes. Mathematical Programming Study, 8, 128–140.

    Google Scholar 

  • Young, H. P. (1988). Condorcet theory of voting. American Political Science Review, 82, 1231–1244.

    Google Scholar 

  • Younger, D. H. (1963). Minimum feedback arc sets for a directed graph. IEEE Transactions on Circuit Theory, 10(2), 238–245.

    Google Scholar 

  • Zermelo, E. (1929). Die Berechnung der Turnier-Ergebnisse als ein maximal Problem der Warscheinlichkeistsrechnung. Mathematische Zeitung, 29, 436–460.

    Google Scholar 

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This is an updated version of the paper that appeared in 4OR, 5(1), 5–60 (2007).

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Charon, I., Hudry, O. An updated survey on the linear ordering problem for weighted or unweighted tournaments. Ann Oper Res 175, 107–158 (2010). https://doi.org/10.1007/s10479-009-0648-7

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