A Bonsaï Branch and Bound Method Applied to Voting Theory

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

“…Several branch and bound methods, more or less sophisticated, have been designed to solve the real-life problems quoted above (and even to enumerate all the optimal solutions) or to solve randomly generated instances with n up to 100 (see references above and for example Bermond and Kodratoff 1976;Burkov and Groppen 1972;Charon et al 1992bCharon et al , 1996bCharon et al , 1997aHudry 2001b, 2006;Christof and Reinelt 2001;Cook and Saipe 1976;de Cani 1972;Flueck and Korsh 1974;Grötschel et al 1984aGrötschel et al , 1984bGrötschel et al , 1985aGrötschel et al , 1985bGuénoche 1977Guénoche , 1986Guénoche , 1988Guénoche , 1995Guénoche , 1996Guénoche et al 1994;Hudry 1989Hudry , 1998Jünger 1985;Kaas 1981;Korte andOberhofer 1968, 1969;Lenstra Jr. 1973;Borchers 1996, 2000;Phillips 1967Phillips , 1969Reinelt 1985;Remage and Thompson 1966;Tüshaus 1983;Wessels 1981;Woirgard 1997;Younger 1963). For instance, Korte andOberhofer (1968, 1969) solved random tournaments (with the same probability for the two orientations of each arc) with 13 vertices.…”

“…For instance, the use of a heap to code the best-first strategy branch and bound tree in Charon et al (1996b) reduces the complexity from O(n!) in Barthélemy et al (1989) down to O(n log n) to find the best leaf to expand.…”

“…In the tournament of Fig. 2, for which the missing arcs are assumed to be oriented from the left to the right, it is easy to show that b is the only Slater winner while its score is not the highest one (see Charon et al 1996c for more details). On the other hand, we may check easily (for instance from the list of the different patterns of non-isomorphic tournaments given in Moon 1968) that:…”

A survey on the linear ordering problem for weighted or unweighted tournaments

“…Several branch and bound methods, more or less sophisticated, have been designed to solve the real-life problems quoted above (and even to enumerate all the optimal solutions) or randomly generated instances with n up to 100 (see references above and for example Bermond and Kodratoff 1976;Burkov and Groppen 1972;de Cani 1972;Charon et al 1992bCharon et al , 1996bCharon et al , 1997aHudry 2001b, 2006;Christof and Reinelt 2001;Cook and Saipe 1976;Flueck and Korsh 1974;Grötschel et al 1984aGrötschel et al ,b, 1985aGuénoche 1977Guénoche , 1986Guénoche , 1988Guénoche , 1995Guénoche , 1996Guénoche et al 1994;Hudry 1989Hudry , 1998Jünger 1985;Kaas 1981;Korte andOberhofer 1968, 1969;Lenstra Jr. 1973;Mitchell andBorchers 1996, 2000;Phillips 1967Phillips , 1969Reinelt 1985;Remage and Thompson 1966;Tüshaus 1983;Wessels 1981;Woirgard 1997;Younger 1963). For instance, Korte andOberhofer (1968, 1969) solved random tournaments (with the same probability for the two orientations of each arc) with 13 vertices.…”

“…For instance, the use of a heap to code the best-first strategy branch and bound tree in Charon et al (1996b) reduces the complexity from O(n!) in Barthélemy et al (1989) down to O(nlog n) to find the best leaf to expand.…”