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Complexity of Cayley Distance and other General Metrics on Permutation Groups

Summary: Complexity of Cayley Distance and other General Metrics on
Permutation Groups
Thaynara Arielly de Lima and 1,2
Mauricio Ayala-Rinc´on
Grupo de Teoria da Computa¸c~ao, Departamentos de 1
Matem´atica e 2
Ci^encia da Computa¸c~ao
Universidade de Bras´ilia, Bras´ilia D.F., Brazil
Abstract. Permutation groups arise as important structures in group theory because many al-
gebraic properties about them are well-known, which makes modelling natural phenomena by
permutations of practical interest. This paper reviews the complexity of some problems involving
permutation groups. Usability of the involved algebraic notions is illustrated by problems such as
genome rearragement by reversals for which it is well-known that for the case of unsigned and
signed sorting by reversals the time complexity is, respectively, NP-hard and P. Reversal distance
is a particular metric and in this work more general metrics on permutation groups are considered
emphasizing on the Cayley distance. In particular, we point out an error in one of the polynomial
reductions applied in Pinch's approach atempting to proof that the subgroup distance problem for
Cayley distance is NP-complete and following his approach we present a simplified and correct


Source: Ayala-Rincón, Mauricio - Departamento de Matemática, Universidade de Brasília


Collections: Mathematics