Integral cycle bases for cyclic timetabling

Cyclic railway timetables are typically modeled by a constraint graph GG with a cycle period time TT, in which a periodic tension xx in GG corresponds to a cyclic timetable. In this model, the periodic character of the tension xx is guaranteed by requiring periodicity for each cycle in a strictly fundamental cycle basis, that is, the set of cycles generated by the chords of a spanning tree of GG.We introduce the more general concept of integral cycle bases for characterizing periodic tensions. We characterize integral cycle bases using the determinant of a cycle basis, and investigate further properties of integral cycle bases.The periodicity of a single cycle is modeled by a so-called cycle integer variable. We exploit the wider class of integral cycle bases to find tighter bounds for these cycle integer variables, and provide various examples with tighter bounds. For cyclic railway timetabling in particular, we consider Minimum Cycle Bases for constructing integral cycle bases with tight bounds.