New facets of the STS polytope generated from known facets of the ATS polytope

While it had been known for a long time how to transform an asymmetric traveling salesman (ATS) problem on the complete graph with nn vertices into a symmetric traveling salesman (STS) problem on an incomplete graph with 2n2n vertices, no method was available for using this correspondence to derive facets of the symmetric polytope from facets of the asymmetric polytope until the work of E. Balas and M. Fischetti in [Lifted cycle inequalities for the asymmetric traveling salesman problem, Mathematics of Operations Research 24 (2) (1999) 273–292] suggested an approach. The original Balas–Fischetti method uses a standard sequential lifting procedure for the computation of the coefficient of the edges that are missing in the incomplete STS graph, which is a difficult task when addressing classes of (as opposed to single) inequalities. In this paper we introduce a systematic procedure for accomplishing the lifting task. The procedure exploits the structure of the tight STS tours and organizes them into a suitable tree structure. The potential of the method is illustrated by deriving large new classes of facet-defining STS inequalities.