The asymmetric travelling salesman problem: on generalizations of disaggregated Miller–Tucker–Zemlin constraints

Abstract

In this paper we show that a multicommodity flow (MCF) model can be aggregated into a node-oriented model which in turn, can be seen as a disaggregation of the well-known Miller–Tucker–Zemlin model. Several outcomes of this node-oriented aggregation are also discussed: (i) the derivation of an “augmented” MCF model with a tighter linear programming (LP) relaxation and which is obtained by adding to MCF a disaggregated version of the Desrochers and Laporte inequalities together with a suitable set of linking constraints and (ii) the derivation of generalizations of the disaggregated Miller–Tucker–Zemlin constraints for paths. These generalized constraints can then be used to show that the LP relaxation of the new and tighter MCF model implies an exponentially sized set of lifted circuit inequalities (simple FD inequalities) which are known to be facet defining for the asymmetric travelling salesman polytope. Generalizations of the disaggregated Desrochers and Laporte inequalities which tighten the LP relaxation of the augmented MCF model are also proposed.