# Separating over classes of TSP inequalities defined by 0 node-lifting in polynomial time

## Abstract

Many important cutting planes nave been discovered for the traveling salesman problem. Until recently (see [5] and [2]), little was known in the way of exact algorithms for separating these inequalities in polynomial time in the size of the fractional point *x** which is being separated. Any facet-defining inequality can be neatly categorized by the simple inequality which it is a 0 node-lifting of. Given a class of inequalities consisting of simple inequalities occuring on a fixed sized graph together with all their O node-liftings, an algorithm is presented here that separates over this class of inequalities in polynomial time. This algorithm uses a relaxation of the TSP which we will call cycle-shrink. Cycle-shrink is a compact description of the subtour polytope, and has some other theoretically interesting properties.

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