Set TSP problem

In combinatorial optimization, the set TSP, also known as the generalized TSP, group TSP, One-of-a-Set TSP, Multiple Choice TSP or Covering Salesman Problem, is a generalization of the traveling salesman problem, whereby it is required to find a shortest tour in a graph which visits all specified subsets of the vertices of a graph. The subsets of vertices must be disjoint, since the case of overlapping subsets can be reduced to the case of disjoint ones. The ordinary TSP is a special case of the set TSP when all subsets to be visited are singletons. Therefore, the set TSP is also NP-hard.
There is a transformation for an instance of the set TSP to an instance of the standard asymmetric TSP. The idea is to connect each subset into a directed cycle with edges of zero weight, and inherit the outgoing edges from the original graph shifting by one vertex backwards along this cycle. The salesman, when visiting a vertex v in some subset, walks around the cycle for free and exits it from the vertex preceding v by an outgoing edge corresponding to an outgoing edge of v in the original graph.
The Set TSP has a lot of interesting applications in several path planning problems. For example, a two vehicle cooperative routing problem could be transformed into a set TSP, tight lower bounds to the Dubins TSP and generalized Dubins path problem could be computed by solving a Set TSP.

Illustration from the cutting stock problem

The one-dimensional cutting stock problem as applied in the paper / plastic film industries, involves cutting jumbo rolls into smaller ones. This is done by generating cutting patterns typically to minimise waste. Once such a solution has been produced, one may seek to minimise the knife changes, by re-sequencing the patterns, or moving rolls left or right within each pattern. These moves do not affect the waste of the solution.
[file:generalised TSP knife changes.png|500 px|border]
In the above figure, patterns are rows; knife changes are indicated by the small white circles; for example, patterns 2-3-4 have a roll of size 42.5 on the left - the corresponding knife does not have to move. Each pattern represents a TSP set, one of whose permutations must be visited. For instance, for the last pattern, which contains two repeated sizes, there are 5! / = 30 permutations. The number of possible solutions to the above instance is 12! × ⁶ × ⁴ × ² / ⁹ × ≈ 5.3 × 10³⁵. The above solution contains 39 knife changes, and has been obtained by a heuristic; it is not known whether this is optimal. Transformations into the regular TSP, as described in would involve a TSP with 5,520 nodes.