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 (TSP), 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.[1] 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.[2] 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,[3] tight lower bounds to the Dubins TSP and generalized Dubins path problem could be computed by solving a Set TSP.[4][5]
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 (up and down in the figure), 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
In the above figure, patterns (width no more than 198) 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 (twice each), there are 5! / (2! × 2!) = 30 permutations. The number of possible solutions to the above instance is 12! × (5!)6 × (6!)4 × (7!)2 / ((2!)9 × (3!)2) ≈ 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 [2] would involve a TSP with 5,520 nodes.
See also
- Fagnano's problem of finding the shortest tour that visits all three sides of a triangle
References
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- ^ C.E. Noon, "The Generalized Traveling Salesman Problem", PhD dissertation, 1988.
- ^ a b Page Module:Citation/CS1/styles.css has no content.Noon, Charles E.; Bean, James C. (February 1993). "An efficient transformation of the generalized traveling salesman problem". INFOR: Information Systems and Operational Research. 31 (1): 39–44. doi:10.1080/03155986.1993.11732212. hdl:2027.42/6833.
- ^ Page Module:Citation/CS1/styles.css has no content.Satyanarayana G. Manyam, Sivakumar Rathinam, Swaroop Darbha, David Casbeer, Yongcan Cao, Phil Chandler (2016). "GPS Denied UAV Routing with Communication Constraints". Journal of Intelligent & Robotic Systems. 84 (1–4): 691–703. doi:10.1007/s10846-016-0343-2. S2CID 33884807.
{{cite journal}}: CS1 maint: multiple names: authors list (link) - ^ Page Module:Citation/CS1/styles.css has no content.Satyanarayana G. Manyam, Sivakumar Rathinam (2016). "On Tightly Bounding the Dubins Traveling Salesman's Optimum". Journal of Dynamic Systems, Measurement, and Control. 140 (7): 071013. arXiv:1506.08752. doi:10.1115/1.4039099. S2CID 16191780.
- ^ Page Module:Citation/CS1/styles.css has no content.Satyanarayana G. Manyam, Sivakumar Rathinam, David Casbeer, Eloy Garcia (2017). "Tightly Bounding the Shortest Dubins Paths Through a Sequence of Points". Journal of Intelligent & Robotic Systems. 88 (2–4): 495–511. doi:10.1007/s10846-016-0459-4. S2CID 30943273.
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