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**Previous:** Problem 53: Minimum-Turn Cycle Cover in Planar Grid Graphs

- Statement
What is the complexity of finding a shortest tour in a solid planar grid graph? A

*planar grid graph*is a graph whose vertices are any set of points on the planar integer lattice and whose edges connect every pair of vertices at unit distance. Distances between nodes correspond to induced shortest-path distances in the graph, which corresponds to “Manhattan” distances. A grid graph is*solid*if it does not have any holes, i.e., its complement in the planar integer lattice is connected.- Origin
[IPS82] show that the problem is NP-complete in general planar grid graphs.

- Status/Conjectures
Open.

- Motivation
- Partial and Related Results
[UL97] show that Hamiltonicity of a solid grid graph can be decided in polynomial time. Thus we can decide whether there is a tour of length equal to the number of vertices. In contrast, deciding Hamiltonicity is NP-hard in general planar grid graphs [IPS82].

[ABD+01] observe that finding the shortest tour is polynomially solvable when restricted to

*thin*grid graphs, i.e., grid graphs that do not contain an induced 2 \times 2 square. This problem asks about replacing the thin restriction with the solid restriction.- Related Open Problems
Minimum-Turn Cycle Cover in Planar Grid Graphs (Problem 53)

- Appearances
Mentioned in [ABD+01].

- Categories
traveling salesman; optimization; point sets; graphs

- Entry Revision History
S. P. Fekete, 20 Dec. 2003; E. Demaine, 16 May 2004.

- [ABD+01]
E. M. Arkin, M. A. Bender, E. D. Demaine, S. P. Fekete, J. S. B. Mitchell, and S. Sethia. Optimal covering tours with turn costs. In

*Proc. 13th ACM-SIAM Sympos. Discrete Algorithms*, pages 138–147, 2001.- [IPS82]
A. Itai, C. H. Papadimitriou, and J. L. Szwarcfiter. Hamilton paths in grid graphs.

*SIAM J. Comput.*, 11:676–686, 1982.- [UL97]
Christopher Umans and William Lenhart. Hamiltonian cycles in solid grid graphs. In

*Proc. 38th Annu. IEEE Sympos. Found. Comput. Sci.*, pages 496–507, 1997.