CSCE: 620 Open Problem
Traveling Salesman Problem in Solid Grid
Graphs
presentation by
Ozgur Gonen
Problem 54 from The Open Problems Project
http://maven.smith.edu/~orourke/TOPP/
Solid Grid Graph
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A planar grid graph is a graph with vertices on the planar
integer lattice and edges connecting every pair of vertices
at unit distance.
finite subgraph of two-dimensional infinite integer grid
Solid Grid Graph
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Solid grid grid graph; bounded faces have area one
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a grid graph is solid if it does not have any holes.
–
Hole; missing vertices in the infinite grid.
Hamiltonian Path
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A path in an undirected graph that visits
each vertex exactly once
Hamiltonian Cycle also returns to the
starting vertex
HC is a special case of TSP; infinity for
non- adjacent cities
Both NP-complete
Problem Statement

What is the complexity of finding a shortest
tour (HC) in a solid planar grid graph?
NP-Complete:
A. Itai, C. H. Papadimitriou, and J. L.
Szwarcfiter, 1982
−
Polynomial time algorithm for rectangular
solid grid graphs O(nm)
Findings
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C. Umans, W, Lenhart. 1997
show that Hamiltonicity of a solid grid graph
can be decided in polynomial time.
O(n^3) algorithm that finds Hamiltonian
cycles in a solid graph by cycle merging
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Input G, find a 2-factor; spanning graph for deg2
−
Disjoint cycles that cover G
−
Repeatedly reduce the # components until HC is
found
Findings
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E. M. Arkin, M. A. Bender, E. Demaine, S.
P. Fekete, J. S. B. Mitchell, and S. Sethia
2001
finding the shortest tour is polynomially
solvable for thin grid graphs (i.e., grid
graphs that do not contain an induced 2×2
square)
O(n^3) algorithm
References
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ABD+01 E. M. Arkin, M. A. Bender, E. 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.
A. Itai, C. H. Papadimitriou, and J. L. Szwarcfiter.
Hamilton paths in grid graphs. SIAM J. Comput., 11:676686, 1982.
Christopher Umans and William Lenhart. Hamiltonian
cycles in solid grid graphs. In Proc. 38th Annu. IEEE
Sympos. Found. Comput. Sci., pages 496-507, 1997.