A linear-time algorithm for the longest path problem in rectangular grid graphs

The longest path problem is a well-known NP-hard problem and so far it has been solved polynomially only for a few classes of graphs. In this paper, we give a linear-time algorithm for finding a longest path between any two given vertices in a rectangular grid graph.

[1]  Christopher Umans,et al.  Hamiltonian cycles in solid grid graphs , 1997, Proceedings 38th Annual Symposium on Foundations of Computer Science.

[2]  Hong Shen,et al.  An efficient algorithm for constructing Hamiltonian paths in meshes , 2002, Parallel Comput..

[3]  Stavros D. Nikolopoulos,et al.  The Longest Path Problem Is Polynomial on Interval Graphs , 2009, MFCS.

[4]  Ryuhei Uehara,et al.  On Computing Longest Paths in Small Graph Classes , 2007, Int. J. Found. Comput. Sci..

[5]  Gregory Gutin,et al.  Finding a Longest Path in a Complete Multipartite Digraph , 1993, SIAM J. Discret. Math..

[6]  Derek G. Corneil,et al.  A Simple Polynomial Algorithm for the Longest Path Problem on Cocomparability Graphs , 2010, SIAM J. Discret. Math..

[7]  David R. Karger,et al.  On approximating the longest path in a graph , 1997, Algorithmica.

[8]  Reinhard Diestel,et al.  Graph Theory , 1997 .

[9]  Harold N. Gabow,et al.  Finding paths and cycles of superpolylogarithmic length , 2004, STOC '04.

[10]  Harold N. Gabow,et al.  Finding Long Paths, Cycles and Circuits , 2008, ISAAC.

[11]  Zhao Zhang,et al.  Algorithms for long paths in graphs , 2007, Theor. Comput. Sci..

[12]  Jayme Luiz Szwarcfiter,et al.  Hamilton Paths in Grid Graphs , 1982, SIAM J. Comput..

[13]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[14]  Andreas Björklund,et al.  Finding a Path of Superlogarithmic Length , 2002, ICALP.

[15]  Rajeev Motwani,et al.  Finding large cycles in Hamiltonian graphs , 2005, SODA '05.

[16]  A. J. M. van Gasteren,et al.  On computing a longest path in a tree , 2002, Inf. Process. Lett..