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An approximation algorithm for the longest cycle problem in solid grid graphs. (English) Zbl 1333.05091
Summary: Although, the Hamiltonicity of solid grid graphs are polynomial-time decidable, the complexity of the longest cycle problem in these graphs is still open. In this paper, by presenting a linear-time constant-factor approximation algorithm, we show that the longest cycle problem in solid grid graphs is in APX. More precisely, our algorithm finds a cycle of length at least $$\frac{2 n}{3} + 1$$ in 2-connected $$n$$-node solid grid graphs.

MSC:
 05C12 Distance in graphs 05C38 Paths and cycles 05C45 Eulerian and Hamiltonian graphs 05C85 Graph algorithms (graph-theoretic aspects) 68W25 Approximation algorithms
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 [1] Alon, Noga; Yuster, Raphael; Zwick, Uri, Color-coding, J. ACM, 42, 4, 844-856, (1995) · Zbl 0885.68116 [2] Bansal, Nikhil; Fleischer, Lisa K.; Kimbrel, Tracy; Mahdian, Mohammad; Schieber, Baruch; Sviridenko, Maxim, Further improvements in competitive guarantees for QoS buffering, (Proceedings of the International Colloquium on Automata, Languages and Programming, (2004), Springer), 196-207 · Zbl 1098.68516 [3] Björklund, Andreas; Husfeldt, Thore, Finding a path of superlogarithmic length, SIAM J. Comput., 32, 6, 1395-1402, (2003) · Zbl 1041.68066 [4] Björklund, Andreas; Husfeldt, Thore; Khanna, Sanjeev, Approximating longest directed paths and cycles, (Proceedings of the International Colloquium on Automata, Languages and Programming, (2004), Springer), 222-233 · Zbl 1098.68094 [5] Bulterman, R. W.; van der Sommen, F. W.; Zwaan, G.; Verhoeff, T.; van Gasteren, A. J.M.; Feijen, W. H.J., On computing a longest path in a tree, Inform. Process. Lett., 81, 2, 93-96, (2002) · Zbl 1032.68671 [6] Chen, Guantao; Gao, Zhicheng; Yu, Xingxing; Zang, Wenan, Approximating longest cycles in graphs with bounded degrees, SIAM J. Comput., 36, 3, 635-656, (2006) · Zbl 1118.05047 [7] Feder, Tomás; Motwani, Rajeev, Finding large cycles in Hamiltonian graphs, (Proceedings of the 16th annual ACM-SIAM symposium on Discrete Algorithms, (2005), Society for Industrial and Applied Mathematics), 166-175 · Zbl 1297.05140 [8] Gabow, Harold N., Finding paths and cycles of superpolylogarithmic length, SIAM J. Comput., 36, 6, 1648-1671, (2007) · Zbl 1135.68044 [9] Gabow, Harold N.; Nie, Shuxin, Finding a long directed cycle, ACM Trans. Algorithms, 4, 1, (2008), 7:1-7:21 · Zbl 1183.05078 [10] Gabow, Harold N.; Nie, Shuxin, Finding long paths, cycles and circuits, (Proceedings of the 19th annual International Symposium on Algorithms and Computation, (2008), Springer), 752-763 · Zbl 1183.05078 [11] Gutin, G., Finding a longest path in a complete multipartite digraph, SIAM J. Discrete Math., 6, 2, 270-273, (1993) · Zbl 0773.05071 [12] Ioannidou, Kyriaki; Mertzios, George B.; Nikolopoulos, Stavros D., The longest path problem is polynomial on interval graphs, (Proceedings of 34th International Symposium on Mathematical Foundations of Computer Science, (2009), Springer), 403-414 · Zbl 1250.68128 [13] Itai, A.; Papadimitriou, C. H.; Szwarcfiter, J. L., Hamiltonian paths in grid graphs, SIAM J. Comput., 11, 4, 676-686, (1982) · Zbl 0506.05043 [14] Karger, David; Motwani, Rajeev; Ramkumar, G. D.S., On approximating the longest path in a graph, Algorithmica, 18, 1, 82-98, (1997) · Zbl 0876.68083 [15] Kehsavarz Kohjerdi, F.; Bagheri, A.; Asgharian Sardroud, A., A linear-time algorithm for the longest path problem in rectangular grid graphs, Discrete Appl. Math., 160, 3, 210-217, (2012) · Zbl 1237.05115 [16] Mertzios, George B.; Corneil, Derek G., A simple polynomial algorithm for the longest path problem on cocomparability graphs, SIAM J. Discrete Math., 26, 3, 940-963, (2012) · Zbl 1256.05237 [17] Uehara, Ryuhei; Uno, Yushi, Efficient algorithms for the longest path problem, (Proceedings of the 15th annual International Symposium on Algorithms and Computation, (2004), Springer), 871-883 · Zbl 1116.05318 [18] Uehara, R.; Uno, Y., On computing longest paths in small graph classes, Internat. J. Found Comput. Sci., 18, 05, 911-930, (2007) · Zbl 1202.68291 [19] Umans, C.; Lenhart, W., Hamiltonian cycles in solid grid graphs, (Proceedings of 38th Annual Symposium on Foundations of Computer Science, (1997), IEEE), 496-505 [20] Zhang, W.; Liu, Y., Approximating the longest paths in grid graphs, Theoret. Comput. Sci., 412, 39, 5340-5350, (2011) · Zbl 1222.68089
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