Abstract
A certifying algorithm for a problem is an algorithm that provides a certificate with each answer that it produces. The certificate is a piece of evidence that proves the answer has no compromised by a bug in the implementation. A Hamiltonian cycle in a graph is a simple cycle in which each vertex of the graph appears exactly once. The Hamiltonian cycle problem is to test whether a graph has a Hamiltonian cycle. A path cover of a graph is a family of vertex-disjoint paths that covers all vertices of the graph. The path cover problem is to find a path cover of a graph with minimum cardinality. The scattering number of a noncomplete connected graph G = (V,E) is defined by s(G) = max {ω(G − S) − |S|: S ⊆ V and \(\omega(G-S)\geqslant 1\}\), in which ω(G − S) denotes the number of components of the graph G − S. The scattering number problem is to determine the scattering number of a graph. A recognition problem of graphs is to decide whether a given input graph has a certain property. To the best of our knowledge, most published certifying algorithms are to solve the recognition problems for special classes of graphs. This paper presents O(n)-time certifying algorithms for the above three problems, including Hamiltonian cycle problem, path cover problem, and scattering number problem, on interval graphs given a set of n intervals with endpoints sorted. The certificates provided by our algorithms can be authenticated in O(n) time.
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References
Arikati, S.R., Rangan, C.: Pandu: Linear Algorithm for Optimal Path Cover Problem on Interval Graphs. Inform. Process. Lett. 35, 149–153 (1990)
Ascheuer, N.: Hamiltonian Path Problems in the On-Line Optimization of Flexible Manufacturing Systems. Technique Report TR 96-3, Konrad-Zuse-Zentrum für Informationstechnik, Berlin (1996)
Boesch, F.T., Gimpel, J.F.: Covering the Points a Digraph with Point-Disjoint Paths and its Application to Code Optimization. J. ACM 24, 192–198 (1977)
Bondy, J.A., Murty, U.S.R.: Graph Theory with Applications, New York (1976)
Chang, M.S., Peng, S.L., Liaw, J.L.: Deferred-Query: an Efficient Approach for Some Problems on Interval Graphs. Networks 34, 1–10 (1999)
Chu, F.P.M.: A Simple Linear Time Certifying LBFS-Based Algorithm for Recognizing Trivially Perfect Graphs and their Complements. Inform. Process. Lett. 107, 7–12 (2008)
Chvátal, V.: Tough Graphs and Hamiltonian Circuits. Discrete Math. 5, 215–228 (1973)
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, San Francisco (1979)
Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs. Annals of Discrete Mathematics 57 (2004)
Heggernes, P., Kratsch, D.: Linear-Time Certifying Algorithms for Recognizing Split Graphs and Related Graph Classes. Nordic J. Comput. 14, 87–108 (2007)
Hell, P., Huang, J.: Certifying LexBFS Recognition Algorithms for Proper Interval Graphs and Proper Interval Bigraphs. SIAM J. Discrete Math. 18, 554–570 (2005)
Jung, H.A.: On a Class of Posets and the Corresponding Comparability Graphs. J. Combin. Theory Ser. B 24, 125–133 (1978)
Kratsch, D., McConnell, R.M., Mehlhorn, K., Spinrad, J.P.: Certifying Algorithms for Recognizing Interval Graphs and Permutation Graphs. In: Proceedings of the 14th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2003), pp. 158–167 (2003)
Kratsch, D., McConnell, R.M., Mehlhorn, K., Spinrad, J.P.: Certifying Algorithms for Recognizing Interval Graphs and Permutation graphs. SIAM J. Comput. 36, 326–353 (2006)
Manacher, G.K., Mankus, T.A., Smith, C.J.: An Optimum Θ(nlogn) Algorithm for Finding a Canonical Hamiltonian Path and a Canonical Hamiltonian Circuit in a Set of Intervals. Inform. Process. Lett. 35, 205–211 (1990)
Ntafos, S.C., Louis Hakimi, S.: On Path Cover Problems in Digraphs and Applications to Program Testing. IEEE Trans. Software Engrg. 5, 520–529 (1979)
Pinter, S., Wolfstahl, Y.: On Mapping Processes to Processors. Internat. J. Parallel Programming 16, 1–15 (1987)
Shih, W.K., Chern, T.C., Hsu, W.L.: An O(n 2logn) Time Algorithm for the Hamiltonian Cycle Problem on Circular-Arc Graphs. SIAM J. Comput. 21, 1026–1046 (1992)
Toussaint, G.T.: Pattern Recognition and Geometrical Complexity. In: Proceedings of the 5th International Conference on Pattern Recognition, Miami Beach, pp. 1324–1347 (1980)
Waterman, M.S., Griggs, J.R.: Interval Graphs and Maps of DNA. Bull. Math. Biol. 48, 189–195 (1986)
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Hung, RW., Chang, MS. (2010). Certifying Algorithms for the Path Cover and Related Problems on Interval Graphs. In: Taniar, D., Gervasi, O., Murgante, B., Pardede, E., Apduhan, B.O. (eds) Computational Science and Its Applications – ICCSA 2010. ICCSA 2010. Lecture Notes in Computer Science, vol 6017. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-12165-4_25
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DOI: https://doi.org/10.1007/978-3-642-12165-4_25
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