Abstract
This paper describes a predicate calculus in which graph problems can be expressed. Any problem possessing such an expression can be solved in linear time on any recursively constructed graph, once its decomposition tree is known. Moreover, the linear-time algorithm can be generatedautomatically from the expression, because all our theorems are proved constructively. The calculus is founded upon a short list of particularly primitive predicates, which in turn are combined by fundamental logical operations. This framework is rich enough to include the vast majority of known linear-time solvable problems.
We have obtained these results independently of similar results by Courcelle [11], [12], through utilization of the framework of Bernet al. [6]. We believe our formalism is more practical for programmers who would implement the automatic generation machinery, and more readily understood by many theorists.
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References
S. Arnborg, Efficient algorithms for combinatorial problems on graphs with bounded decomposability,BIT,25 (1985), 2–23.
S. Arnborg, D. G. Corneil, and A. Proskurowski, Complexity of finding embeddings in ak-tree,SIAM Journal of Algebraic and Discrete Methods,8 (1987), 277–284.
A. Arnborg, J. Lagergren, and D. Seese, Problems easy for tree-decomposable graphs, (extended abstract)Proceedings of the 15th ICALP, Lecture Notes in Computer Science, Vol. 317. Springer-Verlag, Berlin (1988), pp. 38–51; to appear inJournal of Algorithms.
A. Arnborg and A. Proskurowski, Linear time algorithms forNP-hard problems on graphs embedded ink-trees, TRITA-NA-8404, Royal Institute of Technology, Sweden (1984).
M. Bauderon and B. Courcelle, Graph expressions and graph rewritings,Mathematical Systems Theory,20 (1987), 83–127.
M. W. Bern, E. L. Lawler, and A. L. Wong, Linear time computation of optimal subgraphs of decomposable graphs,Jornal of Algorithms,8 (1987), 216–235.
H. L. Bodlaender, Dynamic programming on graphs with bounded tree-width, Technical report, Laboratory for Computer Science,M.I.T. (1987); extended abstract inProceedings of ICALP (1988).
H. L. Bodlaender, Planar graphs with bounded tree-width, RUU-CS-88-14, University of Utrecht (1988).
H. L. Bodlaender,NC-algorithms for graphs with small treewidth,Proceedings of the Workshop on Graph-Theoretic Concepts in Computer Science (J. van Leeuwen, ed.) (1988), Lecture Notes in Computer Science, Vol. 344, Springer-Verlag, Berlin (1988), pp. 1–10.
H. L. Bodlaender, Polynomial algorithms for graph isomorphism and chromatic index on partialk-trees,Proceedings of the 1st Scandinavian Workshop on Algorithm Theory (1988), pp. 223–232.
B. Courcelle, Recognizability and second-order definability for sets of finitegraphs, Report 1-8634, Universite de Bordeaux (1987).
B. Courcelle, An algebraic and logical theory of graphs, a survey, unpublished manuscript (1988).
M. R. Garey, R. L. Graham, D. S. Johnson, and D. E. Knuth, Complexity results for bandwidth minimization,SIAM Journal of Applied Mathematics,34 (1978), 477–495.
M. R. Garey and D. S. Johnson,Computers and Intractability, Freeman, San Francisco (1979).
D. S. Johnson, TheNP-completeness column: an ongoing guide,Journal of Algorithms,6 (1985), 434–451.
S. Mahajan and J. G. Peters, Algorithms for regular properties in recursive graphs,Proceedings of the 25th Annual Allerton Conference on Communications, Control, and Computing (1987), pp. 14–23.
R. L. Rardin and R. G. Parker, Tree subgraph isomorphism isNP-complete on series-parallel graphs, unpublished manuscript (1985).
M. B. Richey, Combinatorial optimization on series-parallel graphs: algorithms and complexity, Ph.D. Thesis, School of Industrial and Systems Engineering, Georgia Institute of Technology (1985).
T. Schaefer, Complexity of some two-person perfect information games,Journal of Computer and System Sciences,16 (1978), 185–225.
P. Scheffler and D. Seese, A combinatorial and logical approach to linear-time computability, extended abstract (1986).
M. M. Syslo, The subgraph isomorphism problem for outerplanar graphs,Theoretical Computer Science,17 (1982), 91–97.
K. Takamizawa, T. Nishizeki, and N. Saito, Linear-time computability of combinatorial problems on series-parallel graphs,Journal of the Association for Computing Machinery,29 (1982), 623–641.
T. V. Wimer, Linear algorithms onk-terminal graphs, Ph.D. Thesis, Report No. URI-030, Clemson University (1987).
T. V. Wimer, S. T. Hedetniemi, and R. Laskar, A methodology for constructing linear graph algorithms,Congressus Numerantium,50 (1985), 43–60.
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Communicated by Greg N. Frederickson.
R. B. Borie was supported by a National Science Foundation Graduate Fellowship. C. A. Tovey was supported by a Presidential Young Investigator Award from the National Science Foundation (ECS-8451032) and a matching grant from the Digital Equipment Corporation.
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Borie, R.B., Parker, R.G. & Tovey, C.A. Automatic generation of linear-time algorithms from predicate calculus descriptions of problems on recursively constructed graph families. Algorithmica 7, 555–581 (1992). https://doi.org/10.1007/BF01758777
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DOI: https://doi.org/10.1007/BF01758777