Abstract
Using a variation of the interpretability concept we show that all graph properties definable in monadic second order logic (MS properties) with quantification over vertex and edge sets can be decided in linear time for classes of graphs of fixed bounded tree-width, giving an alternative proof of a recent result by Courcelle. We allow graphs with directed and/or undirected edges, labeled on edges and/or vertices with labels taken from a finite set. We extend MS properties to Extended Monadic Second-order (EMS) problems involving counting or summing evaluations given with the graph over sets definable in monadic second order logic. Our tecnique allowes us to solve also some EMS problems in linear time or in polynomial or pseudopolynomial time for classes of graphs of fixed bounded tree-width. Most problems for wich linear time algorithms for graphs of bounded tree width where previously known to exist, and many others, are EMS problems.
Research supported by the Swedish Natural Sciences Research Council and the Swedish Board for Technological Development.
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References
Aho, Hopcroft and Ullman, Design and Analysis of Computer Algorithms Addison-Wesley 1972.
S. Arnborg, Reduced State Enumeration-Another Algorithm for Reliability Evaluation, IEEE Trans. Reliability R-27 (1978), 101–105.
S. Arnborg, Efficient Algorithms for Combinatorial Problems on Graphs with Bounded Decomposability — A Survey, BIT 25 (1985), 2–33.
S. Arnborg, D.G. Corneil and A. Proskurowski, Complexity of Finding Embeddings in a k-tree, SIAM J. Alg. and Discr. Methods 8(1987), 277–284.
S. Arnborg and A. Proskurowski, Characterization and Recognition of Partial 3-trees, SIAM J. Alg. and Discr. Methods 7(1986), 305–314.
S. Arnborg and A. Proskurowski, Linear Time Algorithms for NP-hard Problems on Graphs Embedded in k-trees, to app..
J. Barwise(ed.), Handbook of Mathematical Logic, Amsterdam, North-Holland 1977.
M.W. Bern, E.L. Lawler, and A.L. Wong, Linear time computation of optimal subgraphs of decomposable graphs,J. of Algorithms 8(1987) 216–235.
U. Bertele and F. Brioschi, Nonserial Dynamic Programming. Academic Press, New York, 1972.
H.L. Bodlaender, Dynamic Programming on Graphs with Bounded Tree-width, Ph D thesis, MIT 1987.
H.L. Bodlaender, Classes of Graphs with Bounded Tree-width, preprint, Dec 1986.
K. Compton and C.W. Henson, A new method for proving lower bounds on the computational complexity of first-order theories, Manuscript.
D.G. Corneil, H. Lerchs and L. Stewart Burlingham, Complement Reducible Graphs, Discrete Appl. Math. 3 (1981), 163–174.
B. Courcelle, Recognizability and Second-Order Definability for Sets of Finite Graphs,Preprint, Universite de Bordeaux, I-8634, Jan 1987.
J.E. Doner, Decidability of the Weak Second-Order theory of two Successors, Abstract 65T-468, Notices Amer. Math. Soc. 12(1965), 819, ibid. (1966), 513
R. Fagin, Generalized First-Order Spectra and Polynomial-Time Recognizable Sets, in Complexity and Computation, edited by R. Karp, SIAM-AMS Proceedings 7.
M.R. Garey and D.S. Johnson, Computers and Intractability, W.H. Freeman and Company, San Francisco (1979).
Y. Gurevich, Monadic Second-Order Theories, Ch XIII of Model-Theoretic Logics, Ed. Barwise and Feferman, Springer-Verlag, New York 1985, 479–506
S.T. Hedetniemi, Open problems concerning the theory of algorithms on partial k-trees, working paper 1987.
H. Immerman, Languages which capture Complexity Classes, proceedings of the 15th annual ACM Symposiom on the Theory of Computing, 1983 347–354.
D.S. Johnson, The NP-Completeness Column: An Ongoing Guide, J. of Algorithms 6(1984) 434–451.
D.S. Johnson, The NP-Completeness Column: An Ongoing Guide, J. of Algorithms 8(1987) 285–303.
M.O.Rabin, A simple Method of Undecidability proofs and some applications, In Log. Meth. Phil. Sci. Proc Jerusalem, 1964 58–68.
N. Robertson and P.D. Seymour, Graph Minors II. Algorithmic Aspects of Tree Width, Journal of Algorithms 7 (1986), 309–322.
A. Rosenthal, Computing the Reliability of a Complex Network, SIAM J. Appl. Math. 32 (1977) 384–393.
J.R. Schoenfield, Mathematical Logic, Reading 1967, Addison-Wesley.
P.Scheffler Linear-time Algorithms for NP-complete problems restricted to partial k-trees, Manuscript.
P.SchefflerManuscript.
D.Seese, Tree-partite Graphs and the Complexity of Algorithms, Preprint, P-Math 08/86, Akademie der Wissenschaften der DDR, Karl Weierstrass Institut für Mathematik
K. Takamizawa, T. Nishizeki and N. Saito, Linear-Time Computability of Combinatorial Problems on Series-Parallel Graphs, J. ACM 29(1982) 623–641.
J.W. Thatcher, J.B. Wright, Generalized Finite Automata Theory with an Application to a Decision Problem in Second-Order Logic, Mathematical Systems Theory 2(1968), 57–81.
T.V.Wimer, Ph D Thesis.
T.V.Wimer, S.T.Hedetniemi, and R.Laskar, A methodology for constructing linear graph algorithms, DCS, Clemson University, September 1985.
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Arnborg, S., Lagergren, J., Seese, D. (1988). Problems easy for tree-decomposable graphs extended abstract. In: Lepistö, T., Salomaa, A. (eds) Automata, Languages and Programming. ICALP 1988. Lecture Notes in Computer Science, vol 317. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-19488-6_105
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