Easy Problems for Tree-Decomposable Graphs

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 treewidth given a tree-decomposition. This gives 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 over sets definable in monadic second-order logic. Our technique allows us also to solve some EMS problems in linear time or in polynomial or pseudopolynomial time for classes of graphs of fixed bounded treewidth. Moreover, it is shown that each EMS problem is in NC for graphs of bounded treewidth. Most problems for which linear time algorithms for graphs of bounded treewidth were previously known to exist, and many others, are EMS problems.

[1]  Detlef Seese,et al.  Tree-partite graphs and the complexity of algorithms , 1985, FCT.

[2]  S. Arnborg,et al.  Characterization and recognition of partial 3-trees , 1986 .

[3]  Stefan Arnborg,et al.  Linear time algorithms for NP-hard problems restricted to partial k-trees , 1989, Discret. Appl. Math..

[4]  Alfred V. Aho,et al.  The Design and Analysis of Computer Algorithms , 1974 .

[5]  Nobuji Saito,et al.  Linear-time computability of combinatorial problems on series-parallel graphs , 1982, JACM.

[6]  Neil Immerman.  Languages which capture complexity classes , 1983, STOC '83.

[7]  Eugene L. Lawler,et al.  Linear-Time Computation of Optimal Subgraphs of Decomposable Graphs , 1987, J. Algorithms.

[8]  Bruno Courcelle,et al.  The Monadic Second-Order Logic of Graphs. I. Recognizable Sets of Finite Graphs , 1990, Inf. Comput..

[9]  A. Rosenthal.  Computing the Reliability of Complex Networks , 1977 .

[10]  Stefan Arnborg,et al.  Efficient algorithms for combinatorial problems on graphs with bounded decomposability — A survey , 1985, BIT.

[11]  David S. Johnson.  The NP-Completeness Column: An Ongoing Guide , 1986, J. Algorithms.

[12]  Derek G. Corneil,et al.  Complement reducible graphs , 1981, Discret. Appl. Math..

[13]  Stephen T. Hedetniemi,et al.  Linear algorithms on k-terminal graphs , 1987 .

[14]  Craig A. Tovey,et al.  Recursively constructed graph families: membership and linear algorithms , 1988 .

[15]  H. Bodlaender.  Classes of graphs with bounded tree-width , 1986 .

[16]  Paul D. Seymour,et al.  Graph Minors. II. Algorithmic Aspects of Tree-Width , 1986, J. Algorithms.

[17]  S. Arnborg.  Reduced State EnumerationߞAnother Algorithm for Reliability Evaluation , 1978, IEEE Transactions on Reliability.

[18]  Umberto Bertelè,et al.  Nonserial Dynamic Programming , 1972 .

[19]  Neil Immerman,et al.  Languages that Capture Complexity Classes , 1987, SIAM J. Comput..

[20]  Derek G. Corneil,et al.  Complexity of finding embeddings in a k -tree , 1987 .

[21]  Michael R. Genesereth,et al.  Logical foundations of artificial intelligence , 1987 .

[22]  H. Bodlaender.  Planar graphs with bounded treewidth , 1988 .

[23]  Joseph R. Shoenfield,et al.  Mathematical logic , 1967 .