Abstract
Hierarchical graphs are used in order to describe systems with a sequential composition of sub-systems. A hierarchical graph consists of a vector of subgraphs. Vertices in a subgraph may “call” other subgraphs. The reuse of subgraphs, possibly in a nested way, causes hierarchical graphs to be exponentially more succinct than equivalent flat graphs. Early research on hierarchical graphs and the computational price of their succinctness suggests that there is no strong correlation between the complexity of problems when applied to flat graphs and their complexity in the hierarchical setting. That is, the complexity in the hierarchical setting is higher, but all “jumps” in complexity up to an exponential one are exhibited, including no jumps in some problems.
We continue the study of the complexity of algorithms for hierarchical graphs, with the following contributions: (1) In many applications, the subgraphs have a small, often a constant, number of exit vertices, namely vertices from which control returns to the calling subgraph. We offer a parameterized analysis of the complexity and point to problems where the complexity becomes lower when the number of exit vertices is bounded by a constant. (2) We describe a general methodology for algorithms on hierarchical graphs. The methodology is based on an iterative compression of subgraphs in a way that maintains the solution to the problems and results in subgraphs whose size depends only on the number of exit vertices, and (3) We handle labeled hierarchical graphs, where edges are labeled by letters from some alphabet, and the problems refer to the languages of the graphs.
The research leading to these results has received funding from the European Research Council under the European Union’s 7th Framework Programme (FP7/2007-2013, ERC grant no. 278410).
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References
Alur, R., Kannan, S., Yannakakis, M.: Communicating hierarchical state machines. In: Wiedermann, J., Emde Boas, P., Nielsen, M. (eds.) ICALP 1999. LNCS, vol. 1644, pp. 169–178. Springer, Heidelberg (1999). doi:10.1007/3-540-48523-6_14
Alur, R., Yannakakis, M.: Model checking of hierarchical state machines. ACM TOPLAS 23(3), 273–303 (2001)
Aminof, B., Kupferman, O., Murano, A.: Improved model checking of hierarchical systems. Inf. Comput. 210, 68–86 (2012)
Barrett, C., Jacob, R., Marathe, M.: Formal-language-constrained path problems. SIAM J. Comput. 30(3), 809–837 (2000)
Clarke, E.M., Grumberg, O., Peled, D.: Model Checking. MIT Press, Cambridge (1999)
de Roever, W.-P.: The need for compositional proof systems: a survey. In: de Roever, W.-P., Langmaack, H., Pnueli, A. (eds.) COMPOS 1997. LNCS, vol. 1536, pp. 1–22. Springer, Heidelberg (1998). doi:10.1007/3-540-49213-5_1
Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, Heidelberg (2013)
Drusinsky, D., Harel, D.: On the power of bounded concurrency I: finite automata. J. ACM 41(3), 517–539 (1994)
Ford, L.R., Fulkerson, D.R.: Maximal flow through a network. Can. J. Math. 8(3), 399–404 (1956)
Galperin, H., Wigderson, A.: Succinct representations of graphs. Inf. Control 56(3), 183–198 (1983)
Harel, D., Kupferman, O., Vardi, M.Y.: On the complexity of verifying concurrent transition systems. Inf. Comput. 173, 1–19 (2002)
Immerman, N.: Length of predicate calculus formulas as a new complexity measure. In: Proceedings of 20th FOCS, pp. 337–347 (1979)
Kupferman, O., Tamir, T.: Hierarchical network formation games. In: Legay, A., Margaria, T. (eds.) TACAS 2017. LNCS, vol. 10205, pp. 229–246. Springer, Heidelberg (2017). doi:10.1007/978-3-662-54577-5_13
Kupferman, O., Vardi, M.Y., Wolper, P.: An automata-theoretic approach to branching-time model checking. J. ACM 47(2), 312–360 (2000)
Lengauer, T.: The complexity of compacting hierarchically specified layouts of integrated circuits. In: Proceedings of 23rd FOCS, pp. 358–368 (1982)
Lengauer, T., Wagner, K.W.: The correlation between the complexities of the nonhierarchical and hierarchical versions of graph problems. JCSS 44, 63–93 (1990)
Lengauer, T., Wanke, E.: Efficient solutions of connectivity problems on hierarchically defined graphs. SIAM J. Comput. 17(6), 1063–1081 (1988)
Megiddo, N.: Optimal flows in networks with multiple sources and sinks. Math. Program. 7(1), 97–107 (1974)
Mendelzon, A.O., Wood, P.T.: Finding regular simple paths in graph databases. SIAM J. Comput. 24(6), 1235–1258 (1995)
Rothvoß, T.: The matching polytope has exponential extension complexity. In: Proceedings of 46th STOC, pp. 263–272 (2014)
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Faran, R., Kupferman, O. (2017). A Parametrized Analysis of Algorithms on Hierarchical Graphs. In: Pighizzini, G., Câmpeanu, C. (eds) Descriptional Complexity of Formal Systems. DCFS 2017. Lecture Notes in Computer Science(), vol 10316. Springer, Cham. https://doi.org/10.1007/978-3-319-60252-3_9
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DOI: https://doi.org/10.1007/978-3-319-60252-3_9
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